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Name Class Date 16-4

Solving by Graphing and Algebraically

Solve the system of equations.

f(x) = -8x + 48 g(x) = -2(x - 2 ) 2 + 32

A Solve the system of equations by graphing.

Start by graphing the quadratic function. The vertex is ( , ) . Describe the

transformation of the parent quadratic function that produces the graph of g(x).

To make the graph more accurate, plot the points where the x-intercepts occur. The x-intercepts are the solutions of the equation g(x) = 0:

-2 (x - 2) 2 + 32 = 0

-2 (x - 2) 2 =

(x - 2) 2 =

x - 2 = ±

x = ± = or

So, the points ( , 0 ) and ( , 0 ) are on the graph.

Use these points and the vertex to draw the graph.

Now graph the linear function. The y-intercept is ,

and the slope is .

The line and the parabola intersect at two points. Identify the coordinates of those points.

( , ) and ( , )

E X AM P L E1

Nonlinear SystemsGoing DeeperEssential question: How can you solve a system of equations when one equation is linear and the other is quadratic?

To estimate the solution to a system of equations, you can graph both equations on the same coordinate plane and find the intersection points. Or you can solve the equations algebraically using substitution or elimination.

MCC9–12.A.REI.7

Video Tutor

Module 16 375 Lesson 4

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B Solve the system of equations algebraically.

Write the functions in terms of y.

y = -8x + 48 y = -2(x - 2 ) 2 + 32

Both equations are solved for y, so set the right sides equal to each other and solve for x.

-8x + 48 = -2(x - 2 ) 2 + 32

-8x + 48 = Simplify the right side.

8x - 48 = 8x - 48 Add 8x - 48 to both sides.

0 = Simplify both sides.

0 = -2 ( ) ( ) Factor the right side.

x = or x = Use the zero-product property to solve for x.

Substitute these values of x into the equation of the line to find the corresponding y-values.

y = -8(2) + 48 =

y = -8(6) + 48 =

The solutions are ( , ) and ( , ) .

REFLECT

1a. If the linear function was f (x) = 8x + 48, how many solutions would there be? Justify your answer.

1b. When solving algebraically, why do you substitute the x-values into the equation of the line instead of the equation of the parabola?

1c. Explain the relationship between the intersection points of the graphs and the solutions of the system of equations.

1d. Describe how to check that the solutions are correct.

Module 16 376 Lesson 4

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Determining the Possible Number of Solutions

In the previous example, the system of equations had two solutions. You can use a graph to understand other possible numbers of solutions of a system of equations involving a linear equation and a quadratic equation.

The graph of the quadratic function f(x) = - x 2 + 10x - 27 is shown below.

Graph each linear function below on the same coordinate plane as the parabola.

Line 1: g(x) = 2x - 11

Line 2: h(x) = -2x + 14

REFLECT

2a. At how many points do the parabola and Line 1 intersect?

How many solutions are there for the system consisting of the quadratic function

and the first linear function?

2b. At how many points do the parabola and Line 2 intersect?

How many solutions are there for the system consisting of the quadratic function

and the second linear function?

2c. A system of equations consisting of one quadratic equation and one linear equation

can have , , or 2 real solutions.

2d. How many solutions does the following system of equations have? Explain your reasoning.

f (x) = - x 2 + 10x - 27

k(x) = -x + 1

2e. How many solutions does the following system of equations have? Explain your reasoning.

f (x) = - x 2 + 10x - 27

p(x) = -2

E X P L o r E2MCC9–12.A.REI.7

Module 16 377 Lesson 4

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You can use the Intersect feature on a graphing calculator to solve systems of equations.

Solving Systems Using Technology

Use a graphing calculator to solve the system of equations.

f(x) = -4.9 x 2 + 50x + 25 g(x) = 30x

A Enter the functions as Y 1 and Y 2 on a graphing calculator. Then graph both functions. Sketch the graphs on the coordinate plane at the right.

Estimate the solutions of the system from the graph.

B Solve the system directly by using the Intersect feature of the graphing calculator.

Press 2nd and CALC, then select Intersect. Press Enter for the first curve and again for the second curve. For Guess?, press the left or right arrows to move the cursor close to one of the intersections, then press Enter again. Repeat, moving the cursor close to the other intersection to find the second solution. Round your solutions to the nearest tenth.

REFLECT

3a. Are the solutions you get using the Intersect feature of a graphing calculator always exact? Explain.

3b. How can you check the accuracy of your estimated solutions?

3c. Use a graphing calculator to solve the system of equations f (x) and h(x) where h(x) = 30x + 50. What is the result? Explain.

E X AM P L E3MCC9–12.A.REI.7

Module 16 378 Lesson 4

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P r A c t i c E

Solve the system of equations algebraically. Round to the nearest tenth, if necessary.

The graph of a system of equations is shown. State how many solutions the system has. Then estimate the solution(s).

1. f (x) = x 2 - 2

g(x) = -2

3. y = -2 x 2 - 4x + 1

y = - 1 __ 2 x + 3

5. y = x 2 + 4x - 5

y = 3x - 2

2. y = (x - 3 ) 2

y = x

4. f (x) = x 2

g(x) = 1

6. f (x) = -16 x 2 + 15x + 10

g(x) = 14 - x

7.

8.

Module 16 379 Lesson 4

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13. y = - x 2 + 6x + 7

y = 2x + 6

14. f (x) = - x 2 + x - 2

g(x) = 2x - 3

Solve the system of equations using the Intersect feature of a graphing calculator. Round your answers to the nearest tenth.

Estimate the solutions to the system of equations graphically. Confirm the solutions by substituting the values into the equations.

9. f (x) = x 2

g(x) = 1

11. f (x) = -16 x 2 + 15x + 10

g(x) = 14 - x

10. y = x 2 - 1

y = 0.5x - 3

12. f (x) = 3(x - 1 ) 2 + 4

g(x) = -4x + 9

Module 16 380 Lesson 4

Solve each system by graphing. Check your answers.

1. 2 2

2y x xy x

2. 2 6

3y x xy x

_________________________________________ ________________________________________

Solve each system by substitution. Check your answers.

3. 22 4

5 8y x xy x

4. 22 3 26

y x xy x

_________________________________________ ________________________________________

5. 23 2 15

y x xx y

6. 2 16

4y xy x

_________________________________________ ________________________________________

7. 2 1

2 8y xx y

8. 2 3 2

2 4y x xx y

_________________________________________ ________________________________________

9. 22 3 1

2 4y x xx y

10. 2 2 4

3 4y x xx y

_________________________________________ ________________________________________

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16-4Name Class Date

Additional Practice

Module 16 381 Lesson 4

Write the correct answer. 1. A ball is thrown upward with an initial

velocity of 40 feet per second from ground level. The height h in feet of the ball after t seconds is given by

216 40 .h t t At the same time, a balloon is rising at a constant rate of 10 feet per second. Its height h in feet after t seconds is given by h 10t. Find the time it takes for the ball and the balloon to reach the same height.

_____________________________________

3. A skateboard company's monthly sales income can be modeled by the equation

2( ) 0.5 25 500,C s s s where s represents the number of skateboards sold. The monthly cost of running the business is ( ) 25 812.5.C s s How many skateboards must the company sell in a month before the sales income equals or exceeds the cost of running the business?

_____________________________________

Select the best answer. 5. A seagull is flying upwards such that its

height h in feet above the sea after t seconds is given by h 3t. At the same time, the height h in feet of a rock falling off a cliff above the sea after t seconds is given by 216 50h t . Find the approximate time it takes for the rock and the bird to be at the same height. A 1.68 seconds C 3.36 seconds B 3.13 seconds D 16.67 seconds

2. A bird starts flying up from the grass in a park and climbs at a steady rate of 0.5 meters per second. Its height h in meters after t seconds is given by h 0.5t. The equation h 4.9t2 40t 3 models the height h, in meters, of a baseball t seconds after it is hit. Find the time it takes for the ball and the bird to reach the same height.

_____________________________________

4. The deer population in a park can be modeled by the equation

2( ) 4 10 60,P y y y where y is the number of years after 2010. The deer population in another park can be modeled by ( ) 10 80P y y , where y is the number of years after 2010. In which year will the two parks have approximately the same number of deer?

_____________________________________

6. A juggler at a fun park throws a ball upwards such that the ball's height h in feet above the ground after t seconds is given by 216 20 5.h t t At the same time, a scenic elevator begins climbing a tower at a constant rate of 20 feet per second. Its height h in feet after t seconds is given by h 20t. Find the approximate time it takes for the ball and the elevator to reach the same height. F 0.56 seconds H 4 seconds G 1.12 seconds J 11.18 seconds

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Problem Solving

Module 16 382 Lesson 4