Copyright © Cengage Learning. All rights reserved.

Functions and Graphs

3

Copyright © Cengage Learning. All rights reserved.

3.5Graphs of Functions

3Copyright © Cengage Learning. All rights reserved.

3 Functions and Graphs

3.5 Graph of Function

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Graphs of FunctionsIn this section we discuss aids for sketching graphs of certain types of functions. In particular, a function f is called even if f (–x) = f (x) for every x in its domain.

In this case, the equation y = f (x) is not changed if –x is substituted for x, and hence, from symmetry test 1, the graph of an even function is symmetric with respect to the y-axis.

A function f is called odd if f (–x) = –f (x) for every x in its domain. If we apply symmetry test 3 to the equation y = f (x), we see that the graph of an odd function is symmetric with respect to the origin.

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Graphs of FunctionsThese facts are summarized in the first two columns of the next chart.

Even and Odd Functions

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Example 1 – Determining whether a function is even or odd

Determine whether f is even, odd, or neither even nor odd.

(a) f (x) = 3x4 – 2x2 + 5 (b) f (x) = 2x5 – 7x3 + 4x

(c) f (x) = x3 + x2

Solution:In each case the domain of f is . To determine whether f is even or odd, we begin by examining f (–x), where x is any real number.

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Example 1 – Solution(a) f (–x) = 3(–x)4 – 2(–x)2 + 5

= 3x4 – 2x2 + 5

= f (x)

Since f (–x) = f (x), f is an even function.

(b) f (–x) = 2(–x)5 – 7(–x)3 + 4(–x)

= –2x5 + 7x3 – 4x

definition of f

simplify

substitute –x for x in f (x)

simplify

substitute –x for x in f (x)

cont’d

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Example 1 – Solution = –(2x5 – 7x3 + 4x)

= –f (x)

Since f (–x) = –f (x), f is an odd function.

(c) f (–x) = (–x)3 + (–x)2

= –x3 + x2

Since f (–x) ≠ f (x), and f (–x) ≠ –f (x) (note that –f (x) = –x3 – x2), the function f is neither even nor odd.

factor out –1

definition of f

substitute –x for x in f(x)

simplify

cont’d

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Graphs of FunctionsIn the next example we consider the absolute value function f, defined by f (x) = | x |.

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Example 2 – Sketching the graph of the absolute value function

Let f (x) = | x |.

(a) Determine whether f is even or odd.

(b) Sketch the graph of f.

(c) Find the intervals on which f is increasing or is decreasing.

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Example 2 – Solution(a) The domain of f is , because the absolute value of x exists for every real number x. If x is in , then

f (–x) = | –x | = | x | = f (x).

Thus, f is an even function, since f (–x) = f (x).

(b) Since f is even, its graph is symmetric with respect to the y-axis. If x 0, then | x | = x, and therefore the first quadrant part of the graph coincides with the line y = x.

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Example 2 – SolutionSketching this half-line and using symmetry gives us Figure 1.

Figure 1

cont’d

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Example 2 – Solution(c) Referring to the graph, we see that f is decreasing on (– , 0] and is increasing on [0, ).

cont’d

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Graphs of FunctionsIf we know the graph of y = f (x), it is easy to sketch the graphs of

y = f (x) + c and y = f (x) – c

for any positive real number c. As in the next chart, for y = f (x) + c, we add c to the y-coordinate of each point on the graph of y = f (x). This shifts the graph of f upward a distance c. For y = f (x) – c with c > 0, we subtract c from each y-coordinate, thereby shifting the graph of f a distance c downward. These are called vertical shifts of graphs.

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Graphs of Functions

Vertically Shifting the Graph of y = f (x)

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Example 3 – Vertically shifting a graph

Sketch the graph of f :

(a) f (x) = x2 (b) f (x) = x2 + 4 (c) f (x) = x2 – 4

Solution:We shall sketch all graphs on the same coordinate plane.

(a) Since, f (–x) = (–x)2 = x2 = f (x),

the function f is even, and hence its graph is symmetric with respect to the y-axis.

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Example 3 – SolutionSeveral points on the graph of y = x2 are (0, 0), (1, 1), (2, 4), and (3, 9). Drawing a smooth curve through these points and reflecting through the y-axis gives us the sketch in Figure 2.

