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3.3 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS
A power function is a function that can be represented in the form f (x) = kx p where k and p are real numbers, and k is known as the coefficient. Ex. Is f (x) = 2 x a power function? No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function. EX. Which of the following functions are power functions? f (x) = 1 f (x) = x f (x) = x^2 f (x) = x^3 f (x) = 1 𝑥 f (x) = 1 𝑥^2 f (x) = (^) √𝑥 f (x) = (^) √𝑥 3
Identifying End Behavior of Power Functions As the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.
End behavior of power functions in the form f (x) = kx n where n is a non-negative integer depending on the power and the constant.
EX. Graph f(x) = x 8 and g(x) = - x 9 using a graphing calculator. EX. Which of the following are polynomial functions? a) f (x) = 2x^3 ⋅ 3x + 4 b) g(x) = −x(x 2 − 4) c) h(x) = 5 (^) √𝑥 + 2
EX. Given the function f (x) = −3x 2 (x − 1)(x + 4), express the function as a polynomial in general form, and determine the leading term, degree.
Try It # Without graphing the function, determine the maximum number of x- intercepts and turning points for f (x) = 108 − 13x^9 − 8x^4 + 14x^12 + 2x^3
