3.5D Graphing Rational Functions

Transcription

1 3.5D Graphing Rational Functions A. Strategy 1. Find all asymptotes (vertical, horizontal, oblique, curvilinear) and holes for the function. 2. Find the and intercepts. 3. Plot the and intercepts, draw the asymptotes, and plot enough points on the graph to see exactly what is going on. Note: The graph may not cross vertical asymptotes, but may cross others. Also, sometimes knowledge about symmetry (even/odd/neither) can speed up the graphing. B. Examples Example 1: Graph, where µ Solution 1. Asymptotes: Vertical: Horizontal: 2. Intercepts: -intercepts: set ¼: ¼ Disallowed values:, and LCD=. Multiplying by the LCD: ¼ µ 1

2 -intercept: set ¼: ¼µ ¼µ ¼ Now graph an initial rough sketch: Now we need to plot enough points to see what is going on. We pick and to see the behavior near the horizontal asymptote, and pick and to see the behavior near the vertical asymptote. Then pick a few others to see what is going on: ¼ We plot these points on the grid we already made. Then we connect the points using the asymptote behavior. 2

3 Ans Example 2: Graph, where µ Solution 1. Asymptotes: We first have to factor the top and bottom. Top: µ µ Bottom: Rational Root Theorem: factors of factors of Rational Candidates: 3

4 Note: µ µ µ ¼ Thus is a factor. Divide it out using synthetic division: 0 µ µ µ µ µ Thus, µ µ µ. µ µ µ Simplifying, µ µ. µ µ Thus, we have vertical asymptotes with equations and. Also, we have a hole at. Since degree top and degree bottom, and since, we have a horizontal asymptote with equation ¼. 2. Intercepts: -intercepts: set ¼: µ µ ¼ Disallowed values:, and LCD= µ µ. Multiplying by the LCD: ¼ µ -intercept: set ¼: ¼µ ¼ ¼ µ ¼ µ µ µ 4

5 Now graph an initial rough sketch: y x hole on graph here Now plot some more points using the output formula µ µ : µ µ Note: In fact, the point µ from the table above is the location of a hole, since we saw previously that we had a hole at. 5

6 We plot these points on the grid we already made. Then we connect the points using the asymptote behavior. Ans Example 3: Graph, where µ Solution 1. Asymptotes: We first have to factor... Considering, we can t factor it immediately, so we decide to use the quadratic formula. However µ µ µ ¼, so the zeros are complex. Thus we have no vertical asymptotes or holes! Since degree top and degree bottom, and since, we have an oblique or curvilinear asymptote (oblique, in fact, as we see below). 6

7 Now find it by algebraic long division: ¼ µ Hence. Thus defines an oblique asymptote. 2. Intercepts: -intercepts: set ¼: ¼ No disallowed values and LCD=. Multiplying by the LCD: ¼. Rational Root Theorem: factors of factors of Rational Candidates: Note: ¼ Thus is a factor. Divide it out using synthetic division: ¼ ¼ Thus µ µ ¼. 7

8 Now use the quadratic formula on the remaining quadratic: Ô µ µ µ Ô Ô Since these are complex, we only get one -intercept from the factor. Thus we have one -intercept,. -intercept: set ¼: ¼µ ¼ ¼ ¼ ¼. Now graph an initial rough sketch: 8

9 Now plot some more points using the output formula µ : ¼ ¼ We plot these points on the grid we already made. Then we connect the points using the asymptote behavior. Ans 9

10 Example 4: Graph, where µ Solution 1. Asymptotes: µ µ µ µ Simplifying: µ µ µ Thus we have vertical asymptotes with equations and. We also have a hole at ¼. Since degree top and degree bottom, and since, we have an oblique or curvilinear asymptote (curvilinear, in fact, as we see below). Now find it by algebraic long division: ¼ ¼ ¼ ¼ ¼ ¼ µ ¼ ¼ ¼ µ Hence. Thus defines a curvilinear asymptote. 10

11 2. Intercepts: -intercepts: set ¼: ¼ Disallowed values: and LCD= µ µ. Multiplying by the LCD: ¼ µ ¼ µ ¼. However, this intercept is not a point, since we said earlier that we had a hole at ¼. -intercept: set ¼: ¼µ ¼µ ¼ ¼ ¼ This intercept is not a point either, since we found it by setting ¼, which is where we said we had a hole. Now graph an initial rough sketch: hole 11

12 Now plot some more points using the output formula µ. Note: Since this function is even, we need only make our table with negative - values. The positive -values we get for free by reflection. We plot these points on the grid we already made, along with the -axis reflected points (even function). Then we connect the points using the asymptote behavior. Ans hole 12