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.
$f(x) = \frac{x^2 - 16}{-x - 4}$
plot rational function
vertical asymptote horizontal asymptote
x-intercept y-intercept hole
click on a feature then drag it into place.
graph with x-axis from -10 to 10 and y-axis from -5 to 10, grid lines, and axes arrows

Explanation:

Step1: Simplify the function

First, factor the numerator and denominator. The numerator \(x^2 - 16\) is a difference of squares, so it factors to \((x - 4)(x + 4)\). The denominator \(-x - 4\) can be factored as \(-(x + 4)\). So we have:

$$ f(x)=\frac{(x - 4)(x + 4)}{-(x + 4)} $$

We can cancel out the common factor \((x + 4)\) (note that \(x
eq - 4\) to avoid division by zero), so the simplified function is \(f(x)=-(x - 4)=-x + 4\) with a hole at \(x=-4\) (since that's where the original function is undefined but the simplified function is defined).

Step2: Find the hole

To find the coordinates of the hole, we substitute \(x = - 4\) into the simplified function \(f(x)=-x + 4\). So \(f(-4)=-(-4)+4 = 4 + 4=8\). So the hole is at \((-4,8)\).

Step3: Find the x - intercept

Set \(y = f(x)=0\) in the simplified function \(f(x)=-x + 4\). So \(0=-x + 4\), which gives \(x = 4\). So the x - intercept is \((4,0)\).

Step4: Find the y - intercept

Set \(x = 0\) in the simplified function \(f(x)=-x + 4\). So \(f(0)=-0 + 4=4\). So the y - intercept is \((0,4)\).

Step5: Analyze asymptotes

Since we canceled out the common factor \((x + 4)\), there is no vertical asymptote (because the factor that would cause a vertical asymptote was canceled, leaving a linear function). For horizontal asymptote, since the degree of the numerator (after canceling) and the denominator (after canceling, the denominator is effectively 1, degree 0) : the degree of the numerator (after simplification, it's a linear function, degree 1) and the degree of the denominator (degree 0). When the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Answer:

• Hole: \((-4,8)\)
• x - Intercept: \((4,0)\)
• y - Intercept: \((0,4)\)
• Vertical Asymptote: None
• Horizontal Asymptote: None