Question

math 30 pre - calculus
section 9.2 bw
1: find the x and y intercepts, the horizontal and vertical asymptotes (if any), and any points of discontinuity (if they exist). use sign analysis to determine where y is positive or negative. then graph the function.
$f(x)=\frac{1}{x^{2}+2x - 8}$

Explanation:

Step1: Find x-intercept

To find the x-intercept, set \( f(x) = 0 \). So, \( \frac{1}{x^2 + 2x - 8}=0 \). But a fraction is zero only when the numerator is zero (and denominator non - zero). The numerator is 1, which is never zero. So, there is no x - intercept.

Step2: Find y-intercept

Set \( x = 0 \) in \( f(x)=\frac{1}{x^2 + 2x - 8} \). Then \( f(0)=\frac{1}{0 + 0-8}=-\frac{1}{8} \). So, the y - intercept is \( (0,-\frac{1}{8}) \).

Step3: Find points of discontinuity (vertical asymptotes)

Factor the denominator: \( x^2+2x - 8=(x + 4)(x - 2) \). Set the denominator equal to zero: \( (x + 4)(x - 2)=0 \). Solving, we get \( x=-4 \) or \( x = 2 \). These are the vertical asymptotes (points of discontinuity).

Step4: Find horizontal asymptote

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), where the degree of \( N(x) \) is \( n \) and the degree of \( D(x) \) is \( m \). Here, degree of numerator (1) is 0, degree of denominator (2) is 2. Since \( n\lt m \), the horizontal asymptote is \( y = 0 \) (the x - axis).

Step5: Sign analysis

The critical points are \( x=-4 \) and \( x = 2 \). These divide the number line into three intervals:

• Interval 1: \( (-\infty,-4) \). Test \( x=-5 \): \( f(-5)=\frac{1}{(-5)^2+2(-5)-8}=\frac{1}{25 - 10 - 8}=\frac{1}{7}\gt0 \).
• Interval 2: \( (-4,2) \). Test \( x = 0 \): \( f(0)=-\frac{1}{8}\lt0 \).
• Interval 3: \( (2,\infty) \). Test \( x = 3 \): \( f(3)=\frac{1}{9 + 6-8}=\frac{1}{7}\gt0 \).

Answer:

• x - intercept: None.
• y - intercept: \( (0,-\frac{1}{8}) \).
• Vertical asymptotes (points of discontinuity): \( x=-4 \), \( x = 2 \).
• Horizontal asymptote: \( y = 0 \).
• Sign analysis: \( f(x)\gt0 \) for \( x\in(-\infty,-4)\cup(2,\infty) \), \( f(x)\lt0 \) for \( x\in(-4,2) \).