# 5.1: Quadratic Functions

Difficulty Level: At Grade Created by: CK-12

Name: __________________

## Multiplying Binomials

or, “These are a few of my favorite formulae”

1. Multiply: \begin{align*}(x+2)(x+2)\end{align*}

Test your result by plugging \begin{align*}x=3\end{align*} into both my original function, and your resultant function. Do they come out the same?

2. Multiply: \begin{align*}(x+3)(x+3)\end{align*}

Test your result by plugging \begin{align*}x=-1\end{align*} into both my original function, and your resultant function. Do they come out the same?

3. Multiply: \begin{align*}(x+5)(x+5)\end{align*}

Test your result by plugging \begin{align*}x=\frac{1}{2}\end{align*} into both my original function, and your resultant function. Do they come out the same?

4. Multiply: \begin{align*}(x+a)(x+a)\end{align*}

Now, leave \begin{align*}x\end{align*} as it is, but plug \begin{align*}a=3\end{align*} into both my original function, and your resultant function. Do you get two functions that are equal? Do they look familiar?

5. Do not answer these questions by multiplying them out explicitly. Instead, plug these numbers into the general formula for \begin{align*}(x+a)^2\end{align*} that you found in number \begin{align*}4\end{align*}.

a. \begin{align*}(x+4)(x+4)\end{align*}

b. \begin{align*}(y+7)^2\end{align*}

c. \begin{align*}\left (z+ \frac{1}{2} \right )^2 \end{align*}

d. \begin{align*}(m + \sqrt{2} )^2 \end{align*}

e. \begin{align*}(x-3)^2 \end{align*} (*so in this case, \begin{align*}a\end{align*} is \begin{align*}-3\end{align*}.)

f. \begin{align*}(x-1)^2 \end{align*}

g. \begin{align*}(x-a)^2 \end{align*}

6. Earlier in class, we found the following generalization: \begin{align*}(x+a)(x-a)=x^2-a^2\end{align*}. Just to refresh your memory on how we found that, test this generalization for the following cases.

a. \begin{align*}x=10, \ a=0 \end{align*}

b. \begin{align*}x=10, \ a=1 \end{align*}

c. \begin{align*}x=10, \ a=2 \end{align*}

d. \begin{align*}x=10, \ a=3 \end{align*}

7. Test the same generalization by multiplying out \begin{align*}(x+a)(x-a)\end{align*} explicitly.

8. Now, use that “difference between two squares” generalization. As in number \begin{align*}5\end{align*}, do not solve these by multiplying them out, but by plugging appropriate values into the generalization in number \begin{align*}6\end{align*}.

a. \begin{align*} (20+1)(20-1) = \end{align*}

b. \begin{align*} (x+3)(x-3) = \end{align*}

c. \begin{align*} (x+\sqrt{2})(x-\sqrt{2}) \end{align*}

d. \begin{align*} \left (x+\frac{2}{3} \right ) \left (x-\frac{2}{3} \right ) \end{align*}

Name: __________________

Homework: Multiplying Binomials

Memorize these:

\begin{align*} (x+a)^2 & = x^2+2ax+a^2\\ (x-a)^2 & = x^2-2ax+a^2\\ x^2-a^2 & = (x+a)(x-a) \end{align*}

1. In the following drawing, one large square is divided into four regions. The four small regions are labeled with Roman numerals because I like to show off.

\begin{align*}& \quad \text{a} \\ & \text{a} \ \ \text{I} \quad \ \text{II} \\ & \quad \text{III} \ \ \text{IV} \ \ \text{x}\\ & \qquad \quad \text{x}\end{align*}

a. How long is the left side of the entire figure? _______________

b. How long is the bottom of the entire figure? _______________

c. One way to compute the area of the entire figure is to multiply these two numbers (total height times total width). Write this product down here: Area = _______________

d. Now: what is the area of the small region labeled I? _______________

e. What is the area of the small region labeled II? _______________

f. What is the area of the small region labeled III? _______________

g. What is the area of the small region labeled IV? _______________

h. The other way to compute the area of the entire figure is to add up these small regions. Write this sum down here: Area =_________________

i. Obviously, the answer to (c) and the answer to (h) have to be the same, since they are both the area of the entire figure. So write down the equation setting these two equal to each other here: __________

j. Does that look like one of our three formulae?

2. Multiply these out “manually.”

a. \begin{align*} (x+3)(x+4) \end{align*}

b. \begin{align*} (x+3)(x-4) \end{align*}

c. \begin{align*} (x-3)(x-4) \end{align*}

d. \begin{align*} (2x+3)(3x+2) \end{align*}

e. \begin{align*} (x-2)(x^2+2x+4) \end{align*}

f. Check your answer to part (e) by substituting the number \begin{align*}3\end{align*} into both my original function, and your answer. Do they come out the same?

3. Multiply these out using the formulae above.

a. \begin{align*} \left (x+\frac{3}{2} \right )^2 \end{align*}

b. \begin{align*} \left (x-\frac{3}{2} \right )^2 \end{align*}

c. \begin{align*}(x+3)^2 \end{align*}

d. \begin{align*}(3+x)^2 \end{align*}

e. \begin{align*}(x-3)^2 \end{align*}

f. \begin{align*}(3-x)^2 \end{align*}

g. Hey, why did (e) and (f) come out the same? (\begin{align*}x-3\end{align*} isn’t the same as \begin{align*}3-x\end{align*}, is it?)

h. \begin{align*} \left (x+\frac{1}{2} \right ) \left (x+\frac{1}{2} \right )\end{align*}

i. \begin{align*} \left (x+\frac{1}{2} \right ) \left (x-\frac{1}{2} \right )\end{align*}

j. \begin{align*} (\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})\end{align*}

k. Check your answer to part (j) by running through the whole calculation on your calculator: \begin{align*} \sqrt{5}-\sqrt{3} =\end{align*} _______, \begin{align*}\sqrt{5}+\sqrt{3}=\end{align*}_____, multiply them and you get _____.

