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LAB: GRAPHING RATIONAL FUNCTIONS NAME ___________________

Instructor: Don Power

To graph a rational function, follow steps a to h below. For the sake of neatness, I have shown the steps on a graphing calculator screen. However, it is common to encounter functions for which the important features cannot all be shown adequately on a single graphing calculator screen. Therefore, you should do your graph by hand, varying the scale as necessary to show all the important features of the graph.

For an example, I will use the function .

a. Factor, reduce and identify deleted points (values that make deleted factors equal to 0).
Substitute the x-coordinate into the reduced function and calculate y. Plot with an open dot.

Example, reduces to with the deleted point being (3,g(3)), or (3,15/7). Graph the point as an open dot (do this by hand, not with a calculator):

b. Zeros: Solve the reduced function g(x) = 0 and plot with solid dots on the number line (Hint: solve for numerator = 0)

Example: For our g(x), solve 3(x+2)=0, resulting in x=2. Plot (-2,0):

c. Solve for vertical asymptotes; plot on the graph with a dotted line (Hint: solve for denominator = 0). Always use the reduced function g(x).

Example: For our g(x), solve x+4=0, resulting in x = -4. Graph the vertical line x = -4:

d. Identify intervals along the x-axis from -¥ to ¥, (separated by zeros, deleted points, and vertical asymptotes). Write these intervals below the graph; we will use them later for analysis of the function.

For our example, the intervals are (-¥,-4),(-4,-2),(-2,4),(4,¥)

e. Find the y-intercept and plot it. (If the y-int. is zero, also calcuate and plot an additional point for which y¹0.)

Example: For our function, , g(0) = 3/2. Plot (0,3/2):

(-¥,-4),(-4,-2),(-2,4),(4,¥)

f. Identify and sketch end behavior: A rational function can be viewed as a division problem. At the left and right ends of the graph, the function will behave like the quotient only, after dividing, (throw out the remainder, because the remainder will approach 0). [As usual, work with the reduced function g(x)]
Example: =. At the left and right ends of the graph, x increases in absolute value, so the fraction shrinks toward 0. Therefore the curve approaches the line g(x) = 3.
Sketch the line y = 3:

(-¥,-4),(-4,-2),(-2,4),(4,¥)

The possible results are lines (“asymptotes”) or curves ("asympototic curves") which the graph approaches at the right and left ends of the graph. You should sketch horizontal or slanted lines (asymptotes) at this time. There are four possible cases:

Graph approaches horizontal asymptote y = 0 (Denom. has higher degree)
Graph approaches horizontal asymptote y = constant= ratio of leading coefficients (Degree of num and denom are the same)
Graph approaches slanted asymptote y=mx+b (Degree of num is exactly 1 higher than denom)
(Divide and throw out remainder to calculate the exact expression mx+b)
Graph approaches a polynomial curve (degree of num is 2 or more higher than denom)
(For a rough description of the end behavior, y » the ratio of the leading terms)
(For the exact curve, divide and throw out the remainder)

g. Finally, you are ready to sketch the graph.

For a rough graph, use the multiplicity of each factor to sketch (work from the y-intercept outward).

            If the factor has odd multiplicity (that is, the exponent of the factor is odd):
                  The function changes signs at the zero of the factor.
                  Example: (x+5)³ is negative if x < -5 and positive if x > -5
                  Therefore, the graph moves to the opposite side of the x-axis at the zero of the factor.

            If the factor has even multiplicity (that is, the exponent of the factor is even):
                  The function does not change signs at the zero of the factor.
                  Example: (x+5)⁴ is positive regardless of whether x < -5 or x > -5
                  Therefore, the graph stays on the same side of the x-axis at the zero of the factor.

For our example, :

Start from the known point (the y-intercept) and graph the function to the right: it will skip across the deleted point and approach the horizontal asymptote.

Now start from the y-intercept and graph the function to the left.

The curve passes through the x-axis at -2 because the factor (x+2) is in the numerator and has an odd exponent

The curve will then approach the vertical asymptote. On the right side of the asymptote, the curve approaches the asymptote at the bottom

The curve goes off the top of the graph (y increases without bound) on one side of the vertical asymptote at x = -4, but switches and goes off the bottom of the graph (decreases without bound) on the other side of the vertical asymptote.

[For a more precise graph, pick a test point in each interval, calculate and plot the value at that point.]