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Evaluate polynomial functions

Evaluating a polynomial function means finding its output for a specified real input by substituting the input for the variable and applying exponents, multiplication, and addition in the correct order, including when inputs are negative or fractional. The value f(a)f(a) represents the corresponding yy-coordinate and can be calculated from standard, factored, or equivalent forms, supporting tables, graphs, and analysis of zeros and function behavior; complex inputs and more abstract forms are outside this scope.

Detailed Explanation: Evaluate polynomial functions

To evaluate a polynomial function, substitute the given input everywhere the variable appears. Then simplify using the order of operations:

  1. Evaluate exponents.
  2. Multiply coefficients.
  3. Add or subtract.

For negative inputs, use parentheses so the exponent applies to the entire number.

Example: Find f(−2)f(-2) if

f(x)=2x3−3x2+4x−5.f(x)=2x^3-3x^2+4x-5.

Substitute −2-2 for every xx:

f(−2)=2(−2)3−3(−2)2+4(−2)−5.f(-2)=2(-2)^3-3(-2)^2+4(-2)-5.

Evaluate the powers:

(−2)3=−8and(−2)2=4.(-2)^3=-8 \qquad\text{and}\qquad (-2)^2=4.

Now multiply:

f(−2)=2(−8)−3(4)−8−5.f(-2)=2(-8)-3(4)-8-5.

Simplify:

f(−2)=−16−12−8−5=−41.f(-2)=-16-12-8-5=-41.

Therefore,

f(−2)=−41.\boxed{f(-2)=-41}.

This means the graph of the function contains the point (−2,−41)(-2,-41), so the corresponding yy-coordinate is −41-41.

Learn by doing: Evaluate polynomial functions

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