Lesson 8: Analyzing Graphs of Polynomial Functions

Property

Zero of a Function
For any function $f$, if $f(x) = 0$, then $x$ is a zero of the function.

When we find the values of $x$ for which $f(x) = 0$, we are finding the zeros of the function. When $f(x) = 0$, the point $(x, 0)$ is a point on the graph, which is an $x$-intercept.

Examples

• For the function $f(x) = x^2 - x - 8$, find $x$ when $f(x) = 4$. Set $x^2 - x - 8 = 4$, which simplifies to $x^2 - x - 12 = 0$. Factoring gives $(x - 4)(x + 3) = 0$, so $x = 4$ and $x = -3$.

Property

If $f(x)$ is a polynomial function and $f(a)$ and $f(b)$ have opposite signs, then there is at least one real zero between $a$ and $b$. This occurs because the function must cross the x-axis to change from positive to negative values (or vice versa).

Examples

Property

For polynomial functions, local maximum and minimum values occur at turning points where the function changes direction. At a turning point, the function reaches a peak (local maximum) or valley (local minimum) within a specific interval. These turning points are found where the derivative equals zero or is undefined.

Examples