Symmetry of Functions Even, Odd, or Neither?. Even Functions All exponents are even. May contain a constant. f(x) = f(-x) Symmetric about the y-axis.

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Presentation transcript:

Symmetry of Functions Even, Odd, or Neither?

Even Functions All exponents are even. May contain a constant. f(x) = f(-x) Symmetric about the y-axis

All even exponents Example: Both exponents are even. It does not matter what the coefficients are.

May Contain a Constant Example Even exponents (coefficients don’t matter) Constant does not affect even function.

f(x) = f(-x) Given f(x) = 5x² - 7, find f(-x) to determine if f(x) is even, odd, or neither. 1)Substitute –x for x. 2)f(-x) = 5(-x)² - 7 = 5x² -7 3)Because f(x) = f(-x), f(x) = 5x² - 7 is an even function.

f(x) = f(-x) Given f(x) = 4x² - 2x + 1, determine if f(x) is even, odd, or neither. 1)Substitute –x for x. 2)f(-x) = 4(-x)² - 2(-x) + 1 = 4x² +2x + 1 3)f(-x) ≠ f(x) 4)Therefore, f(x) is NOT an even function. (We will revisit to determine what it is.)

Symmetric About the y-axis The following are symmetric about the y- axis.

Odd Functions Only odd exponents. NO constants! f(-x) = -f(x) Symmetric about the origin.

All Odd Exponents Example All odd exponents. Understood 1 exponent

NO Constants Example: Odd exponents NO constants in odd functions!

f(-x) = -f(x) Given f(x) = 4x³ + 2x, find f(-x) and f(- x) to determine if f(x) is even, odd, or neither. f(-x) = 4(-x)³ + 2(-x) = -4x³ - 2x -f(x) = -4x³ - 2x Because f(-x) = -f(x), f(x) is an odd function.

f(-x) = -f(x) Given f(x) = 5x³ + 7x², find f(-x) and f(-x) to determine if f(x) is even, odd, or neither. f(-x) = 5(-x)³ + 7(-x)² = -5x³ + 7x² -f(x) = -5x³ - 7x² f(-x) ≠ -f(x), therefore f(x) is NOT an odd function.

Symmetric About the Origin These graphs are symmetric about the origin.

Neither? Mixture of even and odd exponents. All odd exponents with a constant. f(x) ≠ f(-x) AND f(-x) ≠ -f(x)

Examples of Neither f(x) = 4x³ - 5x² f(x) = 5x³ + 7 Mixture of odd and even exponents. Odd exponents with a constant.

Examples of Neither If f(x) = -3x³ + 2x², determine if f(x) is even, odd, or neither. 1)Find f(-x). 2)f(-x) = -3(-x)³ + 2(-x)² = 3x³ + 2x² 3)Find –f(x). 4)-f(x) = 3x³ - 2x² 5)Because f(-x) ≠ f(x) and f(-x) ≠ -f(x), f(x) is neither even nor odd.