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7.9 – Graphing Rational Functions
- Sign Chart – A chart that records information about a rational function’s values to help draw its graph.
- The zeros of the function are marked on the sign chart, and test numbers are used to find whether the function is negative or positive for values of x between the zeros.
- Singular Point – A point at which the graph of a rational function has a “hole.”
- If a graph has a singular point at x = b, then the function has a factor of (x – b) in both its numerator and denominator.
- Zero – The value of x for which a function F(x) equals zero or crosses the x-axis.
|Steps for Making a Sign Chart
- Step 1: Find all of the function’s zeros and vertical asymptotes, plot them on a number line, and label them as zeros or asymptotes.
- Step 2: Choose one x-value on either side of each asymptote and zero.
- You only need to test one number on either side of each zero or vertical asymptote.
- Check whether the function is positive or negative at these x-values by substituting them into the function’s equation and solving for (x, y).
- If the function is positive at a test number, it will remain positive until it reaches a zero or a vertical asymptote.
- If the function is negative at a test number, it will remain negative until it reaches a zero or an asymptote.
- You must substitute the SAME value for every x in the function’s equation when testing.
- Step 3: Use the results of Step 2 to fill in the signs on your sign chart.
- The sign of a rational function does not always change at every vertical asymptote and zero.
- Step 4: Make a graph.
- The graph of a rational function is curved, not a line.
- Step 1. Zeros and Asymptotes
- In this case, G(x) has a vertical asymptote at x = -1 and a zero at x = 0
- Step 2. Choose values on each side
- x = -2, x = -0.5, and x = 1 can be used as test numbers
- Step 3: Fill in Sign Chart
- Step 4. Graph the zeros, asymptotes, points, and curves
- A graph with a zero and vertical asymptote in the same place.
- It creates a hole in the graph.
- When you factor out the common factor (x + 1), you still need to account for it in your domain!
- So, x ≠ –2, x ≠ –1, and x ≠ 1
- The –1 is for the factored-out as factor (x + 1), but it still exists in the domain.
- Graph the following equation by making a sign chart.
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