Copyright © 2013, 2009, 2005 Pearson Education, Inc. Graphs and Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

2.7 Graphing Techniques Stretching and Shrinking Reflecting Symmetry Even and Odd Functions Translations

Graph each function. (a) STRETCHING OR SHRINKING A GRAPH Example 1 Graph each function. (a) Solution Comparing the table of values for (x) = x and g(x) = 2x, we see that for corresponding x-values, the y-values of g are each twice those of . So the graph of g(x) is narrower than that of (x). x (x) = x g(x) = 2x – 2 2 4 – 1 1

Graph each function. (a) STRETCHING OR SHRINKING A GRAPH Example 1 y x (x) = x g(x) = 2x – 2 2 4 – 1 1 4 3 – 4 – 3 – 2 x 1 2 3 4 – 2 – 3 – 4

(b) STRETCHING OR SHRINKING A GRAPH Example 1 Solution The graph of h(x) is also the same general shape as that of (x), but here the coefficient 1/2 causes a vertical shrink. The graph of h(x) is wider than the graph of (x), as we see by comparing the tables of values. x (x) = h(x) = – 2 2 1 – 1 1/2

(b) STRETCHING OR SHRINKING A GRAPH Example 1 x (x) = h(x) = – 2 2 1 – 1 1/2 x y 1 2 3 – 2 4 – 3 – 4

(c) STRETCHING OR SHRINKING A GRAPH Example 1 Solution Use Property 2 of absolute value to rewrite Property 2 The graph of k(x) = 2x is the same as the graph of g(x) = 2xin part (a).

Vertical Stretching or Shrinking of the Graph of a Function Suppose that a > 0. If a point (x, y) lies on the graph of y = (x), then the point (x, ay) lies on the graph of y = a(x). (a) If a > 1, then the graph of y = a(x) is a vertical stretching of the graph of y = (x). (b) If 0 < a <1, then the graph of y = a(x) is a vertical shrinking of the graph of y = (x).

Horizontal Stretching or Shrinking of the Graph of a Function Suppose that a > 0. If a point (x, y) lies on the graph of y = (x), then the point ( , y) lies on the graph of y = (ax). (a) If 0 < a < 1, then the graph of y = (ax) is horizontal stretching of the graph of y = (x). (b) If a > 1, then the graph of y = (ax) is a horizontal shrinking of the graph of y = (x).

Reflecting Forming the mirror image of a graph across a line is called reflecting the graph across the line.

Graph each function. (a) REFLECTING A GRAPH ACROSS AN AXIS Example 2 Graph each function. (a) Solution Every y-value of the graph of g(x) is the negative of the corresponding y-value of x (x) = g(x) = 1 – 1 4 2 – 2

Graph each function. (a) REFLECTING A GRAPH ACROSS AN AXIS Example 2 x g(x) = 1 – 1 4 2 – 2

Graph each function. (b) REFLECTING A GRAPH ACROSS AN AXIS Example 2 Graph each function. (b) Solution Choosing x-values for h(x) that are the negatives of those used for (x), we see that the corresponding y-values are the same. The graph of h is a reflection of the graph  across the y-axis. x (x) = h(x) = – 4 undefined 2 – 1 1 4

Graph each function. (b) REFLECTING A GRAPH ACROSS AN AXIS Example 2 x h(x) = – 4 undefined 2 – 1 1 4

Reflecting Across an Axis The graph of y = –(x) is the same as the graph of y = (x) reflected across the x-axis. (If a point (x, y) lies on the graph of y = (x), then (x, – y) lies on this reflection. The graph of y = (– x) is the same as the graph of y = (x) reflected across the y-axis. (If a point (x, y) lies on the graph of y = (x), then (– x, y) lies on this reflection.)

Symmetry with Respect to An Axis The graph of an equation is symmetric with respect to the y-axis if the replacement of x with – x results in an equivalent equation. The graph of an equation is symmetric with respect to the x-axis if the replacement of y with – y results in an equivalent equation.

Test for symmetry with respect to the x-axis and the y-axis. TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS Example 3 Test for symmetry with respect to the x-axis and the y-axis. (a) y Solution Replace x with – x. 2 – 2 4 Equivalent x

Test for symmetry with respect to the x-axis and the y-axis. TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS Example 3 Test for symmetry with respect to the x-axis and the y-axis. (a) Solution The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Substituting –y for y does not result in an equivalent equation, and thus the graph is not symmetric with respect to the x-axis.

