Problem 39 - Algebra and Trigonometry

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(s(x)=\frac{4 x-8}{(x-4)(x+1)}\)

Step-by-Step Solution

Answer

X-intercept: (2, 0); Y-intercept: (0, 2); Vertical asymptotes: x=4, x=-1; Horizontal asymptote: y=0.

Step 1: Identify the x-intercepts

The x-intercepts occur where the numerator of the rational function equals zero. For the given function \(s(x) = \frac{4x - 8}{(x - 4)(x + 1)}\), set the numerator \(4x - 8\) equal to zero and solve for \(x\):\[4x - 8 = 0 \]\[4x = 8 \]\[x = 2\] Hence, the x-intercept is at \((2, 0)\).

Step 2: Identify the y-intercepts

The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the function to find the y-intercept:\[s(0) = \frac{4(0) - 8}{(0 - 4)(0 + 1)} = \frac{-8}{-4} = 2\]Thus, the y-intercept is at \((0, 2)\).

Step 3: Determine the vertical asymptotes

Vertical asymptotes occur where the denominator equals zero, and the numerator does not equal zero at the same values. Solve the equation:\[(x - 4)(x + 1) = 0\]This gives us two critical points: \(x = 4\) and \(x = -1\). At these points, the denominator is zero, but check if these points make the numerator zero:- For \(x = 4\): \(4(4) - 8 = 8 eq 0\)- For \(x = -1\): \(4(-1) - 8 = -12 eq 0\)Thus, the vertical asymptotes are at \(x = 4\) and \(x = -1\).

Step 4: Determine the horizontal asymptote

For the rational function \(s(x) = \frac{4x - 8}{(x - 4)(x + 1)}\), compare the degrees of the numerator and denominator:- The degree of the numerator is 1 (\(4x - 8\)).- The degree of the denominator is 2 (\((x-4)(x+1)\) simplifies to \(x^2 - 3x - 4\)).Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is at \(y = 0\).

Step 5: Sketch the graph

On the coordinate plane, plot the intercepts at \((2, 0)\) for the x-intercept and \((0, 2)\) for the y-intercept. Draw the vertical asymptotes as dashed lines at \(x = 4\) and \(x = -1\), and the horizontal asymptote as a dashed line at \(y = 0\). Connect the intercepts with a smooth curve, which respects the asymptotes. Ensure the graph approaches but never touches the asymptotes.

Step 6: Verify using a graphing device

Utilize a graphing calculator or software to input the function \(s(x) = \frac{4x - 8}{(x - 4)(x + 1)}\) and confirm the intercepts and asymptotes. The graph generated should match your manual sketch, validating your solution.

Key Concepts

x-intercepts

Finding the x-intercepts of a rational function like \(s(x) = \frac{4x - 8}{(x - 4)(x + 1)}\) is an important step in understanding its graph. The x-intercepts are the points where the graph of the function crosses the x-axis. At these points, the output of the function is zero, which means the numerator must be zero. To find the x-intercepts, set the numerator equal to zero and solve for x.

• Start by taking the numerator of the function: \(4x - 8\).
• Set it equal to zero: \(4x - 8 = 0\).
• Solve for x by adding 8 to both sides: \(4x = 8\).
• Divide by 4 to isolate x: \(x = 2\).

Therefore, the x-intercept for the function is at the point \((2, 0)\). This means the graph crosses the x-axis when x is 2. Knowing the x-intercepts helps in not only graphing but also understanding the behavior of the rational function as it relates to the x-axis.

asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches. For rational functions, there are two main types we often analyze: vertical and horizontal asymptotes. Each kind of asymptote provides crucial insights into how the function behaves.
**Vertical Asymptotes**These occur at the values of x that make the denominator zero, provided the numerator isn't also zero at those points. For our function, \(s(x) = \frac{4x - 8}{(x - 4)(x + 1)}\), we find the vertical asymptotes by setting the denominator equal to zero:

• \((x - 4)(x + 1) = 0\)
• This gives critical points: \(x = 4\) and \(x = -1\).
- Neither makes the numerator zero, so both are vertical asymptotes.

Thus, there are vertical asymptotes at \(x = 4\) and \(x = -1\). These lines are boundaries the graph won't cross and often indicate a break in the graph.
**Horizontal Asymptotes**Horizontal asymptotes describe the behavior of the graph as x approaches infinity. They are determined by comparing the degrees of the polynomials in the numerator and the denominator.

• If the denominator's degree is greater than the numerator's, as in our case, the horizontal asymptote will be \(y = 0\).

It's important to note that a horizontal asymptote doesn't mean the graph cannot cross it, unlike vertical asymptotes. Understanding the asymptotes is integral to sketching the rational function and comprehending its extreme behavior.

graphing rational functions

Graphing rational functions involves plotting points, drawing asymptotes, and connecting these aspects to form a coherent curve that reflects the behavior of the function. For the function \(s(x) = \frac{4x - 8}{(x - 4)(x + 1)}\), we gather all our analyses, including intercepts and asymptotes, to sketch its graph.
**Steps to Graph:**

• Start by marking the intercepts. Plot the x-intercept at \((2, 0)\) and the y-intercept at \((0, 2)\).
• Draw in the vertical asymptotes as dashed lines at \(x = 4\) and \(x = -1\). These lines outline limits in the x-direction.
• Include the horizontal asymptote as a dashed line at \(y = 0\) to denote the behavior as x moves towards infinity or negative infinity.

Connect these components with a smooth curve. The graph should never touch or cross the vertical asymptotes but can approach and even cross the horizontal one. Each section of the graph between asymptotes should be carefully curved to reflect its expected behavior based on the intercepts and asymptotes identified.
Finally, checking your sketch with a graphing calculator can be a great way to confirm your understanding and ensure the accuracy of your hand-drawn graph. This comprehensive approach makes the graphing process intuitive and accurate.