Maximum And Minimum Of Quadratic Functions Worksheet With Answer Key printable pdf download

Maximum And Minimum Of Quadratic Functions Worksheet With Answer Key

OPTIMIZATION PROBLEMS

MAXIMUM AND MINIMUM OF QUADRATIC FUNCTIONS

The graph of the quadratic function y = ax

+ bx + c is a parabola. If a > 0 , the

parabola is oriented upward and the vertex is the minimum point of the function.

If a < 0 , the parabola is oriented downward and the vertex is the maximum point

of the function. In either case the vertex of the parabola is located halfway

between the roots (the zeros or x-intercepts.) The x-coordinate of the vertex can

be found by: x =

. Substituting the x-value of the vertex into the equation of

the function yields the y-value of the vertex.

The quadratic function may be given or it may need to be created based on the

given information of the situation.

Examples of Optimization Problems

Example 1

What is the minimum value of the function y = 2x

! 8x ! 5 ?

!(!8)

Solution: The x-coordinate of the vertex is x =

= 2 . The y-coordinate of the

2(2)

vertex is y = 2(2)

! 8(2) ! 5 = !13. The coordinates of the vertex are (2, !13) and the

minimum value of the function is –13.

Example 2

With 100 feet of fence, what are the dimensions of the corral of largest area that a farmer

can create if his barn will provide one side of the corral?

barn

Solution: Let x = the width of the corral and then

100 ! 2x is the length. The function which represents

the area is y = (100 ! 2x)x = !2x

+ 100x .

100 – 2x

!100

The vertex of this function is at x =

= 25 and

2(!2)

y = !2(25)

+ 100(25) = 1250 .

The corral of maximum area has width x = 25 feet, length 50 feet, and area 1250 ft

Maximum And Minimum Of Quadratic Functions Worksheet With Answer Key