Graphing Rational Functions Worksheets - Math Monks - Free Printable

To solve the problem of graphing the given rational functions, we need to analyze each function step by step. Rational functions are of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. Key features to identify include:

1. Vertical Asymptotes: Occur where the denominator is zero (unless there is a common factor in the numerator).
2. Horizontal Asymptotes: Depend on the degrees of the numerator and denominator.
3. Intercepts: x-intercepts occur where \( f(x) = 0 \), and y-intercepts occur where \( x = 0 \).
4. Behavior Near Asymptotes and Intercepts.

Let's go through each function:

---

Function 1: \( f(x) = \frac{3}{x-2} \)

#### Step 1: Vertical Asymptote
The denominator is zero when \( x - 2 = 0 \), so \( x = 2 \). This is a vertical asymptote.

#### Step 2: Horizontal Asymptote
The degree of the numerator is 0, and the degree of the denominator is 1. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).

#### Step 3: Intercepts
- x-intercept: Set \( f(x) = 0 \). There is no value of \( x \) that makes the numerator zero, so there is no x-intercept.
- y-intercept: Set \( x = 0 \):
\[
f(0) = \frac{3}{0-2} = -\frac{3}{2}
\]
So, the y-intercept is \( \left( 0, -\frac{3}{2} \right) \).

#### Step 4: Sketch
- The graph approaches the vertical asymptote \( x = 2 \) from both sides.
- The graph approaches the horizontal asymptote \( y = 0 \) as \( x \to \pm \infty \).
- The graph passes through the point \( \left( 0, -\frac{3}{2} \right) \).

---

Function 2: \( f(x) = \frac{x+2}{x-1} \)

#### Step 1: Vertical Asymptote
The denominator is zero when \( x - 1 = 0 \), so \( x = 1 \). This is a vertical asymptote.

#### Step 2: Horizontal Asymptote
The degrees of the numerator and denominator are both 1. The horizontal asymptote is given by the ratio of the leading coefficients:
\[
y = \frac{1}{1} = 1
\]
So, the horizontal asymptote is \( y = 1 \).

#### Step 3: Intercepts
- x-intercept: Set \( f(x) = 0 \). The numerator is zero when \( x + 2 = 0 \), so \( x = -2 \). The x-intercept is \( (-2, 0) \).
- y-intercept: Set \( x = 0 \):
\[
f(0) = \frac{0+2}{0-1} = -2
\]
So, the y-intercept is \( (0, -2) \).

#### Step 4: Sketch
- The graph approaches the vertical asymptote \( x = 1 \) from both sides.
- The graph approaches the horizontal asymptote \( y = 1 \) as \( x \to \pm \infty \).
- The graph passes through the points \( (-2, 0) \) and \( (0, -2) \).

---

Function 3: \( f(x) = \frac{1}{-x+4} \)

#### Step 1: Vertical Asymptote
The denominator is zero when \( -x + 4 = 0 \), so \( x = 4 \). This is a vertical asymptote.

#### Step 2: Horizontal Asymptote
The degree of the numerator is 0, and the degree of the denominator is 1. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).

#### Step 3: Intercepts
- x-intercept: Set \( f(x) = 0 \). There is no value of \( x \) that makes the numerator zero, so there is no x-intercept.
- y-intercept: Set \( x = 0 \):
\[
f(0) = \frac{1}{-0+4} = \frac{1}{4}
\]
So, the y-intercept is \( \left( 0, \frac{1}{4} \right) \).

#### Step 4: Sketch
- The graph approaches the vertical asymptote \( x = 4 \) from both sides.
- The graph approaches the horizontal asymptote \( y = 0 \) as \( x \to \pm \infty \).
- The graph passes through the point \( \left( 0, \frac{1}{4} \right) \).

---

Function 4: \( f(x) = \frac{3}{x} + 1 \)

#### Step 1: Vertical Asymptote
The denominator is zero when \( x = 0 \). This is a vertical asymptote.

#### Step 2: Horizontal Asymptote
The function can be rewritten as \( f(x) = \frac{3}{x} + 1 \). As \( x \to \pm \infty \), \( \frac{3}{x} \to 0 \), so the horizontal asymptote is \( y = 1 \).

#### Step 3: Intercepts
- x-intercept: Set \( f(x) = 0 \):
\[
\frac{3}{x} + 1 = 0 \implies \frac{3}{x} = -1 \implies x = -3
\]
So, the x-intercept is \( (-3, 0) \).
- y-intercept: Set \( x = 0 \). The function is undefined at \( x = 0 \), so there is no y-intercept.

#### Step 4: Sketch
- The graph approaches the vertical asymptote \( x = 0 \) from both sides.
- The graph approaches the horizontal asymptote \( y = 1 \) as \( x \to \pm \infty \).
- The graph passes through the point \( (-3, 0) \).

---

Final Answer

The graphs of the functions are sketched based on the analysis above. The final answer is:

\[
\boxed{
\begin{array}{c}
\text{Graphs of the functions are sketched as described.}
\end{array}
}
\]