Quadratic Functions and Models
MATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Quadratic Functions and Models
Objectives
In this lesson we will learn to:
analyze the graphs of quadratic functions,
write quadratic functions in standard form and use the
results to sketch the graphs of functions,
find the minimum and maximum values of quadratic
functions in real-world applications.
J. Robert Buchanan
Quadratic Functions and Models
Polynomial Functions
The linear function (f (x) = ax + b), the constant function
(f (x) = c), and the squaring function (f (x) = x 2 ) are all
examples of polynomial functions.
Definition
Let n be a nonnegative integer and let an , an−1 , . . . , a2 , a1 , and
a0 be real numbers with an 6= 0. The function given by
f (x) = an x n + an−1 x n−1 + · · · + a2 x 2 + a1 x + a0
is called a polynomial function of x with degree n.
J. Robert Buchanan
Quadratic Functions and Models
Quadratic Functions
A quadratic function is a polynomial function in x of degree 2.
Definition
Let a, b, and c be real numbers with a 6= 0. The function given
by
f (x) = ax 2 + bx + c
is called a quadratic function.
The graph of a quadratic function is a roughly “U”-shaped curve
called a parabola.
J. Robert Buchanan
Quadratic Functions and Models
Upward Opening Parabola
f (x) = ax 2 + bx + c,
a>0
y
axis
vertex
x
J. Robert Buchanan
Quadratic Functions and Models
Downward Opening Parabola
f (x) = ax 2 + bx + c,
a<0
y
vertex
axis
x
J. Robert Buchanan
Quadratic Functions and Models
Anatomy of a Parabola
All parabolas are symmetric with respect to a line called
the axis of symmetry or simply the axis.
The point where the axis intersects the parabola is called
the vertex.
If f (x) = ax 2 + bx + c then
the parabola opens upward when a > 0 and
the parabola opens downward when a < 0.
J. Robert Buchanan
Quadratic Functions and Models
Standard Form
If a quadratic function is written in standard form we can easily
determine the axis of symmetry, the vertex, and whether the
parabola opens upward or downward.
Standard Form
The quadratic function
f (x) = a(x − h)2 + k ,
a 6= 0
is in standard form. The graph of f is a parabola whose axis is
the vertical line x = h and whose vertex is the point (h, k ). If
a > 0 the parabola opens upward, while if a < 0 the parabola
opens downward.
J. Robert Buchanan
Quadratic Functions and Models
Example (1 of 4)
Write the following quadratic function in standard form, graph it,
and determine the axis and vertex.
f (x) = 2x 2 + 6x − 5
J. Robert Buchanan
Quadratic Functions and Models
Example (1 of 4)
Write the following quadratic function in standard form, graph it,
and determine the axis and vertex.
f (x) = 2x 2 + 6x − 5
f (x) = 2(x 2 + 3x) − 5
y
15
= 2(x 2 + 3x + 9/4) − 9/2 − 5
3 2 19
−
= 2 x+
2
2
-5
10
5
-4
-3
-2
1
-1
axis: x = −3/2
-5
vertex: (−3/2, −19/2)
-10
J. Robert Buchanan
Quadratic Functions and Models
2
x
Example (2 of 4)
Write the following quadratic function in standard form, graph it,
and determine the axis and vertex.
f (x) = −x 2 − 4x + 1
J. Robert Buchanan
Quadratic Functions and Models
Example (2 of 4)
Write the following quadratic function in standard form, graph it,
and determine the axis and vertex.
f (x) = −x 2 − 4x + 1
y
f (x) = −(x 2 + 4x) + 1
4
= −(x 2 + 4x + 4) + 4 + 1
= −(x + 2)2 + 5
2
-5
axis: x = −2
-4
-3
-2
1
-1
-2
vertex: (−2, 5)
-4
J. Robert Buchanan
Quadratic Functions and Models
x
Example (3 of 4)
For the following parabola find the vertex, the y -intercept, and
the x-intercepts.
1
f (x) = x 2 + 5x +
4
J. Robert Buchanan
Quadratic Functions and Models
Example (3 of 4)
For the following parabola find the vertex, the y -intercept, and
the x-intercepts.
1
f (x) = x 2 + 5x +
4
5 2
25 25 1
2
−
+ = x+
−6
f (x) = x + 5x +
4
4
4
2
5
Vertex: − , −6
2
J. Robert Buchanan
Quadratic Functions and Models
Example (3 of 4)
For the following parabola find the vertex, the y -intercept, and
the x-intercepts.