Test for symmetry with respect to the x-axis and the y-axis. TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS Example 3 Test for symmetry with respect to the x-axis and the y-axis. (b) Solution Replace y with – y. Equivalent

Test for symmetry with respect to the x-axis and the y-axis. TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS Example 3 Test for symmetry with respect to the x-axis and the y-axis. y (b) 2 – 2 4 Solution The graph is symmetric with respect to the x-axis. It is not symmetric with respect to the y-axis. x

Test for symmetry with respect to the x-axis and the y-axis. TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS Example 3 Test for symmetry with respect to the x-axis and the y-axis. (c) Solution Substitute – x for x then – y for y in x2 + y2 = 16. (– x)2 + y2 = 16 and x2 + (– y)2 = 16. Both simplify to the original equation. The graph, a circle of radius 4 centered at the origin, is symmetric with respect to both axes.

Test for symmetry with respect to the x-axis and the y-axis. TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS Example 3 Test for symmetry with respect to the x-axis and the y-axis. (c) Solution

Test for symmetry with respect to the x-axis and the y-axis. TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS Example 3 Test for symmetry with respect to the x-axis and the y-axis. (d) Solution In 2x + y = 4, replace x with – x to get – 2x + y = 4. Replace y with – y to get 2x – y = 4. Neither case produces an equivalent equation, so this graph is not symmetric with respect to either axis.

Test for symmetry with respect to the x-axis and the y-axis. TESTING FOR SYMMETRY WITH RESPECT TO AN AXIS Example 3 Test for symmetry with respect to the x-axis and the y-axis. (d) Solution

Symmetry with Respect to the Origin The graph of an equation is symmetric with respect to the origin if the replacement of both x with – x and y with – y at the same time results in an equivalent equation.

Use parentheses around – x and – y TESTING FOR SYMMETRY WITH RESPECT TO THE ORIGIN Example 4 Is this graph symmetric with respect to the origin? (a) Solution Replace x with – x and y with – y. Use parentheses around – x and – y Equivalent The graph is symmetric with respect to the origin.

Is this graph symmetric with respect to the origin? TESTING FOR SYMMETRY WITH RESPECT TO THE ORIGIN Example 4 Is this graph symmetric with respect to the origin? (b) Solution Replace x with – x and y with – y. The graph is symmetric with respect to the origin. Equivalent

Important Concepts of Symmetry A graph is symmetric with respect to both x- and y-axes is automatically symmetric with respect to the origin. A graph symmetric with respect to the origin need not be symmetric with respect to either axis. Of the three types of symmetrywith respect to the x-axis, the y-axis, and the origina graph possessing any two types must also exhibit the third type of symmetry.

Symmetry with Respect to: x-axis y-axis Origin Equation is unchanged if: y is replaced with – y x is replaced with – x x is replaced with – x and y is replaced with – y Example x y x y x y

Even and Odd Functions A function  is an even function if (– x) = (x) for all x in the domain of . (Its graph is symmetric with respect to the y-axis.) A function  is an odd function if (– x) = – (x) for all x in the domain of . (Its graph is symmetric with respect to the origin.)

Decide whether each function defined is even, odd, or neither. DETERMINING WHETHER FUNCTIONS ARE EVEN, ODD, OR NEITHER Example 5 Decide whether each function defined is even, odd, or neither. (a) Solution Replacing x with – x gives: Since (– x) = (x) for each x in the domain of the function,  is even.

Decide whether each function defined is even, odd, or neither. DETERMINING WHETHER FUNCTIONS ARE EVEN, ODD, OR NEITHER Example 5 Decide whether each function defined is even, odd, or neither. (b) Solution The function  is odd because (– x) = – (x).

Decide whether each function defined is even, odd, or neither. DETERMINING WHETHER FUNCTIONS ARE EVEN, ODD, OR NEITHER Example 5 Decide whether each function defined is even, odd, or neither. (c) Solution Replace x with – x. Since (– x) ≠ (x) and (– x) ≠ – (x), the function  is neither even nor odd.

