Question

determine each feature of the graph of the given function.
$f(x) = \frac{3x^2 + 21x + 36}{-x - 4}$
answer attempt 1 out of 3
horizontal asymptote: $y = \square$ no horizontal asymptote
vertical asymptote: $x = \square$ no vertical asymptote
$x$-intercept: $(\square, 0)$ no $x$-intercept
$y$-intercept: $(0, \square)$ no $y$-intercept
hole: $(\square, \square)$ no hole

Explanation:

Step1: Simplify the function

First, factor the numerator: \(3x^2 + 21x + 36 = 3(x^2 + 7x + 12)=3(x + 3)(x + 4)\). The denominator is \(-x - 4=-(x + 4)\). So the function becomes \(f(x)=\frac{3(x + 3)(x + 4)}{-(x + 4)}\), where \(x
eq - 4\). Canceling out the common factor \((x + 4)\) (with the restriction \(x
eq - 4\)), we get \(f(x)=-3(x + 3)=-3x - 9\), for \(x
eq - 4\).

Step2: Analyze Horizontal Asymptote

For a rational function \(y = \frac{N(x)}{D(x)}\), if the degree of \(N(x)\) (\(n\)) and degree of \(D(x)\) (\(d\)):

• If \(n>d\), no horizontal asymptote.
• If \(n = d\), horizontal asymptote is \(y=\frac{\text{leading coefficient of }N(x)}{\text{leading coefficient of }D(x)}\).
• If \(n

In the original function, degree of numerator \(N(x)=3x^2+21x + 36\) is \(n = 2\), degree of denominator \(D(x)=-x - 4\) is \(d = 1\). Since \(n>d\), there is no horizontal asymptote.

Step3: Analyze Vertical Asymptote

Vertical asymptotes occur where the denominator is zero and the numerator is not zero (after canceling common factors). After canceling \((x + 4)\), the denominator of the simplified function (excluding the hole) is non - zero for all \(x\) except the hole. But originally, the denominator is zero at \(x=-4\), but we canceled \((x + 4)\) (which means there is a hole at \(x=-4\) instead of a vertical asymptote). So there is no vertical asymptote.

Step4: Analyze x - Intercept

To find the x - intercept, set \(y = 0\) in the simplified function \(f(x)=-3x - 9\) (since the hole does not affect the x - intercept as it's a single point exclusion). Set \(-3x-9 = 0\), then \(-3x=9\), so \(x=-3\). So the x - intercept is \((-3,0)\).

Step5: Analyze y - Intercept

To find the y - intercept, set \(x = 0\) in the simplified function \(f(x)=-3x - 9\). Then \(f(0)=-3(0)-9=-9\). So the y - intercept is \((0,-9)\).

Step6: Analyze Hole

A hole occurs where a common factor is canceled from numerator and denominator. We canceled \((x + 4)\) from numerator and denominator, so the hole is at \(x=-4\). To find the y - coordinate of the hole, substitute \(x=-4\) into the simplified function \(f(x)=-3x - 9\). So \(f(-4)=-3(-4)-9 = 12 - 9=3\). So the hole is at \((-4,3)\).

Answer:

Horizontal Asymptote: No horizontal asymptote
Vertical Asymptote: No vertical asymptote
x - Intercept: \((-3,0)\)
y - Intercept: \((0,-9)\)
Hole: \((-4,3)\)