Mathematics, 31.01.2020 04:49 ronaldo22

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

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Answer from: falishaduncanovmtz2

f(x) = x⁵ – 8x⁴ + 16x³

Does not cross the x-axis at x = 4.

Crosses the x-axis at x = 0

This is about the end behavior of a graph of a function at the end of the x-axis.

We are given the function;

f(x) = x⁵ - 8x⁴ + 16x³

A) As x approaches negative infinity -∞, x⁵ will also approach negative infinity -∞. This is because when we raise a negative number to the power of an odd number, the result remains negative.B) As x approaches positive infinity +∞, x⁵ will also approach positive infinity -∞. This is because when we raise a positive number to the power of an odd number, the result remains positive. Let's now find the roots of this function;

f(x) = x⁵ - 8x⁴ + 16x³

Let's factorize it first to get;

f(x) = x³(x² – 8x + 16)

(x² – 8x + 16) is a perfect square trinomial and can be expressed as (x – 4)(x - 4).

Thus;

f(x) = x³ (x – 4)(x - 4)

C) Since we have found the factorized form to be;

f(x) = x³ (x – 4)(x - 4)

The roots are at f(x) = 0;

The roots are; x³ = 0; (x – 4) = 0 ; (x - 4) = 0

This means the roots of f(x) are; x=0 and x=4. x = 4

This means the graph has a repeated root and so it will touch the x-axis but not at the repeated root of x=4.