4. Now, let’s try going backward. Rewrite the following expressions as \begin{align*}(x + \mathrm{something})^2\end{align*}, or as \begin{align*}(x-\mathrm{something})^2\end{align*}, or as \begin{align*}(x+\mathrm{something})(x-\mathrm{something})\end{align*}. In each case, check your answer by multiplying back to see if you get the original expression.

a. \begin{align*}x^2-8x+16 =\end{align*}____________

Check by multiplying back: _______________

b. \begin{align*} x^2-25 =\end{align*}____________

Check by multiplying back: _______________

c. \begin{align*} x^2+2x+1 =\end{align*}____________

Check by multiplying back: _______________

d. \begin{align*} x^2-20x+100 =\end{align*} ____________

Check by multiplying back: _______________

e. \begin{align*} 4x^2-9 =\end{align*} ____________

Check by multiplying back: _______________

5. Enough squaring: let’s go one higher, and see what \begin{align*}(x+a)^3\end{align*} is!

a. \begin{align*}(x+a)^3\end{align*} means \begin{align*}(x+a)(x+a)(x+a)\end{align*}. You already know what \begin{align*}(x+a)(x+a)\end{align*} is. So multiply that by \begin{align*}(x+a)\end{align*} to find the cubed formula. (Multiply term-by-term, then collect like terms.)

b. Use the formula you just found to find \begin{align*}(y+3)^3\end{align*}

c. Use the same formula to find \begin{align*}(y-3)^3 \end{align*}.

Name: __________________

## Factoring

The first step is always to “pull out” as much as you can

1. Multiply the following, using the distributive property:

\begin{align*}3x (4x^2 + 5x + 2) =\end{align*} ______________________

2. Now, you’re going to do the same thing backward.

a. “Pull out” the common term of 4y from the following expression.

\begin{align*}16y^3 + 4y^2 + 8y = 4y\end{align*} (_________________________________)

b. Check yourself, by multiplying \begin{align*}4y\end{align*} by the term you put in parentheses.

c. Did it work? _______________

For each of the following expressions, pull out the highest common factor you can find.

3. \begin{align*} 9xy+12x =\end{align*} ____________________

4. \begin{align*} 10x^2+9y^2 =\end{align*} ______________________

5. \begin{align*} 100x^3 + 25x^2 =\end{align*} ______________________

6. \begin{align*} 4x^2y + 3y^2x = \end{align*} ______________________

Next, look to apply our three formulae

Factor the following by using our three formulae for \begin{align*}(x+y)^2, (x-y)^2,\end{align*} and \begin{align*}x^2 - y^2 \end{align*}.

7. \begin{align*} x^2-9 = \end{align*} ______________

8. \begin{align*} x^2-10x+25 = \end{align*}______________

9. \begin{align*} x^2+8x+16 = \end{align*}______________

10. \begin{align*} x^2+9 = \end{align*}______________

11. \begin{align*} 3x^2-27 = \end{align*} ______________ (hint: start by pulling out the common factor!)

If all else fails, factor the “old-fashioned way”

12. \begin{align*} x^2+7x+10 = \end{align*}______________

13. \begin{align*}x^2-5x+6 = \end{align*}______________

Check your answer by plugging a number into the original expression, and into your modified expression:

14. \begin{align*} x^2-6x+5 = \end{align*}______________

15. \begin{align*} x^2+8x+6 = \end{align*} ______________

16. \begin{align*} x^2-x-12 = \end{align*} ______________

17. \begin{align*} x^2+x-12 = \end{align*} ______________

18. \begin{align*} x^2+4x-12 = \end{align*}______________

19. \begin{align*} 2x^2+7x+12 = \end{align*} ______________

Name: __________________

Homewoik: Factah Alla Dese Heah Spressions

1. \begin{align*} 3x^3+15x^2+18x \end{align*}

2. \begin{align*} x^2-12x+32 \end{align*}

Check your answer by plugging the number \begin{align*}4\end{align*} into both my original expression, and your factored expression. Did they come out the same?

Is there any number \begin{align*}x\end{align*} for which they would not come out the same?

3. \begin{align*} x^2-4x-32 \end{align*}

4. \begin{align*} 8x^2-18y^4 \end{align*}

5. \begin{align*} x^2+18x+32 \end{align*}

6. \begin{align*} x^2-18x+32 \end{align*}

7. \begin{align*} 2x^2+12x+10 \end{align*}

8. \begin{align*} 100x^2+800x+1000 \end{align*}

9. \begin{align*} x^2+4x-32 \end{align*}

10. \begin{align*} x^4-y^4 \end{align*}

11. \begin{align*} x^2+13x+42 \end{align*}

12. \begin{align*} x^2+16\end{align*}

13. \begin{align*} 4x^2-9 \end{align*}

14. \begin{align*} 3x^2-48\end{align*}

15. \begin{align*} 2x^2+10x+12 \end{align*}

16. \begin{align*} x^2-9 \end{align*}

17. \begin{align*} 2x^2+11x+12 \end{align*}

Name: __________________

## Introduction to Quadratic Equations

For problems \begin{align*}1-5\end{align*}, I have two numbers \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. Tell me everything you can about \begin{align*}x\end{align*} and \begin{align*}y\end{align*} if...

1. \begin{align*} x+y = 0 \end{align*}

2. \begin{align*} xy = 0 \end{align*}

3. \begin{align*} xy = 1 \end{align*}

4. \begin{align*} xy > 0 \end{align*}

5. \begin{align*} xy < 0 \end{align*}

6. OK, here’s a different sort of problem. A swimming pool is going to be built, \begin{align*}3\;\mathrm{yards}\end{align*} long by \begin{align*}5\;\mathrm{yards}\end{align*} wide. Right outside the swimming pool will be a tiled area, which will be the same width all around. The total area of the swimming pool plus tiled area must be \begin{align*}35\;\mathrm{yards.}\end{align*}

a. Draw the situation. This doesn’t have to be a fancy drawing, just a little sketch that shows the \begin{align*}3\end{align*}, the \begin{align*}5\end{align*}, and the unknown width of the tiled area.

b. Write an algebraic equation that gives the unknown width of the tiled area.

c. Solve that equation to find the width.

d. Check your answer—does the whole area come to \begin{align*}35\;\mathrm{yards}\end{align*}?

