Question

Part C: The graph y = P(x) is shown.

What property of an even function do you see in this graph?

Part D
An equation for y = P(x) is shown.
State the algebraic definition of an even function. Show algebraically that P(x) is an even
function for all values of x.

Answer 1

An even function exhibits symmetry about the y-axis. To show that a function is even, we need to prove that y(x) = y(-x) for all values of x. In the given equation y = P(x), we substitute -x for x and show that the equation holds.

Step-by-step explanation:

An even function is a type of function that exhibits symmetry about the y-axis. This means that if we reflect the function across the y-axis, the graph remains unchanged. The algebraic definition of an even function is y(x) = y(-x). To show that a function is even, we need to prove that y(x) = y(-x) for all values of x.

In this case, the equation for y = P(x) is y(x) = −y(-x). To show that P(x) is an even function, we substitute -x for x in the equation and simplify:

y(-x) = −y(-(-x))

y(-x) = −y(x)

Since y(x) = −y(-x) holds for all values of x, we can conclude that P(x) is an even function.