Section 7.5:

The Vertex of a Parabola and Max-Min Applications

7.5 Lecture Guide: The Vertex of a Parabola and Max-Min Applications

Objective: Calculate the vertex of a parabola.

For a parabola defined by 2f x ax bx c , the x-intercepts(if they exist) can be determined by using the quadratic formula. The vertex will be located at the x-value midway between the two x-intercepts. See the figure below.

y

x

2 4,0

2 2

b b ac

a a

2

bx

a

2 4,0

2 2

b b ac

a a

,2 2

b bf

a a

1. The x-intercepts of a parabola are 5,0 and 8,0Determine the x-coordinate of the vertex.

.

2. The x-intercepts of a parabola are and Determine the x-coordinate of the vertex.

. 7,0 15,0

Vertex of the Parabola defined by 2f x ax bx c

Algebraically

,2 2

b bf

a a

Verbally The vertex is either the highest or the lowest point on the parabola.

Numerically The y-values form a symmetric pattern about the vertex.

Example

2, 10, 1 by 10, 25, 5

Vertex: 3.5, 20.25

Finding the Vertex of the Parabola defined by 2f x ax bx c

Step 1. Determine the x-coordinate using 2

bx

a

Step 2. Then evaluate 2

bf

a

to determine the y-coordinate.

3. Determine the vertex of the parabola defined by

23 12 1f x x x

4. Determine the vertex of the parabola defined by

22 6 5f x x x

Use the given equation to calculate the x and y-intercepts and the vertex of each parabola.

5. 22 5 3y x x

(a) y-intercept (b) x-intercepts

(c) Vertex

Use the given equation to calculate the x and y-intercepts and the vertex of each parabola.

6.

(a) y-intercept (b) x-intercepts

(c) Vertex

2 7 1y x x

Use your graphing calculator to determine the minimum/maximum value of f x and the x-value at which thisminimum/maximum occurs. Use a window of 10,10,1 by 50,50,5

7.2 4 21y x x

for each graph. See Calculator Perspective 7.5.1.

Sketch of calculator graph:

-50

50

-10 10

y

x

Max/min value:

x-value where max/min occurs:

Use your graphing calculator to determine the minimum/maximum value of f x and the x-value at which thisminimum/maximum occurs. Use a window of 10,10,1 by 50,50,5

8. 2 5 36y x x

for each graph. See Calculator Perspective 7.5.1.

Sketch of calculator graph:

-50

50

-10 10

y

x

Max/min value:

x-value where max/min occurs:

Objective: Solve problems involving a maximum or minimum value.

9. The equation 216 80 3y x x gives the height y of abaseball in feet x seconds after it was hit.

(a) Use the equation to determine how many seconds into the flight the maximum height is reached.

(b) Determine the maximum height the ball reached.

9. The equation 216 80 3y x x gives the height y of abaseball in feet x seconds after it was hit.

(c) Do your results agree with what you can observe from the graph?

0

20

40

60

80

100

120

0 1 2 3 4 5

y

x

Time (sec)

Hei

gh

t (f

t)

10. A rancher has 240 yards of fencing available to enclose 3 sides of a rectangular corral. A river forms one side of the corral.

(a) If x yards are used for the two parallel sides, how much fencing remains for the side parallel to the river? Give this length in terms of x.

x

L

RIVER

x

L = __________________

10. A rancher has 240 yards of fencing available to enclose 3 sides of a rectangular corral. A river forms one side of the corral.

(b) Express the total area of the fenced corral as a function of x. Hint: Area = (Length)(Width)

x

L

RIVER

x

A x __________________

10. A rancher has 240 yards of fencing available to enclose 3 sides of a rectangular corral. A river forms one side of the corral.

(c) What is the maximum area that can be enclosed with this fencing?

x

L

RIVER

x

Maximum area = __________________

Write the equation of a quadratic function for each parabola shown.

11.

-5

5

-6 4x

y

1, 4

3,0 1,0

-5

5

-2 8

x

y

Write the equation of a quadratic function for each parabola shown.

12.

1,0 5,0

3, 4

Objective: Construct an equation with given solutions.

13. Construct a quadratic equation in x that has solutions of 6x and

8

9x

14. Construct a quadratic equation in x that has solutions of and 3x 3x

15. Construct a quadratic equation in x that has solutions of and 2 3x 2 3x

16. Construct a cubic equation in x that has solutions of and 3x 0x 5x ,,