Solve for \begin{align*}x\end{align*} by factoring.

7. \begin{align*} x^2+5x+6 = 0 \end{align*}

Check your answers by plugging them into the original equation. Do they both work?

8. \begin{align*} 2x^2-16x+15 = 0 \end{align*}

9. \begin{align*} x^3+4x^2-21x = 0 \end{align*}

10. \begin{align*} 3x^2-27 = 0 \end{align*}

Solve for \begin{align*}x\end{align*}. You may be able to do all these without factoring. Each problem is based on the previous problem in some way.

11. \begin{align*} x^2 = 9 \end{align*}

12. \begin{align*} (x-4)^2 = 9 \end{align*}

13. \begin{align*} x^2-8x+16 = 9 \end{align*}

14. \begin{align*} x^2-8x = -7 \end{align*}

Name: __________________

Homework: Introduction to Quadratic Equations

If a ball is thrown up into the air, the equation for its position is:

\begin{align*} h(t) = h_o + v_ot-16t^2 \end{align*}

where...

• \begin{align*}h\end{align*} is the height—given as a function of time, of course—measured in feet.
• \begin{align*}t\end{align*} is the time, measured in seconds.
• \begin{align*} h_o \end{align*} is the initial height that it had when it was thrown—or, to put it another way, \begin{align*} h_o \end{align*} is height when \begin{align*}t=0\end{align*}.
• \begin{align*} v_o \end{align*} is the initial velocity that it had when it was thrown, measured in feet per second—or, to again put it another way, \begin{align*} v_o \end{align*} is the velocity when \begin{align*}t=0\end{align*}.

This is sometimes called the equation of motion for an object, since it tells you where the object is (its height) at any given time.

Use that equation to solve the questions below.

1. I throw a ball up from my hand. It leaves my hand \begin{align*}3\;\mathrm{feet}\end{align*} above the ground, with a velocity of \begin{align*}35\;\mathrm{feet}\end{align*} per second. (So these are the initial height and velocity, \begin{align*}h_o\end{align*} and \begin{align*}v_o\end{align*}.)
1. Write the equation of motion for this ball. You get this by taking the general equation I gave you above, and plugging in the specific \begin{align*} h_o \end{align*} and \begin{align*} v_o \end{align*} for this particular ball.
2. How high is the ball after two seconds? (In other words, what \begin{align*}h\end{align*} value do you get when you plug in \begin{align*}t=2\end{align*}?)
3. What \begin{align*}h\end{align*} value do you get when you plug \begin{align*}t=0\end{align*} into the equation? Explain in words what this result means.
2. I throw a different ball, much more gently. This one also leaves my hand \begin{align*}3\;\mathrm{feet}\end{align*} above the ground, but with a velocity of only \begin{align*}2\;\mathrm{feet}\end{align*} per second.
1. Write the equation of motion for this ball.
2. How long does it take the ball to reach the ground? (In other words, what \begin{align*}t\end{align*} value will make \begin{align*}h\end{align*} come out zero?)
3. How long does it take the ball to come back to my hand (which is still \begin{align*}3\;\mathrm{feet}\end{align*} above the ground)?
3. A spring leaps up from the ground, and hits the ground again after \begin{align*}3\;\mathrm{seconds}\end{align*}. What was the velocity of the spring as it left the ground?
4. I drop a ball from a \begin{align*}100\;\mathrm{ft}\end{align*} building. How long does it take to reach the ground?
5. Finally, one straight equation to solve for \begin{align*}x\end{align*}:

\begin{align*} (x-2)(x-1) = 12 \end{align*}

Name: _________________

Completing the Square

Solve for \begin{align*}x\end{align*}.

1. \begin{align*} x^2 = 18 \end{align*}

2. \begin{align*} x^2 = 0 \end{align*}

3. \begin{align*} x^2 = -60 \end{align*}

4. \begin{align*} x^2+8x+12 = 0 \end{align*}

a. Solve by factoring.

b. Now, we’re going to solve it a different way. Start by adding four to both sides.

c. Now, the left side can be written as \begin{align*}(x+\mathrm{something})^2\end{align*}. Rewrite it that way, and then solve from there.

d. Did you get the same answers this way that you got by factoring?

Fill in the blanks

5. \begin{align*}(x-3)^2= x^2-6x+ \underline{\;\;\;\;\;}\end{align*}

6. \begin{align*}\left (x - \frac{3}{2} \right )^2 = x^{2-} \underline{\;\;\;\;\;} x + \underline{\;\;\;\;\;}\end{align*}

7. \begin{align*}(x+\underline{\;\;\;\;\;})^2 = x^2 + 10x + \underline{\;\;\;\;\;} \end{align*}

8. \begin{align*}(x-\underline{\;\;\;\;\;})^2 = x^2-18x+\underline{\;\;\;\;\;}\end{align*}

Solve for \begin{align*}x\end{align*} by completing the square

9. \begin{align*} x^2-20x + 90 = 26\end{align*}

10. \begin{align*} 3x^2+2x-4 = 0 \end{align*} (Hint: start by dividing by \begin{align*}3\end{align*}. The \begin{align*}x^2\end{align*} term should never have a coefficient when you are completing the square.)

Name: __________________

Homework: Completing the Square

1. A pizza (a perfect circle) has a \begin{align*}3"\end{align*} radius for the real pizza part (the part with cheese). But they advertise it as having an area of \begin{align*}25 \pi\end{align*} square inches, because they include the crust. How wide is the crust?

