# 5.1: Quadratic Functions

**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*}______________

Check your answer by multiplying back:

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*}

Check your answer by multiplying back:

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.

- 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*}.)- 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.
- 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*}?)
- 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.

- 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.
- Write the equation of motion for this ball.
- 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?)
- 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)?

- 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?
- I drop a ball from a \begin{align*}100\;\mathrm{ft}\end{align*} building. How long does it take to reach the ground?
- 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*}

*Your turn!*

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*.

- \begin{align*}y = x^2 \end{align*}
- \begin{align*}y = x^2+6x+5 \end{align*}
- Plot at least three points to verify your answer to \begin{align*}^{\#}2\end{align*}.
- \begin{align*}y = x^2-8x+16 \end{align*}
- \begin{align*}y = x^2-7 \end{align*}
- \begin{align*} y+x^2 = 6x+3 \end{align*}
- Use a graph on the calculator to verify your answer to \begin{align*}^{\#}6\end{align*}.