Question

plot all of the existing five features of the following rational function (some may not be needed). if you get a fraction or decimal then plot as close to the true location as possible. $f(x) = \frac{-2x - 4}{x^2 - 4}$ plot rational function vertical asymptote horizontal asymptote x-intercept y-intercept hole click on a feature then drag it into place.

Explanation:

Step1: Simplify the function

First, factor the numerator and the denominator:

• Numerator: \(-2x - 4 = -2(x + 2)\)
• Denominator: \(x^2 - 4 = (x - 2)(x + 2)\)

So, \(f(x)=\frac{-2(x + 2)}{(x - 2)(x + 2)}\). We can cancel out the common factor \((x + 2)\) (note that \(x
eq - 2\) to avoid division by zero), so the simplified function is \(f(x)=\frac{-2}{x - 2}\) with a hole at \(x=-2\) (since \((x + 2)\) was a common factor).

Step2: Find Vertical Asymptote

Vertical asymptotes occur where the denominator of the simplified function is zero (and the numerator is not zero there). For \(f(x)=\frac{-2}{x - 2}\), set \(x - 2=0\), so \(x = 2\) is the vertical asymptote.

Step3: Find Horizontal Asymptote

For rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). Here, the degree of the numerator (after canceling) is \(0\) and the degree of the denominator is \(1\), so horizontal asymptote is \(y = 0\).

Step4: Find x - Intercept

To find the x - intercept, set \(y = f(x)=0\). For \(f(x)=\frac{-2}{x - 2}\), \(\frac{-2}{x - 2}=0\) has no solution because the numerator is \(-2
eq0\). So, there is no x - intercept.

Step5: Find y - Intercept

To find the y - intercept, set \(x = 0\) in the original function (before canceling, to check for the hole and intercept).
\(f(0)=\frac{-2(0)-4}{0^2 - 4}=\frac{-4}{-4}=1\). But we also need to check the hole. The hole is at \(x=-2\), when \(x = 0\), we can use the simplified function \(f(0)=\frac{-2}{0 - 2}=\frac{-2}{-2}=1\). So the y - intercept is at \((0,1)\).

Step6: Find Hole

The hole occurs at the value of \(x\) where the common factor was canceled, i.e., \(x=-2\). To find the y - coordinate of the hole, substitute \(x = - 2\) into the simplified function (or the original function before canceling, but simplified is easier). Using \(f(x)=\frac{-2}{x - 2}\), when \(x=-2\), \(f(-2)=\frac{-2}{-2 - 2}=\frac{-2}{-4}=\frac{1}{2}\). So the hole is at \((-2,\frac{1}{2})\).

Answer:

• Vertical Asymptote: \(x = 2\)
• Horizontal Asymptote: \(y = 0\)
• y - Intercept: \((0,1)\)
• Hole: \((-2,\frac{1}{2})\)
• No x - Intercept