Question

determine each feature of the graph of the
function.
$f(x) = \frac{4x + 8x}{3x^2 - 12}$

Explanation:

Step1: Analyze the function

The function is \( f(x)=\frac{4x^{2}+8x}{3x^{2}-12} \). First, factor numerator and denominator.
Numerator: \( 4x^{2}+8x = 4x(x + 2) \)
Denominator: \( 3x^{2}-12=3(x^{2}-4)=3(x - 2)(x + 2) \)
So, \( f(x)=\frac{4x(x + 2)}{3(x - 2)(x + 2)} \), \( x
eq - 2,2 \) (excluded values from denominator). Cancel \( (x + 2) \) (for \( x
eq - 2 \)): \( f(x)=\frac{4x}{3(x - 2)}=\frac{4x}{3x-6} \), \( x
eq - 2,2 \)

Step2: Find vertical asymptote

Vertical asymptotes occur where denominator is zero (and numerator non - zero) after canceling common factors. Denominator of simplified function \( 3x - 6 = 0\Rightarrow x = 2 \). At \( x=-2 \), the function has a hole (since \( (x + 2) \) canceled), not a vertical asymptote.

Step3: Find horizontal asymptote

For rational functions \( f(x)=\frac{ax^{n}+...}{bx^{m}+...} \), if \( n = m \), horizontal asymptote is \( y=\frac{a}{b} \). Here, \( n = 1 \), \( m = 1 \), \( a = 4 \), \( b = 3 \), so horizontal asymptote \( y=\frac{4}{3} \)

Step4: Find x - intercept

Set \( f(x)=0 \), \( \frac{4x}{3(x - 2)}=0\Rightarrow 4x = 0\Rightarrow x = 0 \) (and \( x
eq2,-2 \), so valid). So x - intercept at \( (0,0) \)

Step5: Find y - intercept

Set \( x = 0 \), \( f(0)=\frac{4(0)}{3(0 - 2)} = 0 \), so y - intercept at \( (0,0) \) (same as x - intercept)

Step6: Analyze the hole

At \( x=-2 \), find the y - value of the hole. Substitute \( x = - 2 \) into the simplified function (before canceling \( (x + 2) \), we can use the factored form before canceling: \( f(x)=\frac{4x}{3(x - 2)} \) (since \( x
eq - 2 \) when we cancel, but to find the hole, we substitute \( x=-2 \) into the simplified numerator and denominator after canceling the common factor). So \( y=\frac{4(-2)}{3(-2 - 2)}=\frac{-8}{3(-4)}=\frac{-8}{-12}=\frac{2}{3} \). So hole at \( (-2,\frac{2}{3}) \)

Answer:

• Hole: At \( x=-2 \), \( y = \frac{2}{3} \) (hole at \( (-2,\frac{2}{3}) \))
• Vertical Asymptote: \( x = 2 \)
• Horizontal Asymptote: \( y=\frac{4}{3} \)
• x - intercept and y - intercept: \( (0,0) \)