2. According to NBA rules, a basketball court must be precisely \begin{align*}94\;\mathrm{feet}\end{align*} long and \begin{align*}50\;\mathrm{feet}\end{align*} wide. (That part is true—the rest I’m making up.) I want to build a court, and of course, bleachers around it. The bleachers will be the same depth (*by “depth” I mean the length from the court to the back of the bleachers) on all four sides. I want the total area of the room to be \begin{align*}8,000\;\mathrm{square \ feet}\end{align*}. How deep must the bleachers be?

3. Recall that the height of a ball thrown up into the air is given by the formula:

\begin{align*} h(t) = h_o+v_ot-16t_2 \end{align*}

I am standing on the roof of my house, \begin{align*}20\;\mathrm{feet}\end{align*} up in the air. I throw a ball up with an initial velocity of \begin{align*}64\;\mathrm{feet/sec}\end{align*}. You are standing on the ground below me, with your hands \begin{align*}4\;\mathrm{feet}\end{align*} above the ground. The ball travels up, then falls down, and then you catch it. How long did it spend in the air?

Solve by completing the square

4. \begin{align*} x^2+6x+8 = 0 \end{align*}

5. \begin{align*} x^2-10x+30 = 5 \end{align*}

6. \begin{align*} x^2+8x+20 = 0 \end{align*}

7. \begin{align*} x^2+x = 0 \end{align*}

8. \begin{align*} 3x^2-18x+12 = 0 \end{align*}

9. Consider the equation \begin{align*}x^2+4x+4=c \end{align*} where \begin{align*}c\end{align*} is some constant. For what values of \begin{align*}c\end{align*} will this equation have...

a. Two real answers?

b. One real answer?

c. No real answers?

10. Solve by completing the square: \begin{align*}x^2+6x+a=0\end{align*}. (\begin{align*}a\end{align*} is a constant.)

Name: _________________

The “Generic” Quadratic Equation

OK, let’s say I wanted to solve a quadratic equation by completing the square. Here are the steps I would take, illustrated on an example problem. (These steps are exactly the same for any problem that you want to solve by completing the square.)

Note that as I go along, I simplify things—for instance, rewriting \begin{align*}3 \frac{1}{2} + 9 \end{align*} as \begin{align*}12 \frac{1}{2}\end{align*}, or \begin{align*}\sqrt{12 \frac{1}{2}}\end{align*} as \begin{align*}\frac{5}{\sqrt{2}}\end{align*}. It is always a good idea to simplify as you go along!

\begin{align*}& \text{Step} && \text{Example}\\ & \text{The problem itself} && 2x^2-3x-7 = 9x\\ & \text{Put all the}\ x \ \text{terms on one side, and the number on the other} && 2x^2-12x = 7\\ & \text{Divide both sides by the coefficient of} \ x^2 && x^2-6x = 3\frac{1}{2}\\ & \text{Add the same number to both sides. What number?} && x^2-6x\underline{\;\;+9} = 3\frac{1}{2}\underline{\;\;+9} \\ & \text{Half the coefficient of} \ x, \ \text{squared.}\ \text{(The coefficient of)} \ x \ \text{is -6}.\\ & \text{Half of that is} -3. \ \text{So we add} \ 9 \ \text{to both sides.)}\\ & \text{Rewrite the left side as a perfect square} && (x-3)^2 = 12\frac{1}{2} \\ & \text{Square root-but with a plus or minus''!}\\ & \text{(Remember, if}\ x^2 \ \text{is 25,}\ x \ \text{may be 5 or -5!)}&& x-3 = \pm \sqrt{12 \frac{1}{2}} = \pm \sqrt{\frac{25}{2}} = \pm \frac{5}{\sqrt{2}}\\ & \text{Finally, add or subtract the number next to the}\ x && x = 3 \pm \frac{5}{\sqrt{2}} (\approx -0.5, 6.5)\end{align*}

Now, you’re going to go through that same process, only you’re going to start with the “generic” quadratic equation:

\begin{align*} ax^2+bx+c = 0\end{align*}

As you know, once we solve this equation, we will have a formula that can be used to solve any quadratic equation—since every quadratic equation is just a specific case of that one!

Walk through each step. Remember to simplify things as you go along!

1. Put all the \begin{align*}x\end{align*} terms on one side, and the number on the other.

2. Divide both sides by the coefficient of \begin{align*} x^2 \end{align*}.

3. Add the same number to both sides. What number? Half the coefficient of \begin{align*}x\end{align*}, squared.

• What is the coefficient of \begin{align*}x\end{align*}?
• What is \begin{align*} \frac{1}{2} \end{align*} of that?
• What is that squared?

OK, now add that to both sides of the equation.

4. This brings us to a “rational expressions moment”—on the right side of the equation you will be adding two fractions. Go ahead and add them!

5. Rewrite the left side as a perfect square.

6. Square root—but with a “plus or minus”! (*Remember, if \begin{align*}x^2=25, x\end{align*} may be \begin{align*}5\end{align*} or \begin{align*}-5\end{align*}!)

7. Finally, add or subtract the number next to the \begin{align*}x\end{align*}.

Did you get the good old quadratic formula? If not, go back and see what’s wrong. If you did, give it a try on these problems! (Don’t solve these by factoring or completing the square, solve them using the quadratic formula that you just derived!)

8. \begin{align*} 4x^2+5x+1 = 0 \end{align*}

9. \begin{align*} 9x^2+12x+4 = 0 \end{align*}

10. \begin{align*} 2x^2+2x+1 = 0 \end{align*}

11. In general, a quadratic equation may have two real roots, or it may have one real root, or it may have no real roots. Based on the quadratic formula, and your experience with the previous three problems, how can you look at a quadratic equation \begin{align*} ax^2+bx+c=0 \end{align*} and tell what kind of roots it will have?

