To graph the rational function $f(x) = \frac{2x - 1}{-2x^2 - 5x + 3}$, we will use a six-step procedure to analyze the function and identify important features such as asymptotes, intercepts, and behavior near the asymptotes.

Step 1: Factor the Function

First, factor the numerator and denominator as much as possible.

Numerator:

$2x - 1$ is already factored.

Denominator:

To factor $-2x^2 - 5x + 3$:

1. Factor out $-1$: $-2x^2 - 5x + 3 = -(2x^2 + 5x - 3)$.

2. Factor $2x^2 + 5x - 3$:

   $2x^2 + 5x - 3 = (2x - 1)(x + 3)$

Therefore: $f(x) = \frac{2x - 1}{-(2x - 1)(x + 3)}$ which simplifies to: $f(x) = -\frac{1}{x + 3}, \quad x \neq \frac{1}{2}$

Step 2: Find the Domain

The function $f(x) = -\frac{1}{x + 3}$ is undefined when the denominator is zero. Setting $x + 3 = 0$, we find: $x = -3$ Thus, the domain is $x \neq -3$ and $x \neq \frac{1}{2}$.

Step 3: Find the Intercepts

x-Intercept:

To find the x-intercept, set $f(x) = 0$: $-\frac{1}{x + 3} = 0$ No solution exists, so there are no x-intercepts.

y-Intercept:

To find the y-intercept, evaluate $f(0)$: $f(0) = -\frac{1}{0 + 3} = -\frac{1}{3}$ Thus, the y-intercept is $(0, -\frac{1}{3})$.

Step 4: Find the Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, specifically where the simplified denominator is zero. Since $x + 3 = 0$, the vertical asymptote is: $x = -3$

Step 5: Find the Horizontal or Oblique Asymptote

Since the degree of the numerator is 0 (constant) and the degree of the denominator is 1 (linear), the horizontal asymptote is $y = 0$.

Step 6: Analyze the Behavior and Sketch the Graph

1. Behavior Near the Asymptote $x = -3$: As $x \to -3^+$, $f(x) \to +\infty$ and as $x \to -3^-$, $f(x) \to -\infty$.

2. Behavior as $x \to \pm\infty$: Since the horizontal asymptote is $y = 0$, the function approaches zero as $x$ moves far left or far right.

Graph

• Plot the y-intercept at $(0, -\frac{1}{3})$.
• Draw the vertical asymptote as a dashed line at $x = -3$.
• Draw the horizontal asymptote as a dashed line at $y = 0$.

Would you like a visual sketch for this graph?

Relative Questions

1. How do you determine the asymptotes of a rational function in general?
2. What steps are involved in finding x- and y-intercepts for rational functions?
3. How does the degree of the numerator and denominator affect the asymptote types?
4. How do you analyze the behavior of the function near its asymptotes?
5. How does simplifying the function affect the domain?

Tip

Always simplify rational functions before finding their asymptotes and intercepts, as it makes identifying these features easier.