Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{2 x^{2}-5 x+5}{x-2}$$

Step-by-Step Solution

Answer

Following all the steps, one can sketch the graph of the given function. The intercepts are obtained by setting the function equal to zero and solving for x (x-intercepts) and by setting x equal to zero (y-intercept). The vertical asymptote is found by setting the denominator equal to zero and solving for x. The slant asymptote is found by performing polynomial division on the function. All of these steps assist in sketching the graph.

Step 1: Find the Intercepts

Set \(f(x) = 0\), which gives the equation \(\frac{2x^2 - 5x + 5}{x - 2} = 0\). This implies that the numerator \(2x^2 - 5x + 5 = 0\). Solve this for x to get the x-intercepts. Similarly, set x = 0 in the function to get the y-intercept.

Step 2: Identify Vertical Asymptotes

The vertical asymptotes can be found by setting the denominator equal to zero and solving for x. So in this case, we'll have to solve \(x-2=0\) to get a vertical asymptote.

Step 3: Determine Slant Asymptotes

The slant asymptotes can be found when the degree of the numerator is one more than the degree of the denominator. Here, we'll perform polynomial division, which will offer us a linear function. This linear function will be our slant asymptote.

Step 4: Sketch the Graph

Use the information from Steps 1, 2, and 3 to sketch the graph. It might be helpful to plot a few additional points if needed. The x-intercepts, y-intercept, vertical asymptote, and slant asymptote will assist in the sketching of the graph.

Key Concepts

Intercepts of Rational Functions

Understanding the intercepts of a rational function is crucial for graph plotting, as they represent where the graph crosses the axes. The x-intercepts are found by setting the function equal to zero and solving for x, which gives us the points where the graph intersects the x-axis. In the case of the function
\( f(x) = \frac{2x^2 - 5x + 5}{x - 2} \),
we set the numerator to zero and solve for x.
For the y-intercept, simply set x to 0 in the function to see where the graph crosses the y-axis. The y-intercept is given by
\( f(0) = \frac{2(0)^2 - 5(0) + 5}{0 - 2} = -\frac{5}{2} \).
So, for our rational function, the y-intercept is the point (0, -2.5) on the graph.

Vertical Asymptotes

Vertical asymptotes occur when a rational function approaches infinity as x approaches a certain value. To identify vertical asymptotes, we look at where the denominator of the function equals zero, because the function cannot be defined at these points. Importantly, a vertical asymptote can only be where the function is undefined and where there's no hole in the graph. For the function
\( f(x) = \frac{2x^2 - 5x + 5}{x - 2} \),
we find the asymptote by setting the denominator
\( x-2 = 0 \),
which gives
\( x = 2 \).
So, at x = 2, there is a vertical asymptote because the function cannot be evaluated there.

Slant Asymptotes

When the degree of the numerator is exactly one more than the degree of the denominator, a rational function may have a slant asymptote. This occurs because, as x becomes very large in either the positive or negative direction, the graph of the function tends to follow a straight line, which is the slant asymptote. To find this line, we perform polynomial division, which gives us the equation of the slant asymptote.
For our example \( f(x) \), dividing \( 2x^2 - 5x + 5 \) by \( x - 2 \) results in a linear function,
\( 2x - 1 \),
which is our slant asymptote. This means that as x goes to infinity, the graph of \( f(x) \) approaches the line y = 2x - 1.

Polynomial Division

An essential technique in finding slant asymptotes is polynomial division, similar to long division with numbers. To explain further, when the numerator has a higher degree than the denominator, we divide the numerator by the denominator. This division process will yield two results: a quotient, which can be a linear function (if the difference in degree is one) or another polynomial, and a remainder. The quotient gives us the slant asymptote's equation, provided it's a linear function.
In our given function
\( f(x) = \frac{2x^2 - 5x + 5}{x - 2} \),
the quotient after dividing \( 2x^2 - 5x + 5 \) by \( x - 2 \) is \( 2x - 1 \), which indicates a slant asymptote for the graph of the rational function.