WARM UP Evaluate 1. and when and 2. and when and

SYMMETRY

OBJECTIVES Test for symmetry with respect to an axis. Test for symmetry with respect to the origin. Determine whether a function is even or odd. Graph any quadratic function.

SYMMETRY WITH RESPECT TO AXES In this chapter we consider how changes in the equation y = f(x) affect the graph of the function it defines. Then we use this information to graph quadratic functions where. Many examples of line symmetry can be found in nature. A butterfly, a bug, or a leaf all exhibit line symmetry. The idea of line symmetry can be described precisely in mathematical language.

DEFINITION Two points symmetric with respect to a line are called reflections of each other across the line. The line is known as a line of symmetry. lP

EXAMPLES The figure at the right is not symmetric with respect to the vertical line shown, or with respect to a horizontal line. It is symmetric in other respects, as we will see later. Symmetric with respect to the line Not symmetric with respect to the line

TYPES OF SYMMETRY There are types of symmetry in which the x-axis or the y-axis is a line of symmetry. THEOREM 9-1 Two points are symmetric with respect to the x-axis if and only if their y-coordinates are additive inverses and they have the same x-coordinate. Two points are symmetric with respect to the y-axis if and only if their x-coordinates are additive inverses and they have the same y-coordinate.

RELATIONSHIPS The relationship defined by contains (2, 4) and (-2, 4). The first coordinates, 2 and -2 are additive inverses of each other. The second coordinates are the same. For every point (x, y) of the relation, there is another point (-x, y). So the relation is symmetric with respect to the y-axis. (2, 4) (-2, 4) A relation is any set of ordered pairs

SYMMETRY RELATION We now have a means of testing a relation for symmetry with respect to the x- and y- axes when the relation is defined by an equation. THEOREM 9-2 When a relation is defined by an equation. A. Its graph is symmetric with respect to the y-axis if and only if replacing x by –x produces an equivalent equation B. Its graph is symmetric with respect to the x-axis if and only if replacing y by –y produces an equivalent equation.

EXAMPLE 1 Test for symmetry with respect to the axes. To test for symmetry with respect to the y-axis, we replace x by –x and obtain or. This is equivalent to. Therefore, the graph is symmetric with respect to the y-axis. To test for symmetry with respect to the x-axis, we replace y by –y and obtain or. This is not equivalent to. Therefore, the graph is not symmetric with respect to the x-axis.

TRY THIS… Test for symmetry with respect to the axes. a. b.

SYMMETRY WITH RESPECT TO THE ORIGIN Symmetry with respect to the origin is a special kind of point symmetry. Theorem 9-3 Two points are symmetric with respect to the origin if and only if both their x- and y- coordinates are additive inverse of each other. The point symmetric to (3, -5) with respect to the origin is (-3, 5). Theorem 9-4 A graph of a relation defined by an equation is symmetric with respect to the origin if an only if replacing x by –x and replacing y by –y produces an equivalent equation. This gives us a means for testing a relationship for symmetry with respect to the origin when it is defined by an equation.

EXAMPLE 2 Test for symmetry with respect to the origin. We replace x by –x and y by –y. We obtain which is equivalent to, the original equation. Therefore the graph is symmetric with respect to the origin.

TRY THIS… Test each relation for symmetry with respect to the origin. a. b. c.

EVEN AND ODD FUNCTIONS Functions whose graphs are symmetric with respect to the y-axis are called even functions. If y = f(x) defines an even function, then by Theorem 9-1, y = f(-x) will define the same function. Definition A function is an e ven function when f (-x) = f(x) for all x in the domain of f. Functions whose graphs are symmetric with respect to the origin are called odd functions. If y = f(x) defines an odd function, then by Theorem 9-4, -y = f(-x). Definition A function is an o dd function when f (-x) = -f(x) for all x in the domain of f.

EXAMPLE 3 Determine whether is even, odd or neither. Compare f(-x) and f(x). They are the same for all x in the domain, so f is an even function. Compare f(-x) and -f(x). They are not the same for all x in the domain. The function is not an odd function.

TRY THIS… Determine whether each function is odd, even or neither. a. b. c.

TRANSFORMATIONS  An alteration of a relation is called a transformation.  If such an alteration results in moving the graph of the relation without changing its size or shape and without rotating it, the transformation is called a translation.  Consider the following relations and their graphs y = x and y = x +1.  The graphs have the same shape except that the second one is moved up a distance of one unit. 2 2

EXAMPLE  Consider the graph of y = |x| and y = |x| – 2.  Note that y = |x| – 2 is equivalent to y + 2 = |x| or y – (– 2) = |x|. Thus the new equation can be obtained by replacing y by y – (– 2). It will be translated 2 units downward.

CH. 9.1 HOMEWORK Textbook pg. 389 #2, 8, 14, 16, 35, 36, 38 & 40 pg. 393 # 6, 8 & 10