Welcome Ticket: Worksheet You may work in groups and you may use the calculators. You have about 6 minutes to do this.

HW Key: p. 94: 35-42 (all) 35) (a) no (b) yes (c) no 36) (a) yes (b) no (c) no 37) (a) yes (b) no (c) no 38) (a) no (b) no (c) yes 39) (a) yes (b) yes (c) yes 40) (a) no (b) yes (c) no 41) (a) no (b) no (c) yes 42) (a) no (b) yes (c) no

2.1 Graphs of Basic Functions & Relations – Part 2 Unit 2 2.1 Graphs of Basic Functions & Relations – Part 2

Objectives & HW: The students will be able to: Identify symmetry with respect to the y-axis, x-axis, or the origin. Distinguish between odd and even functions. HW: p. 94: 44-52 (even), 54, 60, 68, 70

The following graphs are symmetric with respect to the y-axis:

The following graph is also symmetric with respect to the y-axis:

Algebraic rule for y-axis symmetry: A graph is symmetric with respect to the y-axis if whenever (x, y) is on the graph, then (-x, y) is also on it.

Replace x by –x and see if it produces an equivalent equation: Verify algebraically that y = x6 – x4 + 2 is symmetric with respect to the y-axis: Replace x by –x and see if it produces an equivalent equation: y = x6 – x4 + 2 y = (–x)6 – (–x)4 + 2 Since this is equivalent to the original equation, the graph will be symmetric with respect to the y-axis.

The following graphs are symmetric with respect to the x-axis:

The following graph is also symmetric with respect to the x-axis:

Algebraic rule for x-axis symmetry: A graph is symmetric with respect to the x-axis if whenever (x, y) is on the graph, then (x, -y) is also on it.

Replace y by –y and see if it produces an equivalent equation: Verify algebraically that y2 = x3 – x2 + 2x is symmetric with respect to the x-axis: Replace y by –y and see if it produces an equivalent equation: y2 = x3 – x2 + 2x (–y)2 = (x)3 – (x)2 + 2x Since this is equivalent to the original equation, the graph will be symmetric with respect to the x-axis.

The following graphs are symmetric with respect to the origin:

The following graph is also symmetric with respect to the origin:

Algebraic rule for origin symmetry: A graph is symmetric with respect to the origin if whenever (x, y) is on the graph, then (-x, -y) is also on it.

Verify algebraically that y2 = x4 – x2 + 2 is symmetric with respect to the origin: Replace x by –x and y by –y and see if it produces an equivalent equation: y2 = x4 – x2 + 2 (–y)2 = (–x)4 – (–x)2 + 2 Since this is equivalent to the original equation, the graph will be symmetric with respect to the origin.

Symmetry with respect to Symmetry Tests: Symmetry with respect to Coordinate Test Algebraic Test If (x, y) is on the graph, then (-x, y) is on the graph. Replacing x by –x produces an equi- valent equation. y-axis If (x, y) is on the graph, then (x, -y) is on the graph. Replacing y by –y produces an equi- valent equation. x-axis This is not in the notesheet. Pass out the half sheet separately. Replacing x by –x and y by –y produ- ces an equiv. eq. If (x, y) in on the graph, then (-x, -y) is on the graph. origin

Even Function: A function is even if: f(-x) = f(x) for every value in the domain of f. Its graph is symmetric with respect to the y-axis.

Odd Function: A function is odd if: f(-x) = -f(x) for every value in the domain of f. Its graph is symmetric with respect to the origin.

Determine whether the given function is even, odd, or neither. k(t) = 3t k(-t) = 3(-t) k(-t) = -3t k(-t) = -k(t) Therefore, the function is odd. h(u) = |4u| h(-u) = |4(-u)| h(-u) = |-4u| h(-u) = |4u| h(-u) = h(u) Therefore, the function is even.

Determine whether the given function is even, odd, or neither. f(x) = x(x4 – x2) + 4 f(x) = x5 – x3 + 4 f(–x) = (–x)5 – (–x)3 + 4 f(–x) = –x5 – (–x3) + 4 f(–x) = –x5 + x3 + 4 f(–x) = –(x5 – x3 – 4) f(–x) ≠ –f(x) ≠ f(x) Therefore, the function is neither odd nor even.