Question

sketch the graph of $f(x)=\frac{2x - 1}{x^{2}-3x - 10}$. label all vertical and horizontal asymptotes. show all work by hand. you need to draw your own graph. remember, you are sketching, so it doesnt have to be perfect, but it does still have to be neat and legible. you must show all work in all steps to sketching your graph. you must upload an image of all handwritten work as an attachment in the response box. typed responses will not be accepted. this work can be done using the touchscreen on your computer or taking a picture of your work done on paper. please make sure your submission is legible. no work = no credit.

Explanation:

Step1: Find vertical asymptotes

Set the denominator equal to zero: $x^{2}-3x - 10=0$. Factor the quadratic: $(x - 5)(x+2)=0$. Solving gives $x = 5$ and $x=-2$ as vertical asymptotes.

Step2: Find horizontal asymptote

Since the degree of the denominator ($2$) is greater than the degree of the numerator ($1$), the horizontal asymptote is $y = 0$.

Step3: Find x - intercept

Set the numerator equal to zero: $2x-1=0$, so $x=\frac{1}{2}$ is the x - intercept.

Step4: Find y - intercept

Set $x = 0$ in the function: $f(0)=\frac{2(0)-1}{0^{2}-3(0)-10}=\frac{-1}{-10}=\frac{1}{10}$ is the y - intercept.

Step5: Analyze behavior around asymptotes

As $x\to5^{+}$, $f(x)\to+\infty$; as $x\to5^{-}$, $f(x)\to-\infty$; as $x\to - 2^{+}$, $f(x)\to-\infty$; as $x\to - 2^{-}$, $f(x)\to+\infty$.
Then sketch the graph using the asymptotes, intercepts and behavior around asymptotes.

Answer:

A hand - drawn graph with vertical asymptotes at $x = 5$ and $x=-2$, horizontal asymptote at $y = 0$, x - intercept at $x=\frac{1}{2}$, y - intercept at $y=\frac{1}{10}$ and the appropriate behavior around the asymptotes.