Name: __________________

Homework: Solving Quadratic Equations

1. \begin{align*} 2x^2-5x-3 = 0 \end{align*}

a. Solve by factoring

b. Solve by completing the square

c. Solve by using the quadratic formula \begin{align*}x= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \end{align*}

d. Which way was easiest? Which way was hardest?

e. Check your answers by plugging back into the original equation.

For problems \begin{align*} 2-6 \end{align*}, solve any way that seems easiest.

2. \begin{align*} x^2-5x+30 = 5(x+1) \end{align*}

3. \begin{align*} 3x^2+24x+60 = 0 \end{align*}

4. \begin{align*} \frac{2}{3}x^2+8.5x = \pi x \end{align*}

5. \begin{align*} x^2-x = 0 \end{align*}

6. \begin{align*} 9x^2 = 16 \end{align*}

7. Consider the equation \begin{align*} x^2+8x+c = 0 \end{align*} where \begin{align*}c\end{align*} is some constant. For what values of \begin{align*}c\end{align*} will this equation have...

a. Two real answers?

b. One real answer?

c. No real answers?

8. Starting with the generic quadratic equation \begin{align*} ax^2+bx+c=0 \end{align*}, complete the square and derive the quadratic formula. As much as possible, do this without consulting your notes.

Name: __________________

Sample Test: Quadratic Equations I

1. Multiply:

a. \begin{align*} (x- \frac{3}{2})^2 \end{align*}

b. \begin{align*} (x+ \sqrt{3})^2 \end{align*}

c. \begin{align*} (x-7)(x+7) \end{align*}

d. \begin{align*} (x-2)(x^2-4x+4) \end{align*}

e. \begin{align*} (x+3)(2x-5) \end{align*}

f. Check your answer to part (e) by substituting in the number \begin{align*}1\end{align*} for \begin{align*}x\end{align*} into both the original expression, and your resultant expression. Do they come out the same? (No credit here for just saying, “Yes”—I have to be able to see your work!)

2. Here is a formula you probably never saw, but it is true: for any \begin{align*}x\end{align*} and \begin{align*}a\end{align*}, \begin{align*}(x+a)^4=x^4+4x^3a+6x^2a^2+4xa^3+a^4 \end{align*}. Use that formula to expand the following.

a. \begin{align*} (x+2)^4 =\end{align*}

b. \begin{align*} (x-1)^4 =\end{align*}

3. Factor:

a. \begin{align*} x^2-36 \end{align*}

b. \begin{align*} 2x^2y-72y \end{align*}

c. Check your answer to part (b) by multiplying back. (*I have to see your work!)

d. \begin{align*} x^3-6x^2+9x \end{align*}

e. \begin{align*} 3x^2-27x+24 \end{align*}

f. \begin{align*} x^2+5x+5 \end{align*}

g. \begin{align*} 2x^2+5x+2 \end{align*}

4. Geoff has a rectangular yard which is \begin{align*} 55'\end{align*} by \begin{align*}75 '\end{align*}. He is designing his yard as a big grassy rectangle, surrounded by a border of mulch and bushes. The border will be the same width all the way around. The area of his entire yard is \begin{align*}4125\;\mathrm{square \ feet}\end{align*}. The grassy area will have a smaller area, of course—Geoff needs it to come out exactly \begin{align*} 3264\;\mathrm{square \ feet}\end{align*}. How wide is the mulch border?

5. Standing outside the school, David throws a ball up into the air. The ball leaves David’s hand \begin{align*}4'\end{align*} above the ground, traveling at \begin{align*}30\;\mathrm{feet/sec}\end{align*}. Raven is looking out the window \begin{align*}10'\end{align*} above ground, bored by her class as usual, and sees the ball go by. How much time elapsed between when David threw the ball, and when Raven saw it go by? To solve this problem, use the equation \begin{align*} h(t) = h_o+v_ot-16t^2.\end{align*}

6. Solve by factoring: \begin{align*} 2x^2-11x-30 = 0 \end{align*}

7. Solve by completing the square: \begin{align*} 2x^2+6x+4 = 0 \end{align*}

8. Solve by using the quadratic formula: \begin{align*}-x^2+2x+1 = 0\end{align*}

9. Solve. No credit unless I see your work! \begin{align*}ax^2+bx+c = 0 \end{align*}

Solve any way you want to.

10. \begin{align*} 2x^2+4x+10 = 0 \end{align*}

11. \begin{align*}\left (\frac{1}{2} \right )x^2-x+2 \frac{1}{2} = 0 \end{align*}

12. \begin{align*} x^3 = x \end{align*}

13. Consider the equation \begin{align*} 3x^2-bx+2=0 \end{align*}, where \begin{align*}b\end{align*} is some constant. For what values of \begin{align*}b\end{align*} will this equation have...

a. No real answers:

b. Exactly one answer:

c. Two real answers:

d. Can you find a value of \begin{align*}b\end{align*} for which this equation will have two rational answers—that is, answers that can be expressed with no square root? (Unlike a-c, I’m not asking for all such solutions, just one.)

Extra Credit (5 points): Make up a word problem involving throwing a ball up into the air. The problem should have one negative answer and one positive answer. Give your problem in words—then show the equation that represents your problem—then solve the equation—then answer the original problem in words.

Name: __________________

## Graphing Quadratic Functions

1. Graph by plotting points. Make sure to include positive and negative values of \begin{align*}x\end{align*}! \begin{align*} y=x^2 \end{align*}

\begin{align*}&x\\ &y\end{align*}

Note that there is a little point at the bottom of the graph. This point is called the “vertex.”

Graph each of the following by drawing these as variations of \begin{align*}^{\#}1\end{align*}—that is, by seeing how the various numbers transform the graph of \begin{align*}y=x^2\end{align*}. Next to each one, write down the coordinates of the vertex.

