Problem 46 - 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. \(r(x)=\frac{2 x(x+2)}{(x-1)(x-4)}\)

Step-by-Step Solution

Answer

x-intercepts: (0,0) and (-2,0); vertical asymptotes: x=1, x=4; horizontal asymptote: y=2.

Step 1: Identify the x-intercepts

To find the x-intercepts of the function, set the numerator equal to zero and solve for \(x\):\[2x(x+2) = 0\].This gives \(x = 0\) and \(x = -2\). The x-intercepts are at \((0, 0)\) and \((-2, 0)\).

Step 2: Identify the y-intercept

To find the y-intercept, evaluate the function \(r(x)\) at \(x=0\): \[r(0) = \frac{2(0)(0+2)}{(0-1)(0-4)} = 0\].The y-intercept is at \((0, 0)\).

Step 3: Determine vertical asymptotes

Vertical asymptotes occur where the denominator is zero, and the numerator is not zero. Solve \((x-1)(x-4) = 0\) leading to \(x = 1\) and \(x = 4\). The vertical asymptotes are \(x = 1\) and \(x = 4\).

Step 4: Determine horizontal asymptote

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Both are degree 2. The horizontal asymptote is \(y = \frac{2}{1} = 2\).

Step 5: Sketch the graph

Plot the x-intercepts \((0,0), (-2,0)\), the y-intercept \((0,0)\), vertical asymptotes \(x=1\) and \(x=4\), and the horizontal asymptote \(y=2\). The function will approach the vertical asymptotes but never touch them, crossing the x-axis at the intercepts, and will level off as it approaches the horizontal asymptote.

Step 6: Confirm with graphing device

Using a graphing device, input the function \(r(x) = \frac{2x(x+2)}{(x-1)(x-4)}\) to verify the intercepts and asymptotes. Compare your graph with the plotted sketch from previous steps to ensure accuracy.

Key Concepts

x-intercepts

To understand x-intercepts of a rational function better, you need to set the numerator of the function equal to zero. This step determines where the graph of the function will cross the x-axis. For the given function, we identify the numerator as \(2x(x+2)\). By setting it equal to zero, we solve:

• \(2x(x+2) = 0\)
• This results in two solutions: \(x = 0\) and \(x = -2\).

This means the graph will intersect the x-axis at the points \((0, 0)\) and \((-2, 0)\). It's helpful to remember that every solution to this equation gives us an x-intercept, signifying points where the graph touches or crosses the x-axis. Don't forget that having these intercepts helps you sketch the function's behavior on a graph.

vertical asymptotes

Vertical asymptotes are lines that the graph approaches but never touches or crosses. They show where the function is undefined. To find them, set the denominator of the rational function equal to zero because a rational function is undefined wherever its denominator is zero.For this function, the denominator is \((x-1)(x-4)\). Solving this gives:

• \((x-1)(x-4) = 0\)
• This results in \(x = 1\) and \(x = 4\).

These are our vertical asymptotes. On the graph, you'll notice that as \(x\) approaches 1 and 4, the function's values rise or drop sharply without bound. It's important to mark these on your plot to understand the limits and behavior of the function near these x-values.

horizontal asymptotes

Horizontal asymptotes describe the end behavior of a function as \(x\) approaches infinity or negative infinity. They show where the function levels off. To find horizontal asymptotes in rational functions, compare the degree of the numerator and the degree of the denominator.For the function \(r(x) = \frac{2x(x+2)}{(x-1)(x-4)}\), both the numerator and the denominator are quadratic (degree 2). When the degrees are the same, the horizontal asymptote is found by dividing the leading coefficients. Here, the leading coefficient in the numerator is 2, while it's 1 in the denominator. Thus, the horizontal asymptote is:

• \(y = \frac{2}{1} = 2\)

On a graph, you'll see that as \(x\) moves towards positive or negative infinity, the function approaches the line \(y = 2\). It helps graph the function more accurately, providing a boundary for the graph's arms.

graphing functions

Graphing rational functions blends all of these elements together to give you a visual representation of the function's behavior. Start by plotting the x-intercepts and y-intercept first. For \(r(x)\), these intersections are \((0,0)\) and \((-2,0)\). Next, draw dotted lines for the vertical asymptotes at \(x=1\) and \(x=4\).For the horizontal asymptote, \(y=2\), you can also draw a horizontal dotted line. This indicates where the graph levels off in the long run. When sketching the graph:

• Remember that the graph will approach but not touch the asymptotes.
• Also, it will cross the x-axis at the x-intercepts.
• Use a smooth curve to connect these traits, showing the function's continuous nature between these critical points.

Finally, use a graphing device to double-check your sketch. This step ensures the plotted function aligns with the analytical findings. Graphing gives a complete picture, making the function's behavior and key features clear and understandable.