Skill: Evaluate polynomial functions

Explanation and Free Practice Resources

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 $f(a)$ represents the corresponding $y$ -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.

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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:

Evaluate exponents.
Multiply coefficients.
Add or subtract.

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

Example: Find $f(-2)$ if

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

Substitute $-2$ for every $x$ :

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

Evaluate the powers:

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

Now multiply:

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

Simplify:

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

Therefore,

\boxed{f(-2)=-41}.

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

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Number Sequences - Polynomial, Specific Term

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