Question

What is the end behavior of the graph of f(x) = x5 – 8x4 + 16x3?

Answer: B.) f(x) => -∞ as x => -∞; f(x) => +∞ as x => +∞
The graph touches, but does not cross, the x–axis at x =__
The graph of the function crosses the x–axis at x = ___

Answer 1

Explanation:

f(x) = x⁵ – 8x⁴ + 16x³

As x approaches +∞, the highest term, x⁵, approaches +∞.

As x approaches -∞, x⁵ approaches -∞ (a negative number raised to an odd exponent is also negative).

Now let's factor:

f(x) = x³ (x² – 8x + 16)

f(x) = x³ (x – 4)²

f(x) has roots at x=0 and x=4. x=4 is a repeated root (because it's squared), so the graph touches the x-axis but does not cross at x=4.

The graph crosses the x-axis at x=0.