Math 150 – Fall 2015
Section 4D
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Section 4D – Maximum/Minimum Function Values
Quadratic Functions
Definition. A quadratic function is one that can be written in the form, f (x) =
ax2 + bx + c, where a, b, and c are real numbers and a 6= 0. This if called the general
form of a quadratic function. A quadratic function in the form f (x) = a(x − h)2 + k,
where a 6= 0, is said to be in standard form. The vertex of the parabola is located at
(h, k).
The graph of a quadratic function is a parabola:
If a > 0, the parabola opens up.
k is the minimum functional
value, it occurs when x = h.
If a < 0, the parabola opens up.
k is the maximum functional
value, it occurs when x = h
Definition. If the parabola opens up, the vertex (h, k) is the lowest point on the
parabola. We say that k is the minimum functional value of f or the absolute
minimum value of f . It occurs when x = h. If the parabola opens down, k is
the maximum functional value of f or the absolute maximum value of f , and
occurs when x = h. Maximum and minimum functional values are called extreme
functional values.
Graphing a parabola: For the parabola f (x) = a(x − h)2 + k, we start with a basic
parabola f (x) = x2 and perform the following transformations.
• If a > 0, the graph opens up. If a < 0, the graph opens down (reflection about
the x-axis).
• Horizontal shift h units.
• Vertical shift k units.
• If |a| > 1, vertical stretch by a factor of |a|. If 0 < |a| < 1, vertical shrink by a
factor of |a|.
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Section 4D
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Converting quadratics from general from to standard form
Example 1. Write f (x) = x2 − 7x + 11 in standard form and graph it.
Vertex:
Domain:
Increasing:
Axis of Symmetry:
Range:
Decreasing:
Does f have an absolute maximum or minimum value?
absolute
value of f :
occurs at x =
Example 2. Write f (x) = −3x2 − 4x − 5 in standard form and graph it.
Vertex:
Domain:
Increasing:
Axis of Symmetry:
Range:
Decreasing:
Does f have an absolute maximum or minimum value?
absolute
value of f :
occurs at x =
Theorem. The vertex of the parabola f (x) = ax2 + bx + c is located at x =
−b
2a .
Example 3. Find the maximum or minimum functional value of f (x) = −2x2 +16x−7.
At what x value does this minimum or maximum function value occur?
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Section 4D
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Axis of Symmetry
Theorem. For a quadratic function f (x) = ax2 + bx + c, the vertical line x =
axis of symmetry.
−b
2x
is an
Example 4. Let f (x) = 2x2 + 7x − 4. Find the axis of symmetry and the coordinates
of the vertex.
Example 5. If f (1) = 7 and f (−4) = 7, and f (x) is a quadratic function, what is the
axis of symmetry?
Zeros of Quadratic Functions
Definition. The zeros of a function are the x values that make the function value 0,
i.e., the x-intercepts of the function. To find the zeros of f (x) set f (x) = 0 and solve.
Finding zeros of quadratic funtions: Finding the zeros of a quadratic function is
the same as solving a quadratic equation. We had three methods to do this:
1. Factor
2. Complete the Square
3. Quadratic Formula
Example 6. Does the function f (x) = 3x2 −2x+4 have any zeros in the real numbers?
If so, what are they?
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Section 4D
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Theorem. The expression b2 − 4ac is called the discriminant of the function f (x) =
ax2 + bx + c. The discriminant is the inside of the radcial in the quadratic formula
√
−b ± b2 − 4ac
x=
.
2a
Using the discriminant, we have
• If b2 − 4ac > 0, then f (x) has two zeros.
• If b2 − 4ac = 0, then f (x) has one zero.
• If b2 − 4ac < 0, then f (x) has no zeros in the real numbers.
Applications
We often want to maximize or minimize the value of a function f . If the function f is
a quadratic, then the absolute maximum (if f opens down) or absolute minimum (if f
opens up) is the y-value of the vertex.
Finding max/min: There are two ways to find the absolute maximum/minimum
value for f (x) = ax2 + bx + c:
• Put the quadratic in standard form f (x) = a(x − h)2 + k, and the absolute
maximum/minimum value is k and it occurs at x = h.
• Use the formula: the absolute maximum/minimum occurs when x =
value is f ( −b
2a ).
−b
2a
and its
If a > 0, then the parabola opens up, and it is a minimum functional value of f . If
a < 0, then the parabola opens down, and it is a minimum functional value of f .
Example 7. A baseball is thrown upward at a velocity of 72 ft/sec. If it is released
at a point 6 feet from the ground, its distance s in feet from the ground at t seconds is
given by s(t) = −16t2 + 72t + 6. When will the baseball reach its maximum height and
how high will it go?
Example 8. What is the smallest product of two numbers if their difference is 10?
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Section 4D
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Example 9. A farmer wishes to fence a rectangular area of his farm to hold his cows
and sheep. He also wants to keep the cows and sheep separate, so he wants to run a
fence down the middle to create two equally sized rectangular areas. If he has a total of
6000 feet of fence, what is the maximum area he can enclose? What dimensions should
it have?
Local maximum and minimum values
Definition. A local maximum is a point of the graph that has a larger y value larger
than all nearby points, but not necessarily the largest y value of the entire graph, i.e.,
f (x) > f (x1 ) for all x1 in some interval around x. A local minimum is a point of
the graph that has a smaller y value than all nearby points on the graph. Together the
local maximums and local minimums are called the local extreme points.
Note.
• Local maximum and local minimum values are also called relative maximum values.
• All absolute maximum values of f are also local maximums, and all absolute
minimum values are also local minimum values.
• A graph only has one absolute maximum value and one absolute minimum value
(although it can occur at multiple x-values). Graphs can have multiple local
maximum and local minimum values.