2. \begin{align*}y = x^2+3\end{align*} Vertex:

3. \begin{align*} y = x^2-3\end{align*} Vertex:

4. \begin{align*} y = (x-5)^2\end{align*} Vertex:

5. Plot a few points to verify that your graph \begin{align*}^{\#}4\end{align*} is correct.

6. \begin{align*} y=(x+5)^2 \end{align*} Vertex:

7. \begin{align*} y=2x^2 \end{align*} Vertex:

8. \begin{align*} y=\frac{1}{2}x^2 \end{align*} Vertex

9. \begin{align*}y=-x^2 \end{align*} Vertex:

In these graphs, each problem transforms the graph in several different ways.

10. \begin{align*}y=(x-5)^2-3 \end{align*} Vertex:

11. Make a graph on the calculator to verify that your graph of \begin{align*}^{\#}10\end{align*} is correct.

12. \begin{align*}y=2(x-5)^2-3 \end{align*} Vertex:

13. \begin{align*}y=-2(x-5)^2-3 \end{align*} Vertex:

14. \begin{align*}y=\frac{1}{2}(x+5)^2+3\end{align*} Vertex:

15. Where is the vertex of the general graph \begin{align*}y = a(x-h)^2+k?\end{align*}

16. Graph by plotting points. Make sure to include positive and negative values of \begin{align*}y\end{align*}! \begin{align*}x=y^2 \end{align*}

\begin{align*}& x\\ & y\end{align*}

Graph by drawing these as variations of \begin{align*}^{\#}16\end{align*}—that is, by seeing how the various numbers transform the graph of \begin{align*}x=y^2.\end{align*}

17. \begin{align*}x=y^2+4 \end{align*}

18. \begin{align*}x=(y-2)^2 \end{align*}

19. Plot a few points to verify that your graph of \begin{align*}^{\#}18\end{align*} is correct.

20. \begin{align*}x=-y^2\end{align*}

21. \begin{align*}x=-2(y-2)^2+4\end{align*}

Name: __________________

Graphing Quadratic Functions II

Yesterday we played a bunch with quadratic functions, by seeing how they took the equation \begin{align*}y=x^2\end{align*} and permuted it. Today we’re going to start by making some generalizations about all that.

1. \begin{align*}y = x^2 \end{align*}

a. Where is the vertex?

b. Which way does it open (up, down, left, or right?)

c. Draw a quick sketch of the graph.

2. \begin{align*}y = 2(x-5)^2+7\end{align*}

a. Where is the vertex?

b. Which way does it open (up, down, left, or right?)

c. Draw a quick sketch of the graph.

3. \begin{align*}y = (x+3)^2-8 \end{align*}

a. Where is the vertex?

b. Which way does it open (up, down, left, or right?)

c. Draw a quick sketch of the graph.

4. \begin{align*}y = -(x-6)^2 \end{align*}

a. Where is the vertex?

b. Which way does it open (up, down, left, or right?)

c. Draw a quick sketch of the graph.

5. \begin{align*}y = -x^2+10 \end{align*}

a. Where is the vertex?

b. Which way does it open (up, down, left, or right?)

c. Draw a quick sketch of the graph.

6. Write a set of rules for looking at any quadratic function in the form \begin{align*}y=a(x-h)^2+k \end{align*} and telling where the vertex is and which way it opens.

7. Now, all of those (as you probably noticed) were vertical parabolas. Now we’re going to do the same thing for their cousins, the horizontal parabolas. Write a set of rules for looking at any quadratic function in the form \begin{align*} x=a(y-k)^2+h \end{align*} and telling where the vertex is, and which way it opens.

After you complete \begin{align*}^\#7\end{align*}, stop and let me check your rules before you go on any further.

OK, so far, so good! But you may have noticed a problem already, which is that most quadratic functions that we’ve dealt with in the past did not look like \begin{align*}y=a(x-h)^2+k \end{align*}. They looked more like...well, you know, \begin{align*}x^2-2x-8\end{align*} or something like that. How do we graph that?

Answer: we put it into the forms we now know how to graph.

OK, but how do we do that?

Answer: Completing the square! The process is almost—but not entirely—like the one we used before to solve equations. Allow me to demonstrate. Pay careful attention to the ways in which it is like, and (more importantly) it is not like, the completing the square we did before!

\begin{align*}& \text{Step} && \text{Example}\\ & \text{The function itself} && x^2-6x-8\\ & \text{We used to start by putting the number (-8 in this case) on the other side.} && (x^2-6x)-8\\ & \text{In this case, we don't have another side.}\\ & \text{But I still want to set that}\ -8 \ \text{apart.}\\ & \text{So I'm going to put the rest in parentheses-that's}\\ & \text{where we're going to complete the square.}\\ & \text{Inside the parentheses, add the number you need to complete the square.} && (x^2-6x \underline{+9})\underline{-9}-8 \\ & \text{Problem is, we used to add this number to both sides-but as I said before,}\\ & \text{We have no other side. So I'm going to add it inside the parentheses,} \\ & \text{and at the same time subtract it outside the parentheses,}\\ & \text{so the function is, in total, unchanged.} \\ & \text{Inside the parentheses, you now have a perfect square and can rewrite it as such.}&& (x-3)^2-17\\ & \text{Outside the parentheses, you just have two numbers to combine.} \\ & \text{Voila! You can now graph it!} && \text{Vertex} \ (3,-17) \ \text{opens up}\end{align*}

8. \begin{align*}y = x^2+2x+5 \end{align*}

a. Complete the square, using the process I used above, to make it \begin{align*}y=a(x-h)^2+k \end{align*}.

b. Find the vertex and the direction of opening, and draw a quick sketch.

9. \begin{align*} x = y^2-10y+15 \end{align*}

a. Complete the square, using the process I used above, to make it \begin{align*}x=a(y-k)^2+h \end{align*}.

b. Find the vertex and the direction of opening, and draw a quick sketch.

Name: __________________

Homework: Graphing Quadratic Functions II

Put each equation in the form \begin{align*}y=a(x-h)^2+k\end{align*} or \begin{align*}x=a(y-k)^2+h \end{align*}, and graph.

