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!