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 symmetry

For a function \(y = f(x)\), \(y -\)axis symmetry: \(f(-x)=f(x)\); origin symmetry: \(f(-x)=-f(x)\).
Let \(f(x)=\frac{3x^{2}+x - 6}{2x^{2}-7x}\). Then \(f(-x)=\frac{3(-x)^{2}+(-x)-6}{2(-x)^{2}-7(-x)}=\frac{3x^{2}-x - 6}{2x^{2}+7x}\).
Since \(f(-x)
eq f(x)\) (because \(\frac{3x^{2}-x - 6}{2x^{2}+7x}
eq\frac{3x^{2}+x - 6}{2x^{2}-7x}\)) and \(f(-x)
eq - f(x)\) (because \(-f(x)=-\frac{3x^{2}+x - 6}{2x^{2}-7x}=\frac{-3x^{2}-x + 6}{2x^{2}-7x}
eq\frac{3x^{2}-x - 6}{2x^{2}+7x}\)), so there is neither \(y -\)axis symmetry nor origin symmetry.

Step2: Find \(y -\)intercept

The \(y -\)intercept is found by setting \(x = 0\). But when \(x = 0\), the function \(f(x)=\frac{3x^{2}+x - 6}{2x^{2}-7x}\) has a denominator \(2x^{2}-7x=0\) (since \(2\times0^{2}-7\times0 = 0\)). So there is no \(y -\)intercept.

Step3: Find \(x -\)intercept

The \(x -\)intercept is found by setting \(y = 0\), i.e., \(\frac{3x^{2}+x - 6}{2x^{2}-7x}=0\). This implies \(3x^{2}+x - 6=0\) (since the fraction is \(0\) when the numerator is \(0\) and the denominator is not \(0\)).
Using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(ax^{2}+bx + c = 0\) (here \(a = 3\), \(b=1\), \(c=-6\)), we have \(x=\frac{-1\pm\sqrt{1^{2}-4\times3\times(-6)}}{2\times3}=\frac{-1\pm\sqrt{1 + 72}}{6}=\frac{-1\pm\sqrt{73}}{6}\).

Answer:

• Symmetry: Neither \(y -\)axis symmetry nor origin symmetry.
• \(y -\)intercept: There is no \(y -\)intercept.
• \(x -\)intercept: \(x=\frac{-1+\sqrt{73}}{6},\frac{-1 - \sqrt{73}}{6}\)