Problem 25 - Algebra 2

Question

Graph each rational function. $$ f(x)=\frac{x+4}{x-1} $$

Step-by-Step Solution

Answer

Graph features: vertical asymptote at \( x = 1 \), horizontal asymptote at \( y = 1 \), x-intercept at (-4,0), y-intercept at (0,-4).

Step 1: Identify the Vertical Asymptotes

For a rational function, vertical asymptotes occur where the denominator equals zero, provided the numerator is not also zero at that point. Set the denominator of the function equal to zero: \[ x - 1 = 0 \]Solving this yields \( x = 1 \). So, there is a vertical asymptote at \( x = 1 \).

Step 2: Determine the Horizontal Asymptote

A horizontal asymptote is given by the behavior of the function as \( x \) approaches infinity. For the function \( f(x) = \frac{x+4}{x-1} \), since the degrees of the numerator and the denominator are the same (both 1), the horizontal asymptote equals the ratio of the leading coefficients. Here, both coefficients are 1, so the horizontal asymptote is \( y = 1 \).

Step 3: Find the x-intercept

To find the x-intercept, set \( f(x) = 0 \) and solve for \( x \). This requires the numerator to be zero:\[ x + 4 = 0 \]Solving gives \( x = -4 \). Therefore, the x-intercept is at \( (-4, 0) \).

Step 4: Calculate the y-intercept

Find the y-intercept by evaluating \( f(x) \) at \( x = 0 \): \[ f(0) = \frac{0+4}{0-1} = -4 \]Thus, the y-intercept is at \( (0, -4) \).

Step 5: Sketch the Graph

Draw the vertical asymptote as a dashed line at \( x = 1 \) and a horizontal asymptote at \( y = 1 \). Mark the x-intercept at \( (-4, 0) \) and the y-intercept at \( (0, -4) \). Use these points and asymptotes to sketch the general shape of the graph. Note that as \( x \) approaches the vertical asymptote from the left, \( f(x) \) tends to \(-\infty\) and from the right, it tends to \(+\infty\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are key features in graphing rational functions as they represent the values of \(x\) where the function does not exist. In simpler terms, a vertical asymptote occurs where the denominator of the rational function equals zero. For the function \(f(x) = \frac{x+4}{x-1}\), we find vertical asymptotes by setting the denominator equal to zero:

• Equation: \(x - 1 = 0\)
• Solve: \(x = 1\)

This means there is a vertical asymptote at \(x = 1\). This asymptote indicates that as \(x\) approaches 1 from either side, the function \(f(x)\) will tend to infinity. Make sure to draw this line as a dashed line on your graph to show that the curve approaches it but never touches it.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input \(x\) moves towards positive or negative infinity. For the function \(f(x) = \frac{x+4}{x-1}\), we determine the horizontal asymptote by comparing the degrees of the numerator and denominator:

• The degrees are equal (both are 1).
• The ratio of the leading coefficients (both 1) gives the horizontal asymptote.

Thus, the horizontal asymptote is at \(y = 1\). This tells us that as \(x\) increases or decreases dramatically, the graph of the function will flatten out near the line \(y = 1\). The function never actually reaches this line, but it approaches it closely on both ends of the graph.

X-Intercepts

X-intercepts indicate where the graph crosses or touches the x-axis. For rational functions, this happens where the numerator is equal to zero because the whole fraction, therefore, equals zero. For \(f(x) = \frac{x+4}{x-1}\), set the numerator equal to zero:

• Equation: \(x + 4 = 0\)
• Solve: \(x = -4\)

The x-intercept occurs at \((-4, 0)\). You would mark this point on the graph to show where the function crosses the x-axis. Being aware of x-intercepts is crucial in sketching the graph as they provide clear points through which the curve passes.

Y-Intercepts

Y-intercepts show where the graph crosses the y-axis. To find the y-intercept, substitute \(x = 0\) into the function and solve for \(f(x)\). With \(f(x) = \frac{x+4}{x-1}\):

• Calculate: \(f(0) = \frac{0+4}{0-1} = -4\)

Hence, the y-intercept is located at \((0, -4)\). Plot this point on the graph to represent where the function crosses the y-axis. Understanding the y-intercept provides another anchor for accurately drawing the graph of the function.