Transcript Chapter 02 – Section 01

2
Nonlinear Functions
and Models
Copyright © Cengage Learning. All rights reserved.
2.1
Quadratic Functions and Models
Copyright © Cengage Learning. All rights reserved.
Quadratic Functions and Models
Quadratic Function
A quadratic function of the variable x is a function that
can be written in the form
f(x) = ax2 + bx + c
Function form
y = ax2 + bx + c
Equation form
or
where a, b, and c are fixed numbers (with a ≠ 0).
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Quadratic Functions and Models
Quick Example
f(x) = 3x2 – 2x + 1
a = 3, b = –2, c = 1
Every quadratic function f(x) = ax2 + bx + c (a ≠ 0) has a
parabola as its graph.
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Quadratic Functions and Models
Following is a summary of some features of parabolas that
we can use to sketch the graph of any quadratic function.
Features of a Parabola
The graph of f(x) = ax2 + bx + c (a ≠ 0) is a parabola. If a > 0
the parabola opens upward (concave up) and if a < 0 it
opens downward (concave down):
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Quadratic Functions and Models
Vertex, Intercepts, and Symmetry
Vertex
The vertex is the highest or lowest point of the parabola
(see the above figure). Its x-coordinate is
.
Its y-coordinate is
.
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Quadratic Functions and Models
x-Intercepts (if any)
These occur when f(x) = 0; that is, when
ax2 + bx + c = 0.
Solve this equation for x by either factoring or using the
quadratic formula. The x-intercepts are
If the discriminant b2 – 4ac is positive, there are two
x-intercepts. If it is zero, there is a single x-intercept (at the
vertex). If it is negative, there are no x-intercepts (so the
parabola doesn’t touch the x-axis at all).
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Quadratic Functions and Models
y-Intercept
This occurs when x = 0, so
y = a(0)2 + b(0) + c = c.
Symmetry
The parabola is symmetric with respect to the vertical line
through the vertex, which is the line x =
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Quadratic Functions and Models
Note that the x-intercepts can also be written as
making it clear that they are located symmetrically on either
side of the line x = –b/(2a). This partially justifies the claim
that the whole parabola is symmetric with respect to this
line.
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Example 1 – Sketching the Graph of a Quadratic Function
Sketch the graph of f(x) = x2 + 2x – 8 by hand.
Solution:
Here, a = 1, b = 2, and c = –8.
Because a > 0, the parabola is concave up (Figure 1).
Figure 1
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Example 1 – Solution
cont’d
Vertex: The x coordinate of the vertex is
To get its y coordinate, we substitute the value of x back
into f(x) to get
y = f(–1) = (–1)2 + 2(–1) – 8
=1–2–8
= –9.
Thus, the coordinates of the vertex are (–1, –9).
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Example 1 – Solution
cont’d
x-Intercepts: To calculate the x-intercepts (if any), we solve
the equation
x2 + 2x – 8 = 0.
Luckily, this equation factors as
(x + 4)(x – 2) = 0.
Thus, the solutions are x = –4 and x = 2, so these values
are the x-intercepts.
y-Intercept: The y-intercept is given by c = –8.
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Example 1 – Solution
cont’d
Symmetry: The graph is symmetric around the vertical line
x = –1.
Now we can sketch the curve
as in Figure 2.
(As we see in the figure,
it is helpful to plot additional
points by using the equation
y = x2 + 2x – 8, and
to use symmetry
to obtain others.)
Figure 2
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Applications
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Applications
We know that the revenue resulting from one or more
business transactions is the total payment received. Thus,
if q units of some item are sold at p dollars per unit, the
revenue resulting from the sale is
revenue = price  quantity
R = pq.
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Example 3 – Demand and Revenue
Alien Publications, Inc. predicts that the demand equation
for the sale of its latest illustrated sci-fi novel Episode 93:
Yoda vs. Alien is
q = –2,000p + 150,000
where q is the number of books it can sell each year at a
price of $p per book. What price should Alien Publications,
Inc., charge to obtain the maximum annual revenue?
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Example 3 – Solution
The total revenue depends on the price, as follows:
R = pq
Formula for revenue.
= p(–2,000p + 150,000)
Substitute for q from demand equation.
= –2,000p2 + 150,000p.
Simplify.
We are after the price p that gives the maximum possible
revenue.
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Example 3 – Solution
cont’d
Notice that what we have is a quadratic function of the form
R(p) = ap2 + bp + c, where a = –2,000, b = 150,000, and
c = 0. Because a is negative, the graph of the function is a
parabola, concave down, so its vertex is its highest point
(Figure 5).
Figure 5
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Example 3 – Solution
cont’d
The p coordinate of the vertex is
This value of p gives the highest point on the graph and
thus gives the largest value of R(p).
We may conclude that Alien Publications, Inc., should
charge $37.50 per book to maximize its annual revenue.
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Fitting a Quadratic Function to
Data: Quadratic Regression
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Fitting a Quadratic Function to Data: Quadratic Regression
Here, we see how to use technology to obtain the
quadratic regression curve associated with a set of
points.
The quadratic regression curve is the quadratic curve
y = ax2 + bx + c that best fits the data points in the sense
that the associated sum-of-squares error is a minimum.
Although there are algebraic methods for obtaining the
quadratic regression curve, it is normal to use technology
to do this.
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Example 5 – Carbon Dioxide Concentration
The following table shows the annual mean carbon dioxide
concentration measured at Mauna Loa Observatory in
Hawaii, in parts per million, every 10 years from 1960
through 2010 (t = 0 represents 1960).
a. Is a linear model appropriate for these data?
b. Find the quadratic model
C(t) = at2 + bt + c
that best fits the data.
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Example 5(a) – Solution
To see whether a linear model is appropriate, we plot the
data points and the regression line (Figure 8).
From the graph, we can see
that the given data suggest a
curve and not a straight line:
The observed points are above
the regression line at the ends
but below in the middle. (We
would expect the data points
from a linear relation to fall
randomly above and below
the regression line.)
Figure 8
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Example 5(b) – Solution
cont’d
The quadratic model that best fits the data is the quadratic
regression model. As with linear regression, there are
algebraic formulas to compute a, b, and c, but they are
rather involved.
However, we exploit the fact that
these formulas are built into graphing
calculators, spreadsheets, and other
technology and obtain the regression
curve using technology (see Figure 9):
Figure 9
C(t) = 0.012t2 + 0.85t + 320
Coefficients rounded to two
significant digits
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Example 5(b) – Solution
cont’d
Notice from the previous graphs that the quadratic
regression model appears to give a far better fit than the
linear regression model.
This impression is supported by the values of SSE: For the
linear regression model SSE  58, while for the quadratic
regression model SSE is much smaller, approximately 2.6,
indicating a much better fit.
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