1
f (x) = x 2 + 5x +
4
5 2
25 25 1
2
−
+ = x+
−6
f (x) = x + 5x +
4
4
4
2
5
Vertex: − , −6
2
1
If x = 0 then y = f (0) = , the y -intercept.
4
J. Robert Buchanan
Quadratic Functions and Models
Example (3 of 4)
For the following parabola find the vertex, the y -intercept, and
the x-intercepts.
1
f (x) = x 2 + 5x +
4
5 2
25 25 1
2
−
+ = x+
−6
f (x) = x + 5x +
4
4
4
2
5
Vertex: − , −6
2
1
If x = 0 then y = f (0) = , the y -intercept. If y = 0 then
4
2
5
x+
−6 = 0
2
√
5
x+
= ± 6
2
5 √
x = − ± 6
2
Quadratic Functions and Models
the x-intercepts. J. Robert Buchanan
Example (4 of 4)
Write the standard form of the equation of a parabola that has
vertex at (3, −1) and passes through the point (0, 2).
J. Robert Buchanan
Quadratic Functions and Models
Example (4 of 4)
Write the standard form of the equation of a parabola that has
vertex at (3, −1) and passes through the point (0, 2).
f (x) = a(x − h)2 + k
= a(x − 3)2 − 1
2 = a(0 − 3)2 − 1
2 = 9a − 1
1
a =
3
J. Robert Buchanan
Quadratic Functions and Models
Example (4 of 4)
Write the standard form of the equation of a parabola that has
vertex at (3, −1) and passes through the point (0, 2).
f (x) = a(x − h)2 + k
= a(x − 3)2 − 1
2 = a(0 − 3)2 − 1
2 = 9a − 1
1
a =
3
Thus f (x) =
1
(x − 3)2 − 1.
3
J. Robert Buchanan
Quadratic Functions and Models
Minimum and Maximum Values
If f (x) = ax 2 + bx + c we may write the quadratic function in
standard form by completing the square.
b 2
b2
f (x) = a x +
+ c−
2a
4a
b
b
Thus the coordinates of the vertex are − , f −
.
2a
2a
J. Robert Buchanan
Quadratic Functions and Models
Minimum and Maximum Values
If f (x) = ax 2 + bx + c we may write the quadratic function in
standard form by completing the square.
b 2
b2
f (x) = a x +
+ c−
2a
4a
b
b
Thus the coordinates of the vertex are − , f −
.
2a
2a
Consider the function f (x) = ax 2 + bx + c.
b
1
If a > 0, f has a minimum at x = − . The minimum
2a
b
value is f −
.
2a
b
2
If a < 0, f has a maximum at x = − . The maximum
2a
b
value is f −
.
2a
J. Robert Buchanan
Quadratic Functions and Models
Example
The profit P (in hundreds of dollars) that a company makes
depends on the amount x (in hundreds of dollars) that the
company spends on advertising according to the model
P = 230 + 20x − 0.5x 2 . What expenditure for advertising will
yield maximum profit?
J. Robert Buchanan
Quadratic Functions and Models
Example
The profit P (in hundreds of dollars) that a company makes
depends on the amount x (in hundreds of dollars) that the
company spends on advertising according to the model
P = 230 + 20x − 0.5x 2 . What expenditure for advertising will
yield maximum profit?
In this formula a = 0.5 and b = 20, thus profit is maximized
when
b
20
x =−
=−
= 20.
2a
2(−0.5)
Thus profit is maximized when $2,000 is spent on advertising.
J. Robert Buchanan
Quadratic Functions and Models
Example
The profit P (in hundreds of dollars) that a company makes
depends on the amount x (in hundreds of dollars) that the
company spends on advertising according to the model
P = 230 + 20x − 0.5x 2 . What expenditure for advertising will
yield maximum profit?
In this formula a = 0.5 and b = 20, thus profit is maximized
when
b
20
x =−
=−
= 20.
2a
2(−0.5)
Thus profit is maximized when $2,000 is spent on advertising.
The maximum profit will be
P = 230 + 20(20) − 0.5(20)2 = 430
or $43,000.
J. Robert Buchanan
Quadratic Functions and Models
Homework
Read Section 2.1.
Exercises: 1, 5, 9, 13, . . . , 81, 85
J. Robert Buchanan
Quadratic Functions and Models