Problem 32 - A Graphical Approach to College Algebra

Question

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x-3}{x+4}$$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = -4\), horizontal asymptote at \(y = 1\); x-intercept at (3, 0), y-intercept at (0, -3/4).

Step 1: Identify the Vertical Asymptote

The vertical asymptote occurs where the denominator of the function is zero, as it results in division by zero, which is undefined. To find this, set the denominator equal to zero and solve for \(x\):\(x + 4 = 0 \x = -4\)So, there is a vertical asymptote at \(x = -4\).

Step 2: Identify the Horizontal Asymptote

For rational functions, the horizontal asymptote is determined by the degrees of the numerator and the denominator. Both are the first degree here.The horizontal asymptote can be found by comparing the leading coefficients of the numerator and denominator: \(\frac{a}{b} \rightarrow \frac{1}{1} = 1\)Thus, the horizontal asymptote is \(y = 1\).

Step 3: Determine the X-Intercept

To find the x-intercept, set the numerator equal to zero and solve for \(x\):\(x - 3 = 0 \x = 3\)So, the function has an x-intercept at \((3, 0)\).

Step 4: Determine the Y-Intercept

The y-intercept occurs when \(x = 0\). Substitute 0 for \(x\) and solve:\(f(0) = \frac{0 - 3}{0 + 4} = -\frac{3}{4}\)The y-intercept is at \((0, -\frac{3}{4})\).

Step 5: Sketch the Graph

Now plot the key features:1. Draw a vertical asymptote at \(x = -4\).2. Draw a horizontal asymptote at \(y = 1\).3. Plot the x-intercept at \((3, 0)\).4. Plot the y-intercept at \((0, -\frac{3}{4})\).5. Sketch the curve approaching the asymptotes, passing through the intercepts. This rational function will resemble a hyperbola with branches on either side of the vertical asymptote.

Key Concepts

Vertical Asymptote

A vertical asymptote of a rational function is a line that the graph approaches but never touches or crosses as the values of \( x \) get extremely large or small. This happens because the function becomes undefined at certain points, typically where the denominator of the function equals zero.
To find the vertical asymptote of the function \( f(x) = \frac{x-3}{x+4} \), set the denominator \( x+4 \) to zero and solve for \( x \):

• \( x + 4 = 0 \)
• \( x = -4 \)

This computation shows that there's a vertical asymptote at \( x = -4 \). Keep in mind that the graph of the function will approach this line but never actually meet it, creating the characteristic 'gap' in the graph of the function.

Horizontal Asymptote

Horizontal asymptotes describe the behavior of a function as \( x \) approaches infinity or negative infinity. For rational functions, the horizontal asymptote can usually be determined by looking at the leading terms of the numerator and denominator.
Consider \( f(x) = \frac{x-3}{x+4} \), where both the numerator and the denominator are first-degree polynomials (linear). Since the degrees are equal, the horizontal asymptote is found by dividing the coefficients of the leading terms:

• \( \frac{1}{1} = 1 \)

This results in a horizontal asymptote at \( y = 1 \). This line gives us the height the graph approaches when \( x \) becomes very large or very small.

Intercepts

Intercepts help us to locate points where the function graph crosses the axes. In a rational function, you can find x-intercepts by setting the numerator equal to zero, while y-intercepts are found when the function is evaluated at \( x = 0 \).

**X-Intercept:**

• Set the numerator equal to zero: \( x - 3 = 0 \)
• Solve for \( x \): \( x = 3 \)

This tells us that the graph of the function crosses the x-axis at the coordinate (3, 0).

**Y-Intercept:**

• Evaluate the function at \( x = 0 \): \( f(0) = \frac{0-3}{0+4} = -\frac{3}{4} \)

The y-intercept is at (0, -3/4), marking the point where the graph crosses the y-axis.
By plotting these intercepts, you get essential reference points for sketched curves in the graph of the function. The complete understanding of these intercepts ensures a correctly visualized function graph.