Sketching the Graph of a Polynomial Function

This graph has two x-intercepts. At $x=\mathrm{-3},$ the factor is squared, indicating a multiplicity of 2. The graph will bounce at this x-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-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 Figure 13.

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-axis, so the function must start increasing.

  At $\left(0,90\right),$ the graph crosses the y-axis at the y-intercept. See Figure 14.

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 Figure 15.

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 Figure 16, and verify that the resulting graph looks like our sketch in Figure 15.