**High School Number Sense Lesson 96: Asymptotes**

Today's concept appeared

**12 times**on high school tests this year, with a median placement at

**question # 76**.

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An

**asymptote**of a curve is a line that approaches the curve but never quite reaches it. It may be helpful to review the articles on

**Wikipedia**and

**Khan Academy**before moving on.

There are several types of questions that will appear on number sense tests:

- Finding the
**vertical asymptote** - Finding the
**horizontal asymptote** - Finding the
**number of asymptotes**

**Finding the Vertical Asymptote**

- Set the denominator of the equation to zero
- Solve for its root(s)

**Finding the Horizontal Asymptote**

- Compare the largest power of
**x**in the numerator to the largest power of**x**in the denominator - If the power of the
**x**'s is the same, write down their ratio - If the numerator's
**x**power is larger, the answer is**infinity** - If the denominator's
**x**power is larger, the answer is**0**

**Finding the Number of Asymptotes**

- Test for both vertical and horizontal asymptotes
- Add the number of asymptotes you find for each

**Example 1: The vertical asymptote of y = (4x - 3)/(2x + 1) is x = ___**

- Set the denominator equal to zero:
**2x + 1 = 0** - Solve for x:
**2x = -1**, so x =**-1/2**

**Example 2: The vertical asymptote for f(x) = (4 - 3x)/(7x + 1) is x = ___**

- Set the denominator equal to zero:
**7x + 1 = 0** - Solve for x:
**7x = -1**, so x =**-1/7**

**Example 3: The horizontal asymptote of y = (3x - 6)/(7x + 1) is y = ___**

- The numerator's largest power of x is 1, and its coefficient is
**3** - The denominator's largest power of x is 1, and its coefficient is
**7** - The answer is the ratio of these coefficients:
**3/7**

**Example 4:**

**The horizontal asymptote of y = (2x**

^{2}- 3x + 1)/(7 - 9x^{2}) is y = ___1. The numerator's largest power of x is 2, and its coefficient is

**2**

2. The denominator's largest power of x is 2, and its coefficient is

**-9**

3. The answer is the ratio of these coefficients:

**-2/9**

**Example 5:**

**The graph of (3x + 1)/(9x**

^{2}- 1) has ___ asymptote(s)1. Test for vertical asymptotes by setting the denominator equal to 0:

**9x**, so 9x

^{2}- 1 = 0^{2}= 1, so x

^{2}= 1/9.

2. There are

**2**vertical asymptotes at y = 1/3 and y = -1/3

3. Test for horizontal asymptotes by comparing the largest power of x in the numerator (1) and denominator (2).

4. Since the denominator's x power is larger, there is

**1**horizontal asymptote at x = 0.

5. There are

**3**asymptotes total.

**Up Next for High School: Square31-40**