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The given rational function is:

\[
f(x) = \frac{-10x + 20}{x^2 - 3x - 10}
\]

### Steps to graph the rational function:

#### 1. **Factor the denominator**:
We first need to factor the denominator to find vertical asymptotes and to simplify the function if possible.

The denominator is \(x^2 - 3x - 10\). We factor it:

\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]

So, the rational function becomes:

\[
f(x) = \frac{-10x + 20}{(x - 5)(x + 2)}
\]

#### 2. **Find Vertical Asymptotes**:
Vertical asymptotes occur where the denominator is zero. Set the factored denominator equal to zero:

\[
(x - 5)(x + 2) = 0
\]

So, the vertical asymptotes are at \(x = 5\) and \(x = -2\).

#### 3. **Find Horizontal Asymptotes**:
To find the horizontal asymptote, compare the degrees of the numerator and denominator. 

- The numerator \(-10x + 20\) has degree 1.
- The denominator \(x^2 - 3x - 10\) has degree 2.

Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is:

\[
y = 0
\]

#### 4. **Find the x-intercept(s)**:
To find the x-intercept(s), set the numerator equal to zero and solve for \(x\):

\[
-10x + 20 = 0
\]

Solving this:

\[
x = 2
\]

So, the x-intercept is at \(x = 2\), which corresponds to the point \((2, 0)\).

#### 5. **Find the y-intercept**:
To find the y-intercept, set \(x = 0\) and evaluate \(f(0)\):

\[
f(0) = \frac{-10(0) + 20}{0^2 - 3(0) - 10} = \frac{20}{-10} = -2
\]

So, the y-intercept is \((0, -2)\).

### Final Information:

- **Vertical Asymptotes**: \(x = 5\) and \(x = -2\)
- **Horizontal Asymptote**: \(y = 0\)
- **x-intercept**: \((2, 0)\)
- **y-intercept**: \((0, -2)\)

Would you like to see a sketch of the graph or further clarifications?

### Related Questions:
1. How do we determine slant asymptotes for rational functions?
2. How do you find oblique asymptotes for higher degree functions?
3. Can a rational function have more than one horizontal asymptote?
4. How do we find holes in the graph of a rational function?
5. What happens when the degrees of the numerator and denominator are the same?

#### Tip:
When graphing rational functions, always check for any factor cancellations that might lead to holes in the graph!

Graph the rational function f(x) = (-10x + 20) / (x^2 - 3x - 10). Start by drawing the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote.

Learn how to graph the rational function f(x) = (-10x + 20) / (x^2 - 3x - 10) by finding vertical and horizontal asymptotes, intercepts, and analyzing the behavior of the function. This tutorial includes factoring and detailed graphing steps. Suitable for students in Grades 10-12.

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