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How to Graph Rational Functions From Equations in 7 Easy Steps

# How to Graph Rational Functions From Equations in 7 Easy Steps

## 1. FACTOR the numerator and denominator

- Or have your graphing calculator do it for you

## 2. See if there are any HOLES

- If a term like (x-3) appears in both the numerator and denominator, then cancel them out and note that there is a hole at x=3
- To find the y-coordinate of the hole, plug it into the simplified equation

## 3. Find VERTICAL ASYMPTOTES by finding where factors in the denominator equal zero

- ex: 1/(x+2) has a vertical asymptote at x = -2

## 4. See if the fraction is TOP HEAVY, BOTTOM HEAVY, OR BALANCED for Non-Vertical (Horizontal and Oblique/Slant) Asymptotes

- top-heavy (by one degree, like x? / x) = oblique asymptote
- To find the equation of the oblique asymptote, use long division (ignore the remainder)
- bottom-heavy = horizontal asymptote at y=0
- The rational function will just get infinitely smaller
- balanced = compare leading coefficients for the horizontal asymptote
- ex: 3(x-1) / 2(x+2)(x+1) has a horizontal asymptote at y = 3/2

## 5. Find the x-intercepts where the numerator is equal to zero

- If the numerator is 0, then that means the y-value is 0, which means it is an x-intercept
- You can also find y-intercepts by plugging in x=0

## 6. See if the graph passes through any of the non-vertical asymptotes

- Set the rational function equal to the horizontal or oblique/slant asymptote
- ex: For (x-1) / (x+2), you would set (x-1) / (x+2) = 0
- If there are no solutions, then it does not cross the non-vertical asymptote

## 7. Test each region between the x-intercepts and vertical asymptotes to see if the graph is positive or negative

- Once you do this, you just fill in the curves, connecting your points and making them hug the asymptotes