The graph is a parabola with vertex at the origin and opening upward.

Figure 2

cont’d

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Example 3 – Solution(b) To sketch the graph of y = x2 + 4, we add 4 to the y-coordinate of each point on the graph of y = x2; that is, we shift the graph in part (a) upward 4 units, as shown in the figure.

(c) To sketch the graph of y = x2 – 4, we decrease the y-coordinates of y = x2 by 4; that is, we shift the graph in

part (a) downward 4 units.

cont’d

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Graphs of FunctionsWe can also consider horizontal shifts of graphs. Specifically, if c > 0, consider the graphs of y = f (x) and y = g(x) = f(x – c) sketched on the same coordinate plane, as illustrated in the next chart. Since

g(a + c) = f([ a + c] – c) = f(a),

we see that the point with x-coordinate a on the graph of y = f (x) has the same y-coordinate as the point with x-coordinate a + c on the graph of y = g (x) = f (x – c).

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Graphs of FunctionsThis implies that the graph of y = g(x) = f (x – c) can be obtained by shifting the graph of y = f (x) to the right a distance c.

Similarly, the graph of y = h(x) = f (x + c) can be obtained by shifting the graph of f to the left a distance c, as shown in the chart.

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Graphs of Functions

Horizontally Shifting the Graph of y = f (x)

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Graphs of FunctionsHorizontal and vertical shifts are also referred to as translations.

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Example 4 – Horizontally shifting a graph

Sketch the graph of f :

(a) f (x) = (x – 4)2 (b) f (x) = (x + 2)2

Solution:The graph of y = x2 is sketched in Figure 3.

Figure 3

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Example 4 – Solution(a) Shifting the graph of y = x2 to the right 4 units gives us

the graph of y = (x – 4)2, shown in the figure.

(b) Shifting the graph of y = x2 to the left 2 units leads us the graph of y = (x + 2)2, shown in the figure.

cont’d

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Graphs of FunctionsTo obtain the graph of y = cf (x) for some real number c, we may multiply the y-coordinates of points on the graph of y = f (x) by c. For example, if y = 2f (x), we double the y-coordinates; or if y = , we multiply each y-coordinate by .

This procedure is referred to as vertically stretching the graph of f (if c > 1) or vertically compressing the graph (if 0 < c < 1) and is summarized in the next chart.

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Graphs of Functions

Vertically Stretching or Compressing the Graph of y = f (x)

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Example 5 – Vertically stretching or compressing a graph

Sketch the graph of the equation:

(a) y = 4x2 (b) y =

Solution:(a) To sketch the graph of y = 4x2, we

may refer to the graph of y = x2 in Figure 4 and multiply the y-coordinate of each point by 4.

Figure 4

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Example 5 – Solution This stretches the graph of y = x2 vertically by a factor 4 and gives us a narrower parabola that is sharper at the vertex, as illustrated in the figure.

(b) The graph of y = may be sketched by multiplying the y-coordinates of points on the graph of y = x2 by . This compresses the graph of y = x2 vertically by a factor = 4 and gives us a wider parabola that is flatter at the vertex, as shown in Figure 4.

cont’d

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Graphs of FunctionsWe may obtain the graph of y = –f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by –1.

Thus, every point (a, b) on the graph of y = f (x) that lies above the x-axis determines a point (a, –b) on the graph of y = –f (x) that lies below the x-axis.

Similarly, if (c, d ) lies below the x-axis (that is, d < 0), then (c, –d ) lies above the x-axis. The graph of y = –f (x) is a reflection of the graph of y = f (x) through the x-axis.

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Example 6 – Reflecting a graph through the x-axis

Sketch the graph of y = –x2.

Solution:The graph may be found by plotting points; however, since the graph of y = x2 is familiar to us, we sketch it as in Figure 5 and then multiply the y-coordinates of points by –1.

This procedure gives us the reflectionthrough the x-axis indicated in the figure.

