Chapter 4.1, Problem 76AYU

To determine

To analyze: the given polynomial function

Answer to Problem 76AYU

  

  

Explanation of Solution

Given:

  $f\left(x\right)=\left(x-1\right)\left(x+4\right)\left(x-3\right)$

Calculation:

Step1: Determine the end behavior of the graph of the function

Expand the polynomial to write it in the form

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$

  $\begin{array}{l}f\left(x\right)=\left(x-1\right)\left(x+4\right)\left(x-3\right)\hfill \\ =\left(x-1\right)\left(x^2+4x-3x-/2\right)Multiply\hfill \\ =\left(x-1\right)\left(x^2+x-12\right)\hfill \\ =x^3+x^2-12x-x^2-x+12Combineliketerms\hfill \\ =x^2-13x+12\hfill \end{array}$

The polynomial function $f$ is of degree $3$ . The graph of $f$ behaves like $y=x^2$ for large values of $\left|x\right|$ .

Step2: find the $x-and\text{ y}-intercepts$ of the graph of the function

The y-intercept is $f\left(0\right)=12$ . To find the x-interceptsolve $f\left(x\right)=0$ .

  $\begin{array}{l}f\left(x\right)=0\hfill \\ \left(x-1\right)\left(x+4\right)\left(x-3\right)=0\hfill \\ So\hfill \\ x-1=0orx+4=0\text{ or }x-3=0\hfill \\ x=1orx=-4orx=3\hfill \end{array}$

The x-intercepts are $\text{-4,1 and 3}$

Step3: Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches $x-axis$ ateach $x-intercept$ .

The zeros of $f$ are $-4,1and3$ .. The zero $-4$ is a zero of multiplicity $1$ , so the graph of $f$ crosses the $x-axis$ at $x=-4$ . The zero $1$ is azero of multiplicity $1$ , so the graph of $f$ crosses the $x-axis$ at $x=1$ . The zero $3$ is a zero of multiplicity $1$ , so the graph of $f$ crosses the $x-axis$ at $x=3$ .

Step4: Determine the maximum number of turning points on the graph of the function. Because the polynomial function is of degree $3$ (step1), the graph of the function will have at most $3-1=2$ points.

Step5: Determine the behavior of the graph of $f$ near each $x-intercept$

The three $x-intercepts$ are $-4,1and3$ .

  $\begin{array}{l}Near\text{ 1: }f\left(x\right)=\left(x-1\right)\left(x+4\right)\left(x-3\right)\hfill \\ \approx \left(x-1\right)\left(1+4\right)\left(1-3\right)\hfill \\ =\left(x-1\right)\left(5\right)\left(-2\right)\hfill \\ Alinewithslope-10\hfill \\ =-10\left(x+2\right)\hfill \\ =-10x-10\hfill \\ Near3:f\left(x\right)=\left(x-1\right)\left(x+4\right)\left(x-3\right)\hfill \\ \approx \left(3-1\right)\left(3+4\right)\left(x-3\right)\hfill \\ =\left(2\right)\left(7\right)\left(x-3\right)\hfill \\ Alinewithslope14\hfill \\ =14\left(x-3\right)\hfill \\ =14x-42\hfill \\ Near-4:f\left(x\right)=\left(x-1\right)\left(x+4\right)\left(x-3\right)\hfill \\ \approx \left(-4-1\right)\left(x+4\right)\left(-4-3\right)\hfill \\ =\left(-5\right)\left(x+4\right)\left(-7\right)\hfill \\ Alinewithslope35\hfill \\ =35\left(x+4\right)\hfill \\ =35x+140\hfill \end{array}$

Step 6: Put all the information from Steps 1 through 5 together to obtain the graph of $f$ Figure $1\left(a\right)$ illustrates the information obtained from Steps 1 through 5.

Evaluate $f$ at $-2,0,2$ to help establish the scale on the $y-axis$ . The graph of $f$ is given in Figure $1\left(b\right)$ .

  

  

Conclusion:

Thus, the graph of $f$ is obtained which illustrates the information obtained from Steps 1 through 5.