CPM Homework Help : CALC Problem 1-10

For each function $f(x)$ sketched below, sketch $f(-x)$ and compare it with $f(x)$ . Then describe its symmetry.

Hint 1(a):
What does $f(-x)$ mean?
Compare $f(1)$ and $f(-1)$
Compare $f(2)$ and $f(-2)$
keep going... $f(3)$ and $f(-3)$
More Help (a):
The sketch of $f(-x)$ should be identical to the given graph $f\left(x\right)$ .

Hint 1(b):
On this graph, $f(-x) =-f(x)$ .
More Help (b):
Plot all negative $y$ -values in the positive region, and plot all positive $y$ -values in the negative region.

EVEN AND ODD FUNCTIONS--INFORMALLY
A function that is symmetric with respect to the $y$ -axis, like that in part (a) above, is called an even function. A function that is symmetric with respect to the origin, such as that in part (b), is called an odd function.
Sketch examples of even and odd functions. Include how you can test whether a function is even or odd. Then list some famous even/odd functions from your parent graphs, and the symmetries associated with even and odd functions.

Hint (c):

Even functions are symmetrical ACROSS the $y$ -axis, they have reflective symmetry. Odd functions are symmetrical ABOUT the origin, they have $180°$ rotational symmetry.

Hint 2(c):

Graph (a) is even. Graph (b) is odd.

Hint 3(c):

Famous even functions include: $y = x^2$ , $y = x^4$ , $y = \text{cos}x$ , and all vertical translations and stretches of the graphs above.

Hint 4(c):

Famous odd functions include: $y = x$ , $y = x^3$ , $y = \text{sin}x$ , and all stretches of the graphs above.