Answer :
To analyze and graph the rational function [tex]\( f(x) = \frac{x^2 - 64}{x} \)[/tex], we need to follow several steps.
### a. Find the Slant Asymptote
The slant (or oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. To find the slant asymptote, we perform polynomial long division or synthetic division to divide [tex]\( (x^2 - 64) \)[/tex] by [tex]\( x \)[/tex].
#### Polynomial Long Division:
[tex]\[ \begin{array}{r|ll} x & x^2 - 64 \\ \hline x & x^2 \\ x & x(x) \\ \hline & - \\ & - 64 \\ \end{array} \][/tex]
[tex]\[ \frac{x^2 - 64}{x} = x - \frac{64}{x} \][/tex]
As [tex]\( x \)[/tex] gets larger, [tex]\( \frac{64}{x} \)[/tex] approaches 0, so the slant asymptote is:
[tex]\[ y = x \][/tex]
### b. Seven-Step Strategy for Graphing
#### 1. Determine Symmetry:
[tex]\( f(-x) = \frac{(-x)^2 - 64}{-x} = \frac{x^2 - 64}{-x} = -f(x) \)[/tex]
The function is odd, so it is symmetric about the origin.
#### 2. Find the Intercepts:
- [tex]\( y \)[/tex]-intercept:
To find the [tex]\( y \)[/tex]-intercept, evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{0^2 - 64}{0} \implies \text{undefined} \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept does not exist.
Correct choice:
[tex]\[ \text{B. There is no \( y \)-intercept.} \][/tex]
- [tex]\( x \)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \frac{x^2 - 64}{x} = 0 \implies x^2 - 64 = 0 \implies x^2 = 64 \implies x = \pm 8 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 8 \)[/tex] and [tex]\( x = -8 \)[/tex].
Correct choice:
[tex]\[ \text{A. The \( x \)-intercepts are } \pm 8. \][/tex]
#### 3. Vertical Asymptotes:
Vertical asymptotes occur where the denominator is zero and the numerator is not zero:
[tex]\[ x = 0 \implies \text{Vertical asymptote is } x = 0. \][/tex]
#### 4. Horizontal or Slant Asymptotes:
As previously found, the slant asymptote is [tex]\( y = x \)[/tex].
#### 5. Behavior Near Asymptotes:
- As [tex]\( x \)[/tex] approaches 0 from the left and right, [tex]\( \frac{64}{x} \)[/tex] approaches [tex]\( \pm \infty \)[/tex], respectively.
#### 6. Plot Points and Sketch the Graph:
- The function crosses the x-axis at [tex]\( x = 8 \)[/tex] and [tex]\( x = -8 \)[/tex].
- The vertical asymptote is at [tex]\( x = 0 \)[/tex].
- The slant asymptote is [tex]\( y = x \)[/tex].
#### 7. Fine-Tune the Graph:
- Make sure the graph approaches the vertical asymptote [tex]\( x = 0 \)[/tex] correctly.
- Ensure the graph follows the slant asymptote [tex]\( y = x \)[/tex] as [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex].
By following these steps, you can accurately sketch the graph of the function [tex]\( f(x) = \frac{x^2 - 64}{x} \)[/tex].
### a. Find the Slant Asymptote
The slant (or oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. To find the slant asymptote, we perform polynomial long division or synthetic division to divide [tex]\( (x^2 - 64) \)[/tex] by [tex]\( x \)[/tex].
#### Polynomial Long Division:
[tex]\[ \begin{array}{r|ll} x & x^2 - 64 \\ \hline x & x^2 \\ x & x(x) \\ \hline & - \\ & - 64 \\ \end{array} \][/tex]
[tex]\[ \frac{x^2 - 64}{x} = x - \frac{64}{x} \][/tex]
As [tex]\( x \)[/tex] gets larger, [tex]\( \frac{64}{x} \)[/tex] approaches 0, so the slant asymptote is:
[tex]\[ y = x \][/tex]
### b. Seven-Step Strategy for Graphing
#### 1. Determine Symmetry:
[tex]\( f(-x) = \frac{(-x)^2 - 64}{-x} = \frac{x^2 - 64}{-x} = -f(x) \)[/tex]
The function is odd, so it is symmetric about the origin.
#### 2. Find the Intercepts:
- [tex]\( y \)[/tex]-intercept:
To find the [tex]\( y \)[/tex]-intercept, evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{0^2 - 64}{0} \implies \text{undefined} \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept does not exist.
Correct choice:
[tex]\[ \text{B. There is no \( y \)-intercept.} \][/tex]
- [tex]\( x \)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ \frac{x^2 - 64}{x} = 0 \implies x^2 - 64 = 0 \implies x^2 = 64 \implies x = \pm 8 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 8 \)[/tex] and [tex]\( x = -8 \)[/tex].
Correct choice:
[tex]\[ \text{A. The \( x \)-intercepts are } \pm 8. \][/tex]
#### 3. Vertical Asymptotes:
Vertical asymptotes occur where the denominator is zero and the numerator is not zero:
[tex]\[ x = 0 \implies \text{Vertical asymptote is } x = 0. \][/tex]
#### 4. Horizontal or Slant Asymptotes:
As previously found, the slant asymptote is [tex]\( y = x \)[/tex].
#### 5. Behavior Near Asymptotes:
- As [tex]\( x \)[/tex] approaches 0 from the left and right, [tex]\( \frac{64}{x} \)[/tex] approaches [tex]\( \pm \infty \)[/tex], respectively.
#### 6. Plot Points and Sketch the Graph:
- The function crosses the x-axis at [tex]\( x = 8 \)[/tex] and [tex]\( x = -8 \)[/tex].
- The vertical asymptote is at [tex]\( x = 0 \)[/tex].
- The slant asymptote is [tex]\( y = x \)[/tex].
#### 7. Fine-Tune the Graph:
- Make sure the graph approaches the vertical asymptote [tex]\( x = 0 \)[/tex] correctly.
- Ensure the graph follows the slant asymptote [tex]\( y = x \)[/tex] as [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex].
By following these steps, you can accurately sketch the graph of the function [tex]\( f(x) = \frac{x^2 - 64}{x} \)[/tex].
