correctionkey=nl-d;ca-d name class date 4.3 …...2016/12/20 · explain 1 solving linear-quadratic...

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Explore Investigating Intersections of Lines and Graphs of Quadratic Equations

There are many real-world situations that can be modeled by linear or quadratic functions. What happens when the two situations overlap? Examine graphs of linear functions and quadratic functions and determine the ways they can intersect.

A Examine the two graphs below to consider the ways a line could intersect the parabola.

B Sketch three graphs of a line and a parabola: one showing intersection in one point, one showing intersection in two points, and one not showing intersection.

C So a constant linear function and a quadratic function can intersect at points.

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Module 4 189 Lesson 3

4.3 Solving Linear-Quadratic Systems

Essential question: How can you solve a system composed of a linear equation in two variables and a quadratic equation in two variables?

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Reflect

1. If a line intersects a circle at one point, what is the relationship between the line and the radius of the circle at that point?

2. Discussion Does a line have to be horizontal to intersect a parabola at exactly one point?

Explain 1 Solving Linear-Quadratic Systems GraphicallyGraph each equation by hand and find the set of points where the two graphs intersect.

Example 1 Solve the given linear-quadratic system graphically.

A ⎧

⎨

⎩ 2x - y = 3

y + 6 = 2 (x + 1) 2

Solve each equation for y.

2x - y = 3 y = 2x - 3 y + 6 = 2 (x + 1) 2 y = 2 (x + 1) 2 - 6

Plot the line and the parabola.

Find the approximate points of intersection: Estimating from the graph, the intersection points appear to be near (-1.5, -5.5) and (0.5, -2.5) .

The exact solutions (which can be found algebraically) are ( -1 - √_ 3 _______ 2 , - √

_ 3 - 4) and ( -1 + √

_ 3 _______ 2 , √

_ 3 - 4 ) ,

or about (-1.37, –5.73) and (0.37, -2.27) .

B ⎧

⎨

⎩ 3x + y = 4.5

y = 1 _ 2 (x - 3) 2

Solve each equation for y.

3x + y = 4.5

y =

y =

Plot the line and the parabola on the axes provided.

Find the approximate point(s) of intersection: .

Note that checking these coordinates in the original system shows that this is an exact solution.

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Your Turn

Solve the given linear-quadratic system graphically.

3. ⎧ ⎨

⎩ y + 3x = 0

y - 6 = -3 x 2

4. ⎧ ⎨

⎩ y + 1 = 1 _ 2 (x - 3) 2

x - y = 6

Explain 2 Solving Linear-Quadratic Systems AlgebraicallyUse algebra to find the solution. Use substitution or elimination.

Example 2 Solve the given linear-quadratic system algebraically.

A ⎧

⎨

⎩ 3x - y = 7

y + 4 = 2 (x + 5) 2

Solve this system using elimination. First line up the terms.

Subtract the second equation from the first to eliminate the y variable.

Solve the resulting equation for x using the quadratic formula.

There is no real number equivalent to ―― -135 , so the system has no solution.

7 + y = 3x4 + y = 2 (x + 5)2

7 + y = 3x

- ( ――――――――― 4 + y = 2 (x + 5 2 ) ) 3 = 3x - 2 (x + 5) 2

3 = 3x - 2 (x + 5)2

3 = 3x - 2 (x 2 + 10x + 25)

3 = 3x - 2 x 2 - 20x - 50

0 = -2x 2 - 17x - 53

2 x 2 + 17x + 53 = 0

x = -17 ± ―――――― 17 2 - 4 ⋅ 2 ⋅ 53 ___ 2 ⋅ 2

= -17 ± ―――― 289 - 424 __4

= -17 ± ―― -135 __ 4

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B ⎧

⎨

⎩ y = 1 _ 4 (x - 3) 2

3x - 2y = 13

Solve the system by substitution. The first equation is already solved for y. Substitute the expression 1 __ 4 (x - 3) 2 for y in the second equation.

3x - 2 ( 1 _ 4 (x - 3) 2 ) = 13

Now, solve for x.

So the line and the parabola intersect at two points. Use the x-coordinates of the intersections to find the points.

Solve 3x - 2y = 13 for y.

Find y when x = 5 and when x = 7.

So the solutions to the system are .

Reflect

5. How can you check algebraic solutions for reasonableness?

3x - 2y = 13

-2y = 13 - 3x

y =

y = - 13 - 3 ⋅ 5 _ 2 y = - 13 - 3 ⋅ 7 _ 2

= - 13 - 15 _ 2 = - 13 - 21 _ 2

= - -2 _ 2 = - - 8 _ 2

= 1 = 4

13 = 3x - 2 ( 1 _ 4 (x - 3) 2 ) 13 = 3x - (x - 3) 2

13 = 3x - 1 _ 2 ( ) 13 = 3x - 1 _ 2 x 2 + 3x - 9 _ 2

13 = - 1 _ 2 x 2 + - 9 _ 2

0 = - 1 _ 2 x 2 + 6x - 35 _ 2

0 = x 2

0 = (x ) (x ) x = ( ) or x = ( )



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Your Turn

Solve the given linear-quadratic system algebraically.

6. ⎧ ⎨

⎩ x - 6 = - 1 __ 6 y 2

2x + y = 6

7. ⎧

⎨ ⎩ x - y = 7

x 2 - y = 7

Explain 3 Solving Real-World ProblemsYou can use the techniques from the previous examples to solve real-world problems.

Example 3 Solve each problem.

A A tour boat travels around an island in a pattern that can be modeled by the equation 36 x 2 + 25 y 2 = 900. A fishing boat approaches the island on a path that can be modeled by the equation 3x - 2y = -8. Is there a danger of collision? If so, where?

Write the system of equations.

⎧

⎨

⎩ 36 x 2 + 25 y 2 = 900

3x - 2y = -8

Solve the second equation for x.

3x - 2y = -8

3x = 2y - 8

x = 2y - 8

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Substitute for x in the first equation.

36 x 2 + 25 y 2 = 900

36 ( 2y - 8 _ 3 ) 2 + 25 y 2 = 900

36 ( 4 y 2 - 32y + 64 __ 9 ) + 25 y 2 = 900

4 (4 y 2 - 32y + 64) + 25 y 2 = 900

16 y 2 - 128y + 256 + 25 y 2 = 900

41 y 2 - 128y - 644 = 0

Solve using the quadratic equation.

y = 128 ± √

___ 128 2 - 4(41)(-644) ___

2(41)

Collisions can occur when y ≈ -2.70 or y ≈ 5.82.

To find the x-values, substitute the y-values into x = 2y - 8 _____ 3 .

x = 2 (-2.70) - 8

__ 3 x = 2 (5.82) - 8

_ 3

= -5.40 - 8 _ 3 = 11.64 - 8 _ 3

= -13.40 _ 3 = 3.64 _ 3

≈ -4.47 ≈ 1.21

So the boats could collide at approximately (-4.47, -2.70) or (1.21, 5.82) .

= 128 ± √ ――― 122,000

__ 82

≈ -2.70 or 5.82



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B The range of the signal from a radio station is bounded by a circle described by the equation x 2 + y 2 = 2025. A stretch of highway near the station is modeled by the equation y - 15 = 1 __ 20 x. At which points, if any, does a car on the highway enter and exit the broadcast range of the station?

Write the system of equations.

⎧

⎨ ⎩ x 2 + y 2 = 2025

y - 15 = 1 _ 20 x

Solve the second equation for y.

y - 15 = 1 _ 20 x

y =

Substitute for x in the first equation.

x 2 + y 2 = 2025

x 2 + ( ) 2

= 2025

x 2 + = 2025

x 2 + 3 _ 2 x + 225 = 2025

401 _ 400 x 2 + 3 _ 2 x - = 0

401 x 2 + 600x - 720000 = 0

Solve using the quadratic formula.

y = -600 ± √ ――――――――― 6 00 2 - 4 (401) (-720000)

____ 2 (401)

≈ or (rounded to the nearest hundredth)

To find the y-values, substitute the x-values into y = 1 _ 20 x + 15.

The car will be within the radio station’s broadcast area between .

= 600 ± √__

1,155,240,000 __ 802



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Your Turn

8. An asteroid is traveling toward Earth on a path that can be modeled by the equation y = 1 __ 28 x - 7. It approaches a satellite in orbit on a path that can be modeled by the equation x 2 __ 49 + y 2

__ 51 = 1. What are the approximate coordinates of the points where the satellite and asteroid might collide?

9. The owners of a circus are planning a new act. They want to have a trapeze artist catch another acrobat in mid-air as the second performer comes into the main tent on a zip-line. If the path of the trapeze can be modeled by the parabola y = 1 __ 4 x 2 + 16 and the path of the zip-line can be modeled by y = 2x + 12, at what point can the trapeze artist grab the second acrobat?

Elaborate

10. A parabola opens to the left. Identify an infinite set of parallel lines that will intersect the parabola only once.

11. If a parabola can intersect a line from the set of lines ⎧

⎨ ⎩ x = a | | | a ∈R

⎫

⎬ ⎭ in 0, 1, or 2 points, what do you know about the parabola?

12. Essential Question Check-In How can you solve a system composed of a linear equation in two variables and a quadratic equation in two variables?



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1. How many points of intersection are on the graph?

2. How many points of intersection are there on the graph

of ⎧

⎨

⎩ y = x 2 + 3x - 2

y - x = 4

?

Solve each given linear-quadratic system graphically. If necessary, round to the nearest integer.

3. ⎧

⎨

⎩ y = - (x - 2) 2 + 4

y = -5

4. ⎧

⎨

⎩ y - 3 = (x - 1) 2

2x + y = 5

5. ⎧

⎨

⎩ x = y 2 - 5

-x + 2y = 12

6. ⎧

⎨

⎩ x - 4 = (y + 1) 2

3x - y = 17

Evaluate: Homework and Practice

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7. ⎧

⎨

⎩ (y - 4) 2 + x 2 = -12x - 20

x = y 8. ⎧

⎨

⎩ 5 - y = x 2 + x

y + 1 = 3 _ 4 x

Solve each linear-quadratic system algebraically.

9. ⎧

⎨

⎩ 6x + y = -16

y + 7 = x 2

10. ⎧

⎨

⎩ y - 5 = (x - 2) 2

x + 2y = 6

11. ⎧

⎨

⎩ y 2 - 26 = - x 2

x - y = 6

12. ⎧

⎨

⎩ y - 3 = x 2 -2x

2x + y = 1

13. ⎧

⎨

⎩ y = x 2 + 1

y - 1 = x

14. ⎧

⎨

⎩ y = x 2 + 2x + 7

y - 7 = x

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15. Jason is driving his car on a highway at a constant rate of 60 miles per hour when he passes his friend Alan whose car is parked on the side of the road. Alan has been waiting for Jason to pass so that he can follow him to a nearby campground. To catch up to Jason's passing car, Alan accelerates at a constant rate. The distance d, in miles, that Alan's car travels as a function of time t, in hours, since Jason's car has passed is given by d = 3600 t 2 . Find how long it takes Alan's car to catch up with Jason's car.

16. The flight of a cannonball toward a hill is described by the parabola y = 2 + 0.12x - 0.002x 2 .

The hill slopes upward along a path given by y = 0.15x.

Where on the hill does the cannonball land?

17. Amy throws a quarter from the top of a building at the same time that a balloon is released from the ground. The equation describing the height y above ground of the quarter in feet is y = 64 - 2 x 2 , where x is the time in seconds. The equation describing the elevation of the balloon in feet is y = 6x + 8, where x is the time in seconds. After how many seconds will the balloon and quarter pass each other? Check your solution for reasonableness.



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18. The range of an ambulance service is a circular region bounded by the equation x 2 + y 2 = 400. A straight road within the service area is represented by y = 3x + 20. Find the length of the road, in miles, that lies within the range of the ambulance service (round your answer to the nearest hundredth).

Recall that the distance formula is d = √

____ ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 .

19. Match the equations with their solutions.

⎧

⎨

⎩ y = x - 2

- x 2 + y = 4x - 2

A. (4, 3) (-4, -3)

⎧

⎨ ⎩ y = (x - 2) 2

y = -5x - 8

B. (0, -2) (5, 3)

⎧

⎨

⎩ 4y = 3x

x 2 + y 2 = 25

C. (2, 0)

⎧

⎨

⎩ y = (x - 2) 2

y = 0

D. No solution

20. A student solved the system ⎧

⎨

⎩ y - 7 = x 2 - 5x

y - 2x = 1

graphically and

determined the only solution to be (1, 3). Was this a reasonable

answer? How do you know?

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yH.O.T. Focus on Higher Order Thinking

21. Explain the Error A student was asked to come up with a system of equations, one

linear and one quadratic, that has two solutions. The student gave ⎧

⎨

⎩ y 2 = - (x + 1) 2 + 9

y = x 2 - 4x + 3

as the answer. What did the student do wrong?

22. Analyze Relationships The graph shows a quadratic function and a linear function y = d. If the linear function were changed to y = d + 3, how many solutions would the new system have? If the linear function were changed to y = d - 5, how many solutions would the new system have? Give reasons for your answers.

23. Make a Conjecture Given y = 100x 2 and y = 0.0001x 2 , what can you say about any line that goes through the vertex of each but is not horizontal or vertical?

24. Communicate Mathematical Ideas Explain why a system of a linear equation and a quadratic equation cannot have an infinite number of solutions.

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Lesson Performance Task

Suppose an aerial freestyle skier goes off a ramp with her path represented by the equation y = -0.024 ( x - 25 ) 2 + 40 where x = 0 represents the end of the ramp. If the surface of the mountain is represented by the linear equation y = -0.5x + 25, find the horizontal distance in feet the skier lands from the end of the ramp.



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