Alright class, let's break this down step-by-step.
To graph [tex]\( f(x) = \frac{-3}{x-1} \)[/tex], we need to identify the asymptotes and understand the behavior of the function as [tex]\( x \)[/tex] approaches certain values.
### Step 1: Identify any vertical asymptotes
A vertical asymptote occurs where the denominator of the function is zero, which results in the function value approaching infinity (either positive or negative).
In [tex]\( f(x) = \frac{-3}{x-1} \)[/tex], the denominator is [tex]\( x-1 \)[/tex]. Set the denominator equal to zero to find the vertical asymptote:
[tex]\[ x - 1 = 0 \][/tex]
[tex]\[ x = 1 \][/tex]
So, there is a vertical asymptote at [tex]\( x = 1 \)[/tex].
### Step 2: Identify any horizontal asymptotes
To determine the horizontal asymptote, we need to evaluate the behavior of the function as [tex]\( x \)[/tex] approaches positive or negative infinity.
Since the degree of the numerator (which is 0, as there is no [tex]\( x \)[/tex] term in [tex]\( -3 \)[/tex]) is less than the degree of the denominator (which is 1 in [tex]\( x-1 \)[/tex]), the horizontal asymptote is:
[tex]\[ y = 0 \][/tex]
### Step 3: Plot the key features
Now, let's gather these results and plot the function on a graph:
- The vertical asymptote is a dashed vertical line at [tex]\( x = 1 \)[/tex].
- The horizontal asymptote is a dashed horizontal line at [tex]\( y = 0 \)[/tex].
### Step 4: Understand the behavior of the function
Beyond the asymptotes, we should know how the function behaves near these lines.
- As [tex]\( x \)[/tex] approaches 1 from the left (i.e., [tex]\( x \to 1^- \)[/tex]), [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches 1 from the right (i.e., [tex]\( x \to 1^+ \)[/tex]), [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] (or [tex]\( -\infty \)[/tex]), [tex]\( f(x) \to 0 \)[/tex].
### Step 5: Draw the graph
1. Draw a dashed vertical line at [tex]\( x = 1 \)[/tex].
2. Draw a dashed horizontal line at [tex]\( y = 0 \)[/tex].
3. Sketch the function based on its behavior:
- For [tex]\( x < 1 \)[/tex], the curve will approach the vertical asymptote [tex]\( x = 1 \)[/tex] from below and the horizontal asymptote [tex]\( y = 0 \)[/tex] from a negative value.
- For [tex]\( x > 1 \)[/tex], the curve will approach the vertical asymptote [tex]\( x = 1 \)[/tex] from above and the horizontal asymptote [tex]\( y = 0 \)[/tex] from a positive value.
With all these steps, you should have a clear depiction of the graph of [tex]\( f(x) = \frac{-3}{x-1} \)[/tex] with the vertical asymptote