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Match the features of the graph of the rational function.

[tex]\[ y=\frac{9x^2+81x}{x^3+8x^2-9x} \][/tex]

1. This function has how many [tex]\( x \)[/tex]-intercepts?
2. The horizontal asymptote is along the line where [tex]\( y \)[/tex] always equals what number?
3. The vertical asymptote is along the line where [tex]\( x \)[/tex] always equals what number?
4. How many holes are there?


Sagot :

Let's analyze the rational function [tex]\( y = \frac{9x^2 + 81x}{x^3 + 8x^2 - 9x} \)[/tex].

### Step-by-Step Solution:

1. Finding the [tex]\(x\)[/tex]-intercepts:

The [tex]\(x\)[/tex]-intercepts can be found by setting the numerator equal to zero and solving for [tex]\(x\)[/tex].

Numerator:
[tex]\[ 9x^2 + 81x = 0 \][/tex]

Factor out the common factor:
[tex]\[ 9x(x + 9) = 0 \][/tex]

So, the [tex]\(x\)[/tex]-intercepts are:
[tex]\[ x = -9 \quad \text{and} \quad x = 0 \][/tex]

2. Finding the horizontal asymptote:

Compare the degrees of the numerator and the denominator.

- The degree of the numerator [tex]\(9x^2 + 81x\)[/tex] is 2.
- The degree of the denominator [tex]\(x^3 + 8x^2 - 9x\)[/tex] is 3.

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:
[tex]\[ y = 0 \][/tex]

3. Finding the vertical asymptotes:

The vertical asymptotes occur where the denominator is equal to zero and the numerator is not zero at those values. Set the denominator equal to zero and solve for [tex]\(x\)[/tex].

Denominator:
[tex]\[ x^3 + 8x^2 - 9x = 0 \][/tex]

Factor out the common factor:
[tex]\[ x(x^2 + 8x - 9) = 0 \][/tex]

Solve the factored equation:
[tex]\[ x = 0 \quad \text{or} \quad x^2 + 8x - 9 = 0 \][/tex]

Solve the quadratic equation:
[tex]\[ x = \frac{-8 \pm \sqrt{64 + 36}}{2} = \frac{-8 \pm \sqrt{100}}{2} = \frac{-8 \pm 10}{2} \][/tex]

Therefore:
[tex]\[ x = 1 \quad \text{and} \quad x = -9 \][/tex]

So the vertical asymptotes are:
[tex]\[ x = -9, \quad x = 0, \quad \text{and} \quad x = 1 \][/tex]

4. Finding holes in the graph:

Holes occur where there are common factors in both the numerator and the denominator. The common factor found is [tex]\(x(x + 9)\)[/tex].

Since the common factors are [tex]\(x = -9\)[/tex] and [tex]\(x = 0\)[/tex], there are holes at these points.

5. Counting the number of holes:

There are exactly:
[tex]\[ 2 \quad \text{holes (at} \quad x = -9 \quad \text{and} \quad x = 0) \][/tex]

### Summary:

- [tex]\(x\)[/tex]-intercepts: [tex]\( x = -9 \)[/tex] and [tex]\( x = 0 \)[/tex]
- Horizontal asymptote: [tex]\( y = 0 \)[/tex]
- Vertical asymptotes: [tex]\( x = -9 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 1 \)[/tex]
- Holes: [tex]\( x = -9 \)[/tex] and [tex]\( x = 0 \)[/tex]

There are [tex]\( \boxed{2} \)[/tex] holes in the graph of the given rational function.
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