Topic 5Solving Quadratic Equations by Graphing

Unit 7 Topic 5

ExploreSolving a Quadratic Equation Graphically

A model rocket is launched from the ground with an initial velocity of 60 m/s. If no parachute is deployed, then the function can be used to model the height, h, of the rocket, in metres, after t seconds.

Sasha was asked to find the time the rocket was in the air. Her work is shown below, along with an explanation of her work in the boxes.

25 60h t t

ExploreSolving a Quadratic Equation Graphically

ExploreSolving a Quadratic Equation Graphically

1. Finish Sasha’s solution. At what two times was the height of the rocket 0 m?

2. The graph of is shown on the right. a) Explain how Sasha could have used the graph to solve the equation .

25 60h t t

20 5 60t t

Try this on your own first!!!!Try this on your own first!!!!

ExploreSolving a Quadratic Equation Graphically

1. Finish Sasha’s solution. At what two times was the height of the rocket 0 m?

2. The graph of I s shown on the right. a) Explain how Sasha could have used the graph to solve the equation .

25 60h t t

20 5 60t t

5 0

0

t

t

12 0

12

t

t

Sasha could look at the graph to determine the time when the height is 0 m.

When the y-value is 0 it is the x-intercepts that are read from the graph.

ExploreSolving a Quadratic Equation Graphically

b) Suppose Sasha wanted to know the two times when the height of the rocket was 100 m. This situation could be represented by the equation . Explain how Sasha could have used the graph above to solve the equation .

2100 5 60t t

2100 5 60t t

Sasha could look at the graph to determine the time when the height of the rocket is 100 m.

The rocket reached a height of 100 m at 2 seconds and then when it was coming back down at 10 seconds.

InformationQuadratic equations can also be solved graphically. Two methods for solving an equation graphically are described below. The detailed calculator steps can be found at the end of the workbook.

Find the x-Intercepts of the Corresponding FunctionYou can solve quadratic equations of the form .• • Use the Zero option to find the x-intercepts of the

graph of .

20 ax bx c 2

1y ax bx c

1y

InformationFind the x- Coordinates at the Points of Intersection of a System of EquationsYou can solve quadratic equation of the form .• • • Find the x-coordinate at each point of

intersection of the two graphs.

2 2ax bx c dx ex f 2

1y ax bx c

22y dx ex f

Example 1Solving a Quadratic Equation by Finding the x-intercepts of the Corresponding Quadratic Function

Graphically solve each of the following quadratic equations. If necessary, round to the nearest hundredth.a)

b)

Try this on your own first!!!!Try this on your own first!!!!

2

2

6 0

0.2 2.1 3.6 0

x x

x x

Example 1a: SolutionRound to the nearest hundredth.

21

2

,

6 0

6

2 3

y x x

x

x x2nd trace2: zero

As we proceed through this question, refer to the steps in your workbook.

Since the equations are equal to zero, we need to find the x-values that make the equation equal to zero.

Graph the function and determine the x-intercepts (zeros) of the function.

You will need to repeat this step to find both x-intercepts.

Example 1b: Solution

2

1

2

0.2 2.1

0.2

3.6

2.1 3.

1

6

.

0

5, 12

y x

x

x

x

x

As we proceed through this question, refer to the steps in your workbook.

2nd trace2: zero

You will need to repeat this step to find both x-intercepts.

You will need to adjust the window settings to see both x-intercepts.Possible Window Settings are shown.

Example 2Solving a Quadratic Equation by Finding Points of Intersection of a System of Equations

Graphically solve each of the following quadratic equations. If necessary, round to the nearest hundredth.a)

b)

Try this on your own first!!!!Try this on your own first!!!!

2

2

2 5 3

5 2 6

x x

x x x

Example 2a: Solution

2

1

2

2

2 5

3

no

2

e

5 3

n

x x

y x x

y

x

Since the equations are equal to an expression, we need to find the x-values of the intersection.

There is no intersection which means that there is no solution to the equation.

As we proceed through this question, refer to the steps in your workbook.

Example 2b: Solution

21

2

2

5

5 2

2 6

2.14,5

6

.14

y x x

y x

x

x x x

2nd trace5: intersect

As we proceed through this question, refer to the steps in your workbook.

You will need to repeat this step to find both intersections.

You will need to adjust the window settings to see both intersections.Possible Window Settings are shown.

Example 3Finding the Horizontal Distance of a Diver

Try this on your own first!!!!

A diver dives off the top edge of a 10 m tall cliff. The height of the diver, h, in metres, above the surface of the water is given by the function , where d is the horizontal distance from the top edge of the cliff, in metres. a) Write an equation to represent the situation when the diver enters the water.

22 3 10h d d

Example 3a: Solution

A diver dives off the top edge of a 10 m tall cliff. The height of the diver, h, in metres, above the surface of the water is given by the function , where d is the horizontal distance from the top edge of the cliff, in metres. a) Write an equation to represent the situation when the diver enters the water.

22 3 10h d d

h is the height of the diver above the surface of the water. When the diver enters the water the h = 0 m

2

20

2 3

2 3 10

10h d

d d

d

Example 3b: SolutionThe diver needs to enter the water at a horizontal distance of at least 2 m to safely clear the rocks at the base of the cliff. Graphically determine if the diver safely clears the rocks when he enters the water.

2

1

2

2 3 10

3.

2 0

m

3

1

1

y d d

x d

h d d Horizontal distance (m)x

yH

eig

ht

(m)

The diver safely clears the rocks when he enters the water.

Example 4Finding the Time When a Parachute Opens

A Canadian Cargo plane drops a crate of emergency supplies to aid-workers on the ground. The crate’s height, h, in metres, above the ground t seconds after leaving the aircraft is given by the following two quadratic functions.

represents the height of the crate during free fall. represents the height of the crate with the parachute open.

a) Write an equation to represent the situation when the parachute opens.

Try this on your own first!!!!

24.9 700h t 5 650h t

Example 4a: Solution

represents the height of the crate during free fall. represents the height of the crate with the parachute open.

a) Write an equation to represent the situation when the parachute opens.

24.9 700h t 5 650h t

24.9 700 5 650t t

The crate is free falling and then the parachute opens so the equation to represent this situation is

Example 4b: SolutionGraphically determine how long after the crate leaves the aircraft does the parachute open, to the nearest tenth of a second.

21

2

2

4.9 700

5 650

3.7 sec.

4.9 700 5 650

y x

y x

x

t t

2nd trace5: intersect

Time (sec)x

yH

eig

ht

(m)

The parachute opens 3.7 seconds after it leaves the aircraft.

Need to Know:• A quadratic equation can be solved by graphing.• Below are two methods for solving a quadratic

equation graphically. Method 1: Find the x-Intercepts of the Corresponding FunctionYou can solve quadratic equations of the form , by graphing the corresponding quadratic function, , and finding the x-intercepts of the graph of .

20 ax bx c 2

1y ax bx c

1y

Need to Know:Method 2: Find the x-Coordinates at the Points of Intersection of a System of EquationsYou can solve quadratic equations of the form , by graphing the corresponding quadratic functions, and , and then finding the x-coordinate at each point of intersection of and .

You’re ready! Try the homework from this section.

2 2ax bx c dx ex f 2

1y ax bx c

1y 2

2y dx ex f

2y