Graph g(x) = x– 4. Solution TRANSLATING A GRAPH VERTICALLY Example 6 – 1 1 – 3 x – 4 4 (0, – 4)

Vertical Translations If a function g is defined by g(x) = f(x) + c, where c is a real number, then for every point (x,y) on the graph of f, there will be a corresponding point (x, y + c) on the graph of g. The graph of g will be the same as the graph of f, but translated c units up if c is positive or │c│ units down if c is negative. The graph of g is called a vertical translation of the graph of f.

Vertical Translations

Graph g(x) = x – 4. Solution TRANSLATING A GRAPH HORIZONTALLY Example 7 Graph g(x) = x – 4. Solution y x (x) = g(x) = – 2 2 6 4 x – 4 4 8 (4, 0)

Horizontal Translations If a function g is defined by g(x) = f(x – c), where c is a real number, then for every point (x,y) on the graph of f, there will be a corresponding point (x + c,y) on the graph of g. The graph of g will be the same as the graph of f, but translated c units to the right if c is positive or │c │units to the left is c is negative. The graph of g is called a horizontal translation of the graph of f.

Horizontal Translations

Caution Be careful when translating graphs horizontally Caution Be careful when translating graphs horizontally. To determine the direction and magnitude of horizontal translations, find the value that would cause the expression in parentheses to equal 0. For example, the graph of y = (x – 5)2 would be translated 5 units to the right of y = x2, because x = + 5 would cause x – 5 to equal 0. On the other hand, the graph of y = (x + 5)2 would be translated 5 units to the left of y = x2, because x = – 5 would cause x + 5 to equal 0.

Graph each function. (a) USING MORE THAN ONE TRNASFORMATION Example 8 Solution To graph this function, the lowest point on the graph of y = xis translated 3 units to the left and 1 unit up. The graph opens down because of the negative sign in front of the absolute value expression, making the lowest point now the highest point on the graph. The graph is symmetric with respect to the line x = – 3.

Graph each function. (a) USING MORE THAN ONE TRNASFORMATION Example 8 y 3 x – 6 – 3 – 2

Graph each function. (b) USING MORE THAN ONE TRNASFORMATION Example 8 Solution To determine the horizontal translation, factor out 2. Factor out 2.

Graph each function. (b) USING MORE THAN ONE TRNASFORMATION Example 8 Solution y 4 2 x 2 4

Graph each function. (c) USING MORE THAN ONE TRNASFORMATION Example 8 Graph each function. (c) Solution The graph of g has the same shape as that of y = x2, but it is wider (that is, shrunken vertically), reflected across the x-axis because the coefficient is negative, and then translated 4 units up.

Graph each function. (c) USING MORE THAN ONE TRNASFORMATION Example 8 Solution

GRAPHING TRANSLATIONS GIVEN THE GRAPH OF y = (x) Example 9 A graph of a function defined by y = f(x) is shown in the figure. Use this graph to sketch each of the following graphs.

Solution The graph is the same, translated 3 units up. GRAPHING TRANSLATIONS GIVEN THE GRAPH OF y = (x) Example 9 Graph the function. (a) Solution The graph is the same, translated 3 units up.

Solution The graph must be translated 3 units to the left. GRAPHING TRANSLATIONS GIVEN THE GRAPH OF y = (x) Example 9 Graph the function. (b) Solution The graph must be translated 3 units to the left.

Solution The graph is translated 2 units to the right and 3 units up. GRAPHING TRANSLATIONS GIVEN THE GRAPH OF y = (x) Example 9 Graph the function. (c) Solution The graph is translated 2 units to the right and 3 units up.

Summary of Graphing Techniques In the descriptions that follow, assume that a > 0, h > 0, and k > 0. In comparison with the graph of y = (x): The graph of y = (x) + k is translated k units up. The graph of y = (x) – k is translated k units down. The graph of y = (x + h) is translated h units to the left. The graph of y = (x – h) is translated h units to the right. The graph of y = a(x) is a vertical stretching of the graph of y = (x) if a > 1. It is a vertical shrinking if 0 < a < 1. The graph of y = a(x) is a horizontal stretching of the graph of y = (x) if 0 < a < 1. It is a horizontal shrinking if a > 1. The graph of y = –(x) is reflected across the x-axis. The graph of y = (–x) is reflected across the y-axis.