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Chapter 10: Quadratic Equations and Functions

Difficulty Level: Basic Created by: CK-12

Introduction

As you saw in Chapter 8, algebraic functions not only produce straight lines but curved ones too. A special type of curved function is called a parabola. Perhaps you have seen the shape of a parabola before:

• The shape of the water from a drinking fountain
• The path a football takes when thrown
• The shape of an exploding firework
• The shape of a satellite dish
• The path a diver takes into the water
• The shape of a mirror in a car’s headlamp

Many real life situations model a quadratic equation. This chapter will explore the graph of a quadratic equation and how to solve such equations using various methods.

Chapter Outline

Summary

Define each term.

1. Vertex
2. Standard form for a quadratic equation
3. Model
4. Discriminant

Graph each function. List the vertex (round to the nearest tenth, if possible) and the range of the function.

1. y=x26x+11\begin{align*}y=x^2-6x+11\end{align*}
2. y=4x2+16x19\begin{align*}y=-4x^2+16x-19\end{align*}
3. y=x22x+1\begin{align*}y=-x^2-2x+1\end{align*}
4. y=12x2+8x+6\begin{align*}y=\frac{1}{2} x^2+8x+6\end{align*}
5. y=x2+4x\begin{align*}y=x^2+4x\end{align*}
6. y=14x2+8x4\begin{align*}y=- \frac{1}{4} x^2+8x-4\end{align*}
7. y=(x+4)2+3\begin{align*}y=(x+4)^2+3\end{align*}
8. y=(x3)26\begin{align*}y=-(x-3)^2-6\end{align*}
9. y=(x2)2+2\begin{align*}y=(x-2)^2+2\end{align*}
10. y=(x+5)21\begin{align*}y=-(x+5)^2-1\end{align*}

Rewrite in standard form.

1. x24=5x\begin{align*}x-24=-5x\end{align*}
2. 5+4a=a2\begin{align*}5+4a=a^2\end{align*}
3. 618a2=528\begin{align*}-6-18a^2=-528\end{align*}
4. y=(x+4)2+2\begin{align*}y=-(x+4)^2+2\end{align*}

Solve each equation by graphing.

1. x28x+87=9\begin{align*}x^2-8x+87=9\end{align*}
2. 23x+x2104=4\begin{align*}23x+x^2-104=4\end{align*}
3. 13+26x=x2+11x\begin{align*}13+26x=-x^2+11x\end{align*}
4. x29x=119\begin{align*}x^2-9x=119\end{align*}
5. 32+6x24x=0\begin{align*}-32+6x^2-4x=0\end{align*}

Solve each equation by taking its square roots.

1. x2=225\begin{align*}x^2=225\end{align*}
2. x22=79\begin{align*}x^2-2=79\end{align*}
3. x2+100=200\begin{align*}x^2+100=200\end{align*}
4. 8x22=262\begin{align*}8x^2-2=262\end{align*}
5. 64x2=65\begin{align*}-6-4x^2=-65\end{align*}
6. 703=7x2+3\begin{align*}703=7x^2+3\end{align*}
7. 10+6x2=184\begin{align*}10+6x^2=184\end{align*}
8. 2+6x2=152\begin{align*}2+6x^2=152\end{align*}

Solve each equation by completing the square and then taking its square roots.

1. n24n3=9\begin{align*}n^2-4n-3=9\end{align*}
2. h2+10h+1=3\begin{align*}h^2+10h+1=3\end{align*}
3. x2+14x22=10\begin{align*}x^2+14x-22=10\end{align*}
4. t210t=9\begin{align*}t^2-10t=-9\end{align*}

Determine the maximum/minimum point by completing the square.

1. x220x+28=8\begin{align*}x^2-20x+28=-8\end{align*}
2. a2+263=5\begin{align*}a^2+2-63=-5\end{align*}
3. x2+6x33=4\begin{align*}x^2+6x-33=4\end{align*}

Solve each equation by using the Quadratic Formula.

1. 4x23x=45\begin{align*}4x^2-3x=45\end{align*}
2. 5x+11x2=15\begin{align*}-5x+11x^2=15\end{align*}
3. 3r=12r23\begin{align*}-3r=12r^2-3\end{align*}
4. 2m2+10m=8\begin{align*}2m^2+10m=8\end{align*}
5. 7c2+14c28=7\begin{align*}7c^2+14c-28=-7\end{align*}
6. 3w215=3w\begin{align*}3w^2-15=-3w\end{align*}

In 45-50, for each quadratic equation, determine:

(a) The discriminant

(b) The number of real solutions

(c) Whether the real solutions are rational or irrational

1. 4x24x+1=0\begin{align*}4x^2-4x+1=0\end{align*}
2. 2x2x3=0\begin{align*}2x^2-x-3=0\end{align*}
3. 2x2x1=2\begin{align*}-2x^2-x-1=-2\end{align*}
4. 4x28x+4=0\begin{align*}4x^2-8x+4=0\end{align*}
5. 5x2+10x5=0\begin{align*}-5x^2+10x-5=0\end{align*}
6. 4x2+3x+6=0\begin{align*}4x^2+3x+6=0\end{align*}
7. Explain the difference between y=x2+4\begin{align*}y=x^2+ 4\end{align*} and y=x2+4\begin{align*}y=-x^2+4\end{align*}.
8. Jorian wants to enclose his garden with fencing on all four sides. He has 225 feet of fencing. What dimensions would give him the largest area?
9. A ball is dropped off a cliff 70 meters high.
1. Using Newton’s equation, model this situation.
2. What is the leading coefficient? What does this value tell you about the shape of the parabola?
3. What is the maximum height of the ball?
4. Where is the ball after 0.65 seconds?
5. When will the ball reach the ground?
1. The following table shows the number of hours spent per person playing video games for various years in the United States. xy199524199625199737199843199961200070\begin{align*}& x && 1995 && 1996 && 1997 && 1998 && 1999 && 2000\\ & y && 24 && 25 && 37 && 43 && 61 && 70\end{align*}
1. What seems to be the best function for this data?
2. Find the best fit function.
3. Using your equation, predict the number of hours someone will spend playing video games in 2012.
4. Does this value seem possible? Explain your thoughts.
1. The table shows the amount of money spent (in billions of dollars) in the U.S. on books for various years. xy199016.5199116.9199217.7199318.8199420.8199523.1199624.9199726.3199828.2\begin{align*}& x && 1990 && 1991 && 1992 && 1993 && 1994 && 1995 && 1996 && 1997 && 1998\\ & y && 16.5 && 16.9 && 17.7 && 18.8 && 20.8 && 23.1 && 24.9 && 26.3 && 28.2\end{align*}
1. Find a linear model for this data. Use it to predict the dollar amount spent in 2008.
2. Find a quadratic model for this data. Use it to predict the dollar amount spent in 2008.
3. Which model seems more accurate? Use the best model to predict the dollar amount spent in 2012.
4. What could happen to change this value?
1. The data below shows the number of U.S. hospitals for various years. \begin{align*}& x && 1960 && 1965 && 1970 && 1980 && 1985 && 1990 && 1995 && 2000\\ & y && 6876 && 7123 && 7123 && 6965 && 6872 && 6649 && 6291 && 5810\end{align*}
1. Find a quadratic regression line to fit this data.
2. Use the model to determine the maximum number of hospitals.
4. In what years were there approximately 7,000 hospitals?
5. What seems to be the trend with this data?
1. A pendulum’s distance is measured and recorded in the following table. \begin{align*}& swing && 1 && 2 && 3 && 4 && 5 && 6\\ & length && 25 && 16.25 && 10.563 && 6.866 && 4.463 && 2.901\end{align*}
1. What seems to be the best model for this data?
2. Find a quadratic regression line to fit this data. Approximate the length of the seventh swing.
3. Find an exponential regression line to fit this data. Approximate the length of the seventh swing.

1. True or false? The vertex determines the domain of a quadratic function.
2. Suppose the leading coefficient of a quadratic equation is \begin{align*}a=-\frac{1}{3}\end{align*}. What can you conclude about the shape of the parabola?
3. Find the discriminant of the equation and determine the number of real solutions: \begin{align*}0=-2x^2+3x-2\end{align*}.
4. A ball is thrown upward from a height of four feet with an initial velocity of 45 feet/second.
1. Using Newton’s law, write the equation to model this situation.
2. What is the maximum height of the ball?
3. When will the ball reach 10 feet?
4. Will the ball ever reach 36.7 feet?
5. When will the ball hit the ground?

In 5–9, solve the equation using any method.

1. \begin{align*}2x^2=2x+40\end{align*}
2. \begin{align*}11j^2=j+24\end{align*}
3. \begin{align*}g^2=1\end{align*}
4. \begin{align*}11r^2-5=-178\end{align*}
5. \begin{align*}x^2+8x-65=-8\end{align*}
6. What is the vertex of \begin{align*}y=-(x-6)^2+5\end{align*}? Does the parabola open up or down? Is the vertex a maximum or a minimum?
7. Graph \begin{align*}y=(x+2)^2-3\end{align*}.
8. Evaluate the discriminant. How many real solutions does the quadratic equation have? \begin{align*}-5x^2-6x=1\end{align*}
9. Suppose \begin{align*}D=-14\end{align*}. What can you conclude about the solutions to the quadratic equation?
10. Rewrite in standard form: \begin{align*}y-7=-2(x+1)^2\end{align*}.
11. Graph and determine the function's range and vertex: \begin{align*}y=-x^2+2x-2\end{align*}.
12. Graph and determine the function's range and \begin{align*}y-\end{align*}intercept: \begin{align*}y=\frac{1}{2} x^2+4x+5\end{align*}.
13. The following information was taken from USA Today regarding the number of cancer deaths for various years.
1. Find a linear regression line to fit this data. Use it to predict the number of male deaths caused by cancer in 1999.
2. Find a linear regression line to fit this data. Use it to predict the number of male deaths caused by cancer in 1999.
3. Find an exponential regression line to fit this data. Use it to predict the number of male deaths caused by cancer in 1999.
4. Which seems to be the best fit for this data?
Year Number of Deaths Per 100,000 men
1980 205.3
1985 212.6
1989 217.6
1993 212.1
1997 201.9

Cancer Deaths of Men (Source: USA Today)

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