Figure 5

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Graphs of FunctionsSometimes it is useful to compare the graphs of y = f (x) and y = f (cx) if c ≠ 0. In this case the function values f (x) for

a x b

are the same as the function values f (cx) for

a cx b or, equivalently,

This implies that the graph of f is horizontally compressed (if c > 1) or horizontally stretched (if 0 < c < 1), as summarized in the next chart.

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Graphs of Functions

Horizontally Compressing or Stretching the Graph of y = f (x)

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Graphs of FunctionsIf c < 0, then the graph of y = f (cx) may be obtained by reflecting the graph of y = f (| c |x) through the y-axis.

For example, to sketch the graph of y = f (–2x), we reflect the graph of y = f (2x) through the y-axis. As a special case, the graph of y = f (–x), is a reflection of the graph of y = f (x), through the y-axis.

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Example 7 – Horizontally stretching or compressing a graph

If f(x) = x3 – 4x2, sketch the graphs of y = f (x), y = f (2x), and y =

Solution:We have the following:

y = f (x) = x3 – 4x2

y = f (2x) = (2x)3 – 4(2x)2

= x2(x – 4)

= 8x3 –16x2 = 8x2(x – 2)

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Example 7 – SolutionNote that the x-intercepts of the graph of y = f (2x), are 0 and 2, which are the x-intercepts of 0 and 4 for y = f (x). This indicates a horizontal compression by a factor 2.

The x-intercepts of the graph of are 0 and 8, which are 2 times the x-intercepts for y = f (x). This indicates a horizontal stretching by a factor

cont’d

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Example 7 – SolutionThe graphs, obtained by using a graphing calculator with viewing rectangle [–6, 15] by [–10, 4] , are shown in Figure 6.

Figure 6

cont’d

[–6, 15] by [–10, 4]

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Graphs of FunctionsFunctions are sometimes described by more than one expression, as in the next examples. We call such functions piecewise-defined functions.

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Example 8 – Sketching the graph of a piecewise-defined function

Sketch the graph of the function f if

2x + 5 if x –1 f (x) = x2 if | x | < 1 2 if x 1

Solution:If x –1, then f (x) = 2x + 5 and the graph of f coincides with the line y = 2x + 5 and is represented by the portion of the graph to the left of the line x = –1 in Figure 7. Figure 7

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Example 8 – SolutionThe small dot indicates that the point (–1, 3) is on the graph.

If | x | < 1 (or, equivalently, –1 < x < 1) we use x2 to find values of f, and therefore this part of the graph of f coincides with the parabola y = x2, as indicated in the figure.

Note that the points (–1,1) and (1,1) are not on the graph.

cont’d

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Example 8 – SolutionFinally, if x 1, the values of f are always 2. Thus, the graph of f for x 1 is the horizontal half-line in Figure 7.

Note: When you finish sketching the graph of a piecewise-defined function, check that it passes the vertical line test.

Figure 7

cont’d

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Graphs of FunctionsIf x is a real number, we define the symbol as follows:

= n, where n is the greatest integer such that n x

If we identify with points on a coordinate line, then n is the first integer to the left of (or equal to) x.

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Graphs of FunctionsIllustration: The Symbol

•

•

•

•

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Graphs of Functions•

•

•

•

The greatest integer function f is defined by f (x) = .

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Example 11 – Sketching the graph of the greatest integer function

Sketch the graph of the greatest integer function.

Solution:The x- and y-coordinates of some points on the graph may be listed as follows:

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Example 11 – SolutionWhenever x is between successive integers, the corresponding part of the graph is a segment of a horizontal line. Part of the graph is sketched in Figure 10. The graph continues indefinitely to the right and to the left.

Figure 10

cont’d

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Graphs of FunctionsIn general, if the graph of y = f (x) contains a point P(c, –d) with d positive, then the graph of y = | f (x) | contains the point Q(c, d) —that is, Q is the reflection of P through the x-axis. Points with nonnegative y-values are the same for the graphs of y = f (x) and y = | f (x) |.

We used algebraic methods to solve inequalities involving absolute values of polynomials of degree 1, such as

| 2x – 5 | < 7 and | 5x + 2 | 3.

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Graphs of FunctionsThe processes of shifting, stretching, compressing, and reflecting a graph may be collectively termed transforming a graph, and the resulting graph is called a transformation of the original graph.