find the hole for the function y=(x^2-9x+14)/(x^2-5x+6)

To find the hole in the function \( y = \frac{x^2 - 9x + 14}{x^2 - 5x + 6} \), we follow these steps:

### Step 1: Factor the numerator and denominator
First, factor both the numerator and the denominator:

- Numerator: \( x^2 - 9x + 14 \)
  
  This factors as:

  \[
  x^2 - 9x + 14 = (x - 7)(x - 2)
  \]

- Denominator: \( x^2 - 5x + 6 \)

  This factors as:

  \[
  x^2 - 5x + 6 = (x - 3)(x - 2)
  \]

### Step 2: Identify the common factor
Now, notice that both the numerator and denominator have a common factor of \( (x - 2) \). This common factor indicates a hole in the graph of the function because it can be canceled out, but it still causes the function to be undefined at \( x = 2 \).

### Step 3: Find the location of the hole
The hole occurs at \( x = 2 \) because that's where the common factor \( (x - 2) \) cancels out.

### Step 4: Determine the y-coordinate of the hole
To find the y-coordinate of the hole, substitute \( x = 2 \) into the reduced form of the function (after canceling the common factor).

The reduced form of the function is:

\[
y = \frac{x - 7}{x - 3}
\]

Now substitute \( x = 2 \) into this reduced function:

\[
y = \frac{2 - 7}{2 - 3} = \frac{-5}{-1} = 5
\]

### Final Answer:
The hole in the function occurs at the point \( (2, 5) \).

Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. How do you find vertical asymptotes in rational functions?
2. How do you determine horizontal asymptotes for rational functions?
3. What happens if the common factor is in both the numerator and denominator but doesn’t cancel out?
4. How do holes in rational functions differ from vertical asymptotes?
5. How do you graph a rational function with holes and asymptotes?
6. How do you determine the domain of a rational function with holes?
7. What is the significance of the hole in the context of limits?
8. How do you deal with removable discontinuities in rational functions?

**Tip:** Always factor the numerator and denominator of a rational function to check for common factors, as they often reveal holes in the graph of the function.

Discover how to find the hole in the rational function y=(x^2-9x+14)/(x^2-5x+6). Follow step-by-step instructions to factor polynomials, identify the common factor causing the hole, and determine the exact location of the hole. Suitable for high school students studying rational functions and graphing.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation