Section 1.3: Graphing
By applying transformations to the graph of a function we can obtain the graphs of
2
3
certain
√ relatedxfunctions. You must be able to graph y = c, y = x, y = x , y = x , y = 1/x,
y = x, y = e , y = ln x, y = sin x, y = cos x, and y = tan x without using a calculator.
Theorem: (Vertical and Horizontal Shifts)
Suppose c > 0. To obtain the graph of
• y = f (x) + c, shift the graph of y = f (x) a distance c units upward,
• y = f (x) − c, shift the graph of y = f (x) a distance c units downward,
• y = f (x − c), shift the graph of y = f (x) a distance c units to the right,
• y = f (x + c), shift the graph of y = f (x) a distance c units to the left.
Theorem: (Vertical and Horizontal Stretching and Reflecting)
Suppose c > 1. To obtain the graph of
• y = cf (x), stretch the graph of y = f (x) vertically by a factor of c,
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• y = f (x), compress the graph of y = f (x) vertically by a factor of c,
c
• y = f (cx), compress the graph of y = f (x) horizontally by a factor of c,
• y = f (x/c), stretch the graph of y = f (x) horizontally by a factor of c,
• y = −f (x), reflect the graph of y = f (x) about the x-axis,
• y = f (−x), reflect the graph of y = f (x) about the y-axis.
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Example: Explain how the graph of y = −2(x − 3)2 + 1 can be obtained from the graph of
y = x2 by basic transformations.
Definition: If two quantities differ by a factor of 10, they are said to differ by one order of
magnitude. For example, x = 3 and y = 300 differ by two orders of magnitude. Quantities
which differ my several orders of magnitude are more easily plotted on a logarithmic scale.
That is, a scale according to which multiples of 10 are equally spaced.
Example: Display the numbers 0.001, 0.01, 0.1, 1, 100 , and 10, 000 on a logarithmic scale.
Example: Display the numbers 0.3, 5, 40, 300, and 6000 on a logarithmic scale.
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Definition: A graph in which the vertical axis is on a logarithmic scale and the horizontal
axis is on a linear scale is called a log-linear plot or a semilog plot.
Note: The graph of an exponential function y = bax on a semilog plot is a straight line.
Example: Use a logarithmic transformation to find a linear relationship for y = 4 × 32x .
Example: Suppose that when log y is graphed as a function of x, a straight line results.
Determine the functional relationship determined by the original coordinates (x1 , y1 ) = (1, 4)
and (x2 , y2 ) = (5, 1).
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Example: Using the semilog plot below, find a functional relationship between x and y.
1
y
10
0
10
−1
10
0
0.2
0.4
0.6
x
4
0.8
1
Example: Using the semilog plot below, find a functional relationship between x and y.
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13
y
12
11
10
9
8 0
10
1
2
10
10
x
5
3
10
Definition: A graph in which both the vertical and horizontal axes are on a logarithmic scale
is called a log-log plot or a double log plot.
Note: The graph of a power function y = bxa on a double log plot is a straight line.
Example: Use a logarithmic transformation to find a linear relationship for y = 5x3 .
Example: Suppose that when log y is graphed as a function of log x, a straight line results.
Determine the functional relationship determined by the original coordinates (x1 , y1 ) = (2, 5)
and (x2 , y2 ) = (5, 2).
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Example: Using the double log plot below, find a functional relationship between x and y.
4
10
3
10
2
y
10
1
10
0
10
−1
10
0
1
10
10
x
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