Question

Use the graph of function f(x)=x^3+x^2-5x+1 to identify the end-behavior, the number of turning points and the number of zeros

Answer 1

Solution:

Given:

$f(x)=x^3+x^2-5x+1$

The graph of the function is given below;

The end behavior:

From the graph, the end behavior of the graph shows that as x tends to negative infinity, the function f(x) tends to negative infinity. Also, as x tends to positive infinity, the function f(x) tends to positive infinity.

$\begin{gathered} x\rightarrow-\infty,f(x)\rightarrow-\infty \\ x\rightarrow\infty,f(x)\rightarrow\infty \end{gathered}$

The turning points:

The maximum number of turning points of a polynomial function is always one less than the degree of the function. A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).

$\begin{gathered} The\text{ degree of the function is 3} \\ The\text{ turning points will be }3-1=2 \end{gathered}$

The graph has a maximum and minimum point. Increasing and decreasing, and decreasing and increasing.

Hence, the graph of the function has two turning points.

The number of zeros:

The zeros of a function also referred to as roots or x-intercepts, occur at x-values where the value of the function f(x) = 0.

Hence, the zeros exist at;

$x=-2.866,x=0.211,x=1.655$

Therefore, the function has three zeros.