Section 7.5: The Vertex of a Parabola and Max-Min Applications.
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Transcript of 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
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.
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:
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 = __________________