1. \begin{align*}y = x^2 \end{align*}
2. \begin{align*}y = x^2+6x+5 \end{align*}
3. Plot at least three points to verify your answer to \begin{align*}^{\#}2\end{align*}.
4. \begin{align*}y = x^2-8x+16 \end{align*}
5. \begin{align*}y = x^2-7 \end{align*}
6. \begin{align*} y+x^2 = 6x+3 \end{align*}
7. Use a graph on the calculator to verify your answer to \begin{align*}^{\#}6\end{align*}.
8. \begin{align*} y+x^2 = x^2+6x+9 \end{align*}
9. \begin{align*} y = -2x^2+12x+4 \end{align*}
10. \begin{align*} x = 3y^2+6y \end{align*}
11. \begin{align*} x^2+y^2 = 9 \end{align*}
12. Explain in words how you can look at any equation, in any form, and tell if it will graph as a parabola or not.

Name: _________________

Solving Problems by Graphing Quadratic Functions

Let’s start with our ball being thrown up into the air. As you doubtless recall:

\begin{align*}h(t)=h_o+v_ot-16t^2 \end{align*}

1. A ball is thrown upward from the ground with an initial velocity of \begin{align*}64\;\mathrm{ft/sec}\end{align*}.

a. Write the equation of motion for the ball.

b. Put that equation into standard form for graphing.

c. Now draw the graph. \begin{align*}h\end{align*} (the height, and also the dependent variable) should be on the \begin{align*}y-\end{align*}axis, and \begin{align*}t\end{align*} (the time, and also the independent variable) should be on the \begin{align*}x-\end{align*}axis.

d. Use your graph to answer the following questions: at what time(s) is the ball on the ground?

e. At what time does the ball reach its maximum height?

f. What is that maximum height?

2. Another ball is thrown upward, this time from the roof, \begin{align*}30'\end{align*} above ground, with an initial velocity of \begin{align*} 200 \;\mathrm{ft/sec}\end{align*}.

a. Write the equation of motion for the ball.

b. Put that equation into standard form for graphing, and draw the graph as before.

c. At what time(s) is the ball on the ground?

d. At what time does the ball reach its maximum height?

e. What is that maximum height?

OK, we’re done with the height equation for now. The following problem is taken from a Calculus book. Just so you know.

3. A farmer has \begin{align*} 2400 \;\mathrm{feet}\end{align*} of fencing, and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?

a. We’re going to start by getting a “feeling” for this problem, by doing a few drawings. First of all, draw the river, and the fence around the field by the river, assuming that the farmer makes his field \begin{align*} 2200 \;\mathrm{feet}\end{align*} long. How far out from the river does the field go? What is the total area of the field?

After you do part (a), please stop and check with me, so we can make sure you have the right idea, before going on to part (b).

b. Now, do another drawing where the farmer makes his field only \begin{align*}400\;\mathrm{feet}\end{align*} long. How far out from the river does the field go? What is the total area of the field?

c. Now, do another drawing where the farmer makes his field \begin{align*} 1000\;\mathrm{feet}\end{align*} long. How far out from the river does the field go? What is the total area of the field?

The purpose of all that was to make the point that if the field is too short or too long then the area will be small; somewhere in between is the length that will give the biggest field area. For instance, 1000 works better than \begin{align*}2200\end{align*} or \begin{align*}400\end{align*}. But what length works best? Now we’re going to find it.

d. Do a final drawing, but this time, label the length of the field simply \begin{align*}x\end{align*}. How far out from the river does the field go?

e. What is the area of the field, as a function of \begin{align*}x\end{align*}?

f. Rewrite \begin{align*}A(x)\end{align*} in a form where you can graph it, and do a quick sketch. (Graph paper not necessary, but you do need to label the vertex.)

g. Based on your graph, how long should the field be to maximize the area? What is that maximum area? (Hint: make sure the area comes out bigger than all the other three you already did, or something is wrong!)

Name: _________________

Homework: Solving Problems by Graphing Quadratic Functions

Just as we did in class, we will start with our old friend \begin{align*}h(t)=h_o+v_ot-16t^2\end{align*}.

1. Michael Jordan jumps into the air at a spectacular \begin{align*}24\;\mathrm{feet/second.}\end{align*}

a. Write the equation of motion for the flying Wizard.

b. Put that equation into standard form for graphing, and draw the graph as before.

c. How long does it take him to get back to the ground?

d. At what time does His Airness reach his maximum height?

e. What is that maximum height?

2. Time to generalize! A ball is thrown upward from the ground with an initial velocity of \begin{align*}v_o \end{align*}. At what time does it reach its maximum height, and what is that maximum height?

Some more problems from my Calculus books.

3. Find the dimensions of a rectangle with perimeter \begin{align*}100\;\mathrm{ft}\end{align*} whose area is as large as possible. (Of course this is similar to the one we did in class, but without the river.)

4. There are lots of pairs of numbers that add up to \begin{align*}10\end{align*}: for instance, \begin{align*}8+2,\end{align*} or \begin{align*}9 \frac{1}{2} + \frac{1}{2}\end{align*}. Find the two that have the largest product possible.

5. A pharmaceutical company makes a liquid form of penicillin. If they manufacture \begin{align*}x\;\mathrm{units}\end{align*}, they sell them for \begin{align*}200x \end{align*} dollars (in other words, they charge \begin{align*}\200 \end{align*} per unit). However, the total cost of manufacturing \begin{align*}x\;\mathrm{units}\end{align*} is \begin{align*} 500,000+80x+0.003x^2 \end{align*}. How many units should they manufacture to maximize their profits?

Name: __________________

The first part of this assignment is brought to you by our unit on functions. In fact, this part is entirely recycled from that unit.

The following graph shows the temperature throughout the month of March. Actually, I just made this graph up—the numbers do not actually reflect the temperature throughout the month of March. We’re just pretending, OK?

1. On what days was the temperature exactly \begin{align*} 0^\circ C?\end{align*}

2. On what days was the temperature below freezing?

3. On what days was the temperature above freezing?

4. What is the domain of this graph?

5. During what time periods was the temperature going up?

6. During what time periods was the temperature going down?

7. The following graph represents the graph \begin{align*} y = f(x) \end{align*}.

a. Is it a function? Why or why not?

b. What are the zeros?

c. For what \begin{align*}x-\end{align*}values is it positive?

d. For what \begin{align*}x-\end{align*}values is it negative?

e. Draw the graph \begin{align*}y = f(x)-2 \end{align*}.

f. Draw the graph \begin{align*} y = -f(x) \end{align*}.

OK, your memory is now officially refreshed, right? You remember how to look at a graph and see when it is zero, when it is below zero, and when it is above zero.

Now we get to the actual “quadratic inequalities” part. But the good news is, there is nothing new here! First you will graph the function (you already know how to do that). Then you will identify the region(s) where the graph is positive, or negative (you already know how to do that).

8. \begin{align*} x^2+8x+15>0 \end{align*}

a. Draw a quick sketch of the graph by finding the zeros, and noting whether the function opens up or down.

b. Now, the inequality asks when that function is \begin{align*} >0- \end{align*} that is, when it is positive. Based on your graph, for what \begin{align*} x-\end{align*}values is the function positive?

c. Based on your answer to part (b), choose one \begin{align*}x-\end{align*}value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

9. A flying fish jumps from the surface of the water with an initial speed of \begin{align*}4\;\mathrm{feet/sec}\end{align*}.

a. Write the equation of motion for this fish.

b. Put it in the correct form, and graph it.

c. Based on your graph, answer the question: during what time interval was the fish above the water?

d. During what time interval was the fish below the water?

e. At what time(s) was the fish exactly at the level of the water?

f. What is the maximum height the fish reached in its jump?

Name: __________________

1. \begin{align*} x^2+8x+7 > 2x+3 \end{align*}

a. Put in standard form (with a zero on the right) and graph

b. Based on your graph, for what \begin{align*} x- \end{align*}values is this inequality true?

c. Based on your answer, choose one \begin{align*} x-\end{align*}value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

2. \begin{align*} 2x^2+8x+8 = 0 \end{align*}

a. Graph

b. Based on your graph, for what \begin{align*}x-\end{align*}values is this inequality true?

c. Based on your answer, choose one \begin{align*}x-\end{align*}value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

3. \begin{align*} -2x^2+8x > 9 \end{align*}

a. Put in standard form (with a zero on the right) and graph

b. Based on your graph, for what \begin{align*} x- \end{align*}values is this inequality true?

c. Based on your answer, choose one \begin{align*} x- \end{align*}value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

4. \begin{align*} -x^2+4x+3 > 0\end{align*}

a. Graph the function

b. Based on your graph, for what \begin{align*} x- \end{align*}values is this inequality true?

c. Based on your answer, choose one \begin{align*} x- \end{align*}value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

5. \begin{align*} x^2 > x \end{align*}

a. Put in standard form (with a zero on the right) and graph

b. Based on your graph, for what \begin{align*}x-\end{align*}values is this inequality true?

c. Now, let’s solve the original equation a different way—divide both sides by \begin{align*} x \end{align*}. Did you get the same answer this way? If not, which one is correct? (Answer by trying points.) What went wrong with the other one?

6. \begin{align*} x^2+6x+c < 0 \end{align*}

a. For what values of \begin{align*}c\end{align*} will this inequality be true in some range?

b. For what values of \begin{align*}c\end{align*} will this inequality never be true?

c. For what values of \begin{align*}c\end{align*} will this inequality always be true?

Name: _________________

Sample Test: Quadratics II

1. \begin{align*} x = -3y^2+5 \end{align*}
1. Opens (up / down / left / right)
2. Vertex: ___________
3. Sketch a quick graph on the graph paper
2. \begin{align*} y+9 = 2x^2+8x+8 \end{align*}
1. Put into the standard form of a parabola.
2. Opens (up / down / left / right)
3. Vertex: ___________
4. Sketch a quick graph on the graph paper
3. For what \begin{align*}x-\end{align*}values is this inequality true? \begin{align*} 2x^2+8x+8 < 9 \end{align*}
4. For what \begin{align*}x-\end{align*}values is this inequality true? \begin{align*} -x^2-10x \le 28 \end{align*}
5. A rock is thrown up from an initial height of \begin{align*}4'\end{align*} with an initial velocity of \begin{align*}32\;\mathrm{ft/sec}\end{align*}. As I’m sure you recall, \begin{align*}h(t)=h_o+v_ot-16t^2 \end{align*}.
1. Write the equation of motion for the rock.
2. At what time (how many seconds after it is thrown) does the rock reach its peak? How high is that peak? (Don’t forget to answer both questions...)
3. During what time period is the rock above ground?
6. A hot dog maker sells hot dogs for \begin{align*} \2 \end{align*} each. (So if he sells \begin{align*} x \end{align*} hot dogs, his revenue is \begin{align*} 2x \end{align*}.) His cost for manufacturing \begin{align*}x\end{align*} hot dogs is \begin{align*} 100+\frac{1}{2} x + \frac{1}{1000}x^2 \end{align*}. (Which I just made up.)
1. Profit is revenue minus cost. Write a function \begin{align*}P(x)\end{align*} that gives the profit he will make, as a function of the number of hot dogs he makes and sells.
2. How many hot dogs should he make in order to maximize his profits? What is the maximum profit?
3. How many hot dogs does he need to make, in order to make any profit at all? (The answer will be in the form “as long as he makes more than this and less than that or, in other words, between this and that, he will make a profit.”)

Extra credit:

a) Find the vertex of the parabola \begin{align*} y=ax^2+bx+c \end{align*}. This will, of course, give you a generic formula for finding the vertex of any vertical parabola.

b) Use that formula to find the vertex of the parabola \begin{align*} y=-3x^2+5x+6 \end{align*}

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