3.4 Graphs of polynomial functions  (Page 5/13)

This graph has two $x\text{-}$ intercepts. At $x=\mathrm{-3},$ the factor is squared, indicating a multiplicity of 2. The graph will bounce at this $x\text{-}$ intercept. At $x=5,$ the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.

The y -intercept is found by evaluating $f(0).$

The $y\text{-}$ intercept is $(0,90).$

Additionally, we can see the leading term, if this polynomial were multiplied out, would be $-2x^3,$ so the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity. See [link] .

To sketch this, we consider that:

• As $x\to -\infty$ the function $f(x)\to \infty ,$ so we know the graph starts in the second quadrant and is decreasing toward the $x\text{-}$ axis.
• Since $f\left(-x\right)=\mathrm{-2}{\left(-x+3\right)}^2\left(-x–5\right)$ is not equal to $f\left(x\right),$ the graph does not display symmetry.
• At $\left(-3,0\right),$ the graph bounces off of the $x\text{-}$ axis, so the function must start increasing.

  At $\left(0,90\right),$ the graph crosses the $y\text{-}$ axis at the $y\text{-}$ intercept. See [link] .

Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at $\left(5,0\right).$ See [link] .

As $x\to \infty$ the function $f(x)\to \mathrm{-\infty },$ so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.

Using technology, we can create the graph for the polynomial function, shown in [link] , and verify that the resulting graph looks like our sketch in [link] .