sketch the graph of function, f(x)= (x-5)^2(x^2+25)

Answer 1

You'll notice that the portion "(x - 5)²" was a zero (x-intercept) at x = 5 and that the intercept has a multiplicity of 2 because (x - 5)² = (x - 5)(x - 5) which means the x = 5 is a zero twice. The way this manifests itself in a graph is that the graph just "kisses" the x-axis at a single point (at x = 5).

You'll notice that the portion "(x² + 25)" does not have a zero because when we set it equal to zero, we get x = ±√(-25) and there are no real numbers where that is true. So the graph will not cross the x-axis because of this portion.

So the only place the graph touches the x-axis is at x = 5.

You'll also notice that multiplying the function out will result in a f(x) = x⁴ relationship (x⁴ plus some other stuff). That means the end behavior will match y = x⁴ meaning it goes up without bound as x gets more and more negative and as x gets more and more positive.

You'll also notice that the constant of the function will be 5² times 25 or 625. That means the y-intercept is (0, 625).

So the graph looks like this: