Question

Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1

$f\left(x\right)=20x^4+73x^3+46x^2-52x-24\hspace{0.25em}$

Short answer

For the given polynomial function, we get:

x-intercept$=-2,-\frac{2}{5},\frac{3}{4}$

y-intercept$=-24$

Step-by-step solution

Step 1. End behavior of the function

We have a degree of five from$f\left(x\right)=20x^4+73x^3+46x^2-52x-24\hspace{0.25em}$

Thus, the graph of function has an end behavior similar to$y=20x^4$

Step 2. Determine the intercept of the function

Determine the intercept of function to get the x- intercept.

$\begin{array}{r}20x^4+73x^3+46x^2-52x-24\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ (x+2{)}^2(4x-3\left)\right(5x+2)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ x+2=0;4x-3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5x+2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=0\\ x=-2;4x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=3;\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=-2\\ x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{3}{4};\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=-\frac{2}{5}\end{array}$

Considering$f\left(0\right)=-24$, thus y-intercept is$(0,-24)$

Step 3. Graph the function

Step 4. Maximum and minimum turning point

From the graph above, we can see that the maximum and minimum turning point of the function is:

Maximum: $(-1.048,21.119)$

Minimum:$(-2,0),(0.31,-33.34)$

Step 5. Graph of function with obtained result

Step 6. Domain and range

Therefore, the domain of the function is the set of all real numbers and range is$(-33.34,\infty )$

Step 7. Interval at which the graph is increasing or decreasing

By observing the graph above, we say that the function at the intervals of

$(-2,-1.408),(0.31,\infty )$is increasing.

At intervals of$(-\infty ,-2),(-1.048,0.31)$