Problem 38 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{-9}{(x-3)^{2}}$$

Step-by-Step Solution

Answer

The vertical asymptote of the function is \(x = 3\), the horizontal asymptote is \(y = 0\), it has a y-intercept at (0, 1), and it has no x-intercepts. The graph has a general U-shape, with the branches opening downwards, and mirroring about the vertical line \(x = 3\).

Step 1: Identifying the Vertical Asymptote

To identify any vertical asymptotes of \(h(x)\), set the denominator of the function equal to zero and solve for \(x\). The denominator of the function is \((x-3)^{2}\). Setting this equal to zero gives us \(x-3 = 0\), which in turn gives the solution \(x = 3\). Hence, the vertical asymptote of the function is \(x = 3\).

Step 2: Identifying the Horizontal Asymptote

For the rational function \(h(x)\), the degree of the denominator (which is 2) is greater than the degree of the numerator (which is 0). This means that as \(x\) gets larger and larger in the positive or negative direction, the value of the function gets closer and closer to 0. Hence, the horizontal asymptote of the function is \(y = 0\).

Step 3: Finding the Intercepts

The x-intercept of the function can be found by setting the function equal to zero. However, trying to solve \(h(x) = 0\) will not produce real solutions because the numerator of the function is a constant (-9) instead of a variable. This shows that \(h(x)\) has no x-intercepts. The y-intercept can be found by plugging in x = 0 into the function which gives \(h(0) = \frac{-9}{(0-3)^{2}} = 1\). Hence, the y-intercept of the function is (0, 1).

Step 4: Sketching the graph of the function

The graph of \(h(x) = \frac{-9}{(x-3)^{2}}\) can be sketched using the asymptotes and intercepts found earlier. First, draw the vertical line \(x = 3\) and the horizontal line \(y = 0\), with dashes to indicate they are asymptotes. Then plot the point (0,1), which is the y-intercept of the graph. The graph takes a shape similar to a downward-opening parabola, but with the left and right branches getting closer and closer to the x-axis as x moves away from 3 in either direction. The function never crosses the x-axis as there are no x-intercepts, and also never crosses the vertical asymptote \(x = 3\). Hence, the graph has a general U-shape, mirroring about the vertical line \(x = 3\), but with the branches opening downwards. Also, the graph approaches but never reaches the horizontal line \(y = 0\).

Key Concepts

Vertical Asymptotes

When dealing with rational functions, vertical asymptotes are key indicators of behavior around certain points. They occur where the denominator of the function is zero, leading to undefined values for the function.
In the function \( h(x) = \frac{-9}{(x-3)^2} \), the denominator \((x-3)^2\) becomes zero when \( x=3 \).
This results in a vertical asymptote at \( x=3 \), because as \( x \) approaches this value from either direction, \( h(x) \) heads towards infinity or negative infinity.

• The function cannot have a value where the denominator is zero, leading to breaks in the graph.
• Graphically, vertical asymptotes appear as a vertical line that the function approaches but never intersects or touches.

Vertical asymptotes reveal pivotal points where the function dramatically increases or decreases, signaling crucial insights for graphing and understanding the function's limits.

Horizontal Asymptotes

Horizontal asymptotes are crucial in determining the end behavior of rational functions. They offer a glimpse into the values that a function approaches as \( x \) extends towards positive or negative infinity.
For the function \( h(x) = \frac{-9}{(x-3)^2} \), the degree of the denominator (2) is greater than the degree of the numerator (0).

• This indicates that as \( x \) grows larger in any direction, \( h(x) \) moves closer to 0.
• The horizontal asymptote in this case is \( y = 0 \).

Horizontal asymptotes are vital as they help predict the behavior of a graph at its extremities.
The function \( h(x) \) never truly reaches the horizontal asymptote but continues to inch infinitesimally closer as \( x \) moves outward. Identifying these lines is fundamental when you're interested in the long-term trends of a function.

Intercepts

Intercepts are among the first features plotted when sketching the graph of a function; they provide clear points of reference. The intercepts for a function are where it crosses the x- or y-axis.
For \( h(x) = \frac{-9}{(x-3)^2} \), the process of finding intercepts is straightforward:

• **X-Intercepts:** To find x-intercepts, you set the function \( h(x) \) equal to zero.
  However, because the numerator is a constant \(-9\), it never equals zero.
  Hence, there are no x-intercepts.
• **Y-Intercepts:** To find the y-intercept, set \( x \) to zero in the equation.
  This gives the point \( (0, 1) \) as the y-intercept with \( h(0) = 1 \).

Understanding intercepts is important because they pinpoint where the function touches the axes, providing grounding to the abstract concept the rational function represents.