Question

follow the seven step strategy to graph the following rational function.
$f(x)=\frac{3x^{2}+x - 4}{2x^{2}-7x}$
to graph the function, first determine the symmetry of the graph of $f$. choose the correct answer below.
\bigcirc origin symmetry
\bigcirc y - axis symmetry
\bigcirc neither y - axis symmetry nor origin symmetry
what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\bigcirc a. the y - intercept is \square.
\bigcirc b. there is no y - intercept.
what is/are the x - intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\bigcirc a. the x - intercept(s) is/are \square.
( type an integer or a simplified fraction. use a comma to separate answers as needed. )
\bigcirc b. there is no x - intercept.

Explanation:

Step1: Determine y - intercept

For the y - intercept, set \(x = 0\). The function \(f(x)=\frac{3x^{2}+x - 4}{2x^{2}-7x}\). When \(x = 0\), the denominator \(2x^{2}-7x=0\). So, there is no y - intercept.

Step2: Determine x - intercept

For the x - intercept, set \(y = f(x)=0\), i.e., \(\frac{3x^{2}+x - 4}{2x^{2}-7x}=0\). This implies \(3x^{2}+x - 4 = 0\) (since the denominator cannot be zero for the function to be defined at the x - intercept).
Factor \(3x^{2}+x - 4\): \(3x^{2}+x - 4=3x^{2}+4x - 3x - 4=x(3x + 4)-(3x + 4)=(3x + 4)(x - 1)\)
Set \((3x + 4)(x - 1)=0\). Solving \(3x+4 = 0\) gives \(x=-\frac{4}{3}\), and solving \(x - 1=0\) gives \(x = 1\). We need to check if these values make the denominator zero.
For \(x=-\frac{4}{3}\), \(2x^{2}-7x=2\times(-\frac{4}{3})^{2}-7\times(-\frac{4}{3})=\frac{32}{9}+\frac{28}{3}=\frac{32 + 84}{9}=\frac{116}{9}
eq0\)
For \(x = 1\), \(2x^{2}-7x=2\times1^{2}-7\times1=2 - 7=-5
eq0\)

Answer:

• Y - intercept: B. There is no y - intercept.
• X - intercept: A. The x - intercept(s) is/are \(1,-\frac{4}{3}\)