Answer
To graph the function \( f(x) = \frac{x^{2}-6x+9}{x^{3}-7x^{2}} \), follow these steps:
1. **Vertical Asymptotes**: \( x = 0 \) and \( x = 7 \)
2. **Horizontal Asymptote**: \( y = 0 \)
3. **X-Intercept**: \( (3, 0) \)
4. **Points**:
- \( (-1, -2) \)
- \( (1, -\frac{2}{3}) \)
- \( (8, \frac{25}{64}) \)
Plot these points and draw vertical lines at \( x = 0 \) and \( x = 7 \), and a horizontal line at \( y = 0 \).
Solution
Function by following steps:
- step0: Find the horizontal asymptotes:
\(f\left(x\right)=\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\)
- step1: Evaluate the limits \(\lim _{x\rightarrow +\infty}\left(f\left(x\right)\right)\) and \(\lim _{x\rightarrow -\infty}\left(f\left(x\right)\right):\)
\(\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\right)\\&\lim _{x\rightarrow -\infty}\left(\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\right)\end{align}\)
- step2: Calculate:
\(\begin{align}&0\\&0\end{align}\)
- step3: The finite values are horizontal asymptotes:
\(\begin{align}&f\left(x\right)=0\end{align}\)
Analyze the y intercept of the function \( f(x)=\frac{x^{2}-6 x+9}{x^{3}-7 x^{2}} \)
Function by following steps:
- step0: Find the y-intercept:
\(f\left(x\right)=\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\)
- step1: There is no \(y\)-intercept\(:\)
\(\textrm{No y-intercept}\)
Analyze the x intercept of the function \( f(x)=\frac{x^{2}-6 x+9}{x^{3}-7 x^{2}} \)
Function by following steps:
- step0: Find the \(x\)-intercept/zero:
\(f\left(x\right)=\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\)
- step1: Set \(f\left(x\right)\)=0\(:\)
\(0=\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\)
- step2: Swap the sides:
\(\frac{x^{2}-6x+9}{x^{3}-7x^{2}}=0\)
- step3: Find the domain:
\(\frac{x^{2}-6x+9}{x^{3}-7x^{2}}=0,x \in \left(-\infty,0\right)\cup \left(0,7\right)\cup \left(7,+\infty\right)\)
- step4: Cross multiply:
\(x^{2}-6x+9=\left(x^{3}-7x^{2}\right)\times 0\)
- step5: Simplify the equation:
\(x^{2}-6x+9=0\)
- step6: Factor the expression:
\(\left(x-3\right)^{2}=0\)
- step7: Simplify the expression:
\(x-3=0\)
- step8: Move the constant to the right side:
\(x=0+3\)
- step9: Remove 0:
\(x=3\)
- step10: Check if the solution is in the defined range:
\(x=3,x \in \left(-\infty,0\right)\cup \left(0,7\right)\cup \left(7,+\infty\right)\)
- step11: Find the intersection:
\(x=3\)
Find the vertical asymptotes of \( f(x)=\frac{x^{2}-6 x+9}{x^{3}-7 x^{2}} \).
Function by following steps:
- step0: Find the vertical asymptotes:
\(f\left(x\right)=\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\)
- step1: Evaluate the limit:
\(\begin{align}&\lim _{x\rightarrow 0}\left(\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\right)\\&\lim _{x\rightarrow 7}\left(\frac{x^{2}-6x+9}{x^{3}-7x^{2}}\right)\end{align}\)
- step2: Calculate:
\(\begin{align}&-\infty\\&\textrm{The limit does not exist}\end{align}\)
- step3: \(x=0\) is a vertical asymptote\(:\)
\(\begin{align}&x=0\textrm{ }\textrm{is a vertical asymptote}\\&x=7\textrm{ }\textrm{is a vertical asymptote}\end{align}\)
- step4: List all vertical asymptotes of the function:
\(\begin{align}&x=0\\&x=7\end{align}\)
Let's analyze the function step by step and summarize the findings for the function
\[
f(x) = \frac{x^{2}-6x+9}{x^{3}-7x^{2}}.
\]
### Step 1: Find Vertical Asymptotes
To find the vertical asymptotes, we set the denominator equal to zero:
\[
x^{3} - 7x^{2} = 0.
\]
Factoring gives:
\[
x^{2}(x - 7) = 0.
\]
Thus, the vertical asymptotes are at:
\[
x = 0 \quad \text{and} \quad x = 7.
\]
### Step 2: Find Horizontal Asymptotes
To find the horizontal asymptote, we look at the degrees of the numerator and denominator. The degree of the numerator is 2, and the degree of the denominator is 3. Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is:
\[
f(x) = 0.
\]
### Step 3: Find Intercepts
- **X-Intercept**: Set \(f(x) = 0\):
\[
\frac{x^{2}-6x+9}{x^{3}-7x^{2}} = 0 \implies x^{2}-6x+9 = 0.
\]
Factoring gives:
\[
(x-3)^{2} = 0 \implies x = 3.
\]
So, the x-intercept is at \( (3, 0) \).
- **Y-Intercept**: Set \(x = 0\):
\[
f(0) = \frac{0^{2}-6(0)+9}{0^{3}-7(0)^{2}} \text{ is undefined.}
\]
Thus, there is no y-intercept.
### Step 4: Plot Points
We need to plot at least one point on each side of the vertical asymptotes. Let's choose:
- For \(x < 0\) (e.g., \(x = -1\)):
\[
f(-1) = \frac{(-1)^{2}-6(-1)+9}{(-1)^{3}-7(-1)^{2}} = \frac{1 + 6 + 9}{-1 - 7} = \frac{16}{-8} = -2.
\]
- For \(0 < x < 7\) (e.g., \(x = 1\)):
\[
f(1) = \frac{1^{2}-6(1)+9}{1^{3}-7(1)^{2}} = \frac{1 - 6 + 9}{1 - 7} = \frac{4}{-6} = -\frac{2}{3}.
\]
- For \(x > 7\) (e.g., \(x = 8\)):
\[
f(8) = \frac{8^{2}-6(8)+9}{8^{3}-7(8)^{2}} = \frac{64 - 48 + 9}{512 - 448} = \frac{25}{64}.
\]
### Summary of Findings
- **Vertical Asymptotes**: \(x = 0\) and \(x = 7\)
- **Horizontal Asymptote**: \(f(x) = 0\)
- **X-Intercept**: \( (3, 0) \)
- **Y-Intercept**: None
- **Points**:
- \( (-1, -2) \)
- \( (1, -\frac{2}{3}) \)
- \( (8, \frac{25}{64}) \)
### Graphing
While I cannot plot the graph directly, you can use the information above to sketch the graph of the function. The vertical asymptotes will be vertical lines at \(x = 0\) and \(x = 7\), the horizontal asymptote will be the line \(y = 0\), and you can plot the intercepts and points calculated above to visualize the function's behavior.
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