Graph the function. $$ p(x)=\frac{x^{2}-4 x+4}{x} $$

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We start with the function $$ p(x)=\frac{x^2-4x+4}{x}. $$ ### Step 1. Factor the Numerator Notice that the quadratic in the numerator is a perfect square: $$ x^2-4x+4=(x-2)^2. $$ Thus, the function can be rewritten as $$ p(x)=\frac{(x-2)^2}{x}. $$ ### Step 2. Determine the Domain Since the denominator is $$ x $$, the function is undefined at $$ x=0 $$. Therefore, the domain is: $$ x\in\mathbb{R}\setminus \{0\}. $$ ### Step 3. Rewrite in a Form Suitable for Analyzing Asymptotes Perform polynomial long division or simply rewrite by splitting the fraction: $$ \frac{x^2-4x+4}{x} = \frac{x^2}{x} - \frac{4x}{x} + \frac{4}{x} = x-4+\frac{4}{x}. $$ This form, $$ p(x)=x-4+\frac{4}{x}, $$ will help us find the asymptotic behavior. ### Step 4. Identify the Asymptotes #### Vertical Asymptote Since the function is undefined at $$ x=0 $$ and the term $$ \frac{4}{x} $$ becomes unbounded as $$ x \to 0 $$, there is a vertical asymptote at: $$ x=0. $$ #### Oblique (Slant) Asymptote For large $$ |x| $$, the term $$ \frac{4}{x} $$ approaches 0. Therefore, as $$ x \to \pm\infty $$: $$ p(x) \approx x-4. $$ Thus, the oblique asymptote is: $$ y=x-4. $$ ### Step 5. Find the Intercepts #### $$ x $$-Intercept(s) Set $$ p(x) = 0 $$: $$ x-4+\frac{4}{x}=0. $$ Multiply both sides by $$ x $$ (noting that $$ x\neq 0 $$): $$ x^2-4x+4=0. $$ This factors as: $$ (x-2)^2=0, $$ so the only solution is: $$ x=2. $$ Thus, the $$ x $$-intercept is at $$ (2,0) $$. #### $$ y $$-Intercept The $$ y $$-intercept would be $$ p(0) $$, but since $$ x=0 $$ is not in the domain, there is **no $$ y $$-intercept**. ### Step 6. Sketching the Graph - **For $$ x>0 $$:** - As $$ x \to 0^+ $$, the term $$ \frac{4}{x} $$ tends to $$ +\infty $$, so $$ p(x) \to +\infty $$. - As $$ x \to +\infty $$, $$ p(x)\approx x-4 $$ tends to $$ +\infty $$ in a linear manner. - The graph passes through the point $$ (2, 0) $$. - **For $$ x<0 $$:** - As $$ x \to 0^- $$, $$ \frac{4}{x} $$ tends to $$ -\infty $$, so $$ p(x) \to -\infty $$. - As $$ x \to -\infty $$, $$ p(x)\approx x-4 $$ tends to $$ -\infty $$ linearly. - Evaluate a sample point: for $$ x=-1 $$: $$ p(-1)= -1-4+\frac{4}{-1}=-5-4=-9. $$ - **Asymptote:** The line $$ y=x-4 $$ is the oblique asymptote that the graph approaches as $$ |x| $$ becomes large. ### Step 7. Graph Summary - **Domain:** $$ x\in\mathbb{R}\setminus \{0\} $$ - **Vertical Asymptote:** $$ x=0 $$ - **Oblique Asymptote:** $$ y=x-4 $$ - **$$ x $$-Intercept:** $$ (2,0) $$ - **No $$ y $$-Intercept** since $$ x=0 $$ is excluded. - **Behavior:** - For $$ x>0 $$, the graph approaches $$ +\infty $$ as $$ x\to0^+ $$ and follows the line $$ y=x-4 $$ for large $$ x $$. - For $$ x<0 $$, the graph approaches $$ -\infty $$ as $$ x\to0^- $$ and, for large negative $$ x $$, it tends to follow the line $$ y=x-4 $$. Using these observations, you can sketch the graph with care around the vertical asymptote $$ x=0 $$, plot the intercept at $$ (2,0) $$, and draw the oblique asymptote $$ y=x-4 $$ guiding the end behavior. To graph the function $$ p(x) = \frac{x^2 - 4x + 4}{x} $$: 1. **Simplify the Function:** $$ p(x) = x - 4 + \frac{4}{x} $$ 2. **Determine the Domain:** $$ x \neq 0 $$ 3. **Identify Asymptotes:** - **Vertical Asymptote:** $$ x = 0 $$ - **Oblique Asymptote:** $$ y = x - 4 $$ 4. **Find Intercepts:** - **$$ x $$-Intercept:** $$ (2, 0) $$ - **No $$ y $$-Intercept** 5. **Graph Behavior:** - For $$ x > 0 $$: Approaches $$ +\infty $$ as $$ x \to 0^+ $$ and $$ y = x - 4 $$ for large $$ x $$. - For $$ x < 0 $$: Approaches $$ -\infty $$ as $$ x \to 0^- $$ and $$ y = x - 4 $$ for large negative $$ x $$. Plot these features to sketch the graph accurately.

Graph the function. \[ p(x)=\frac{x^{2}-4 x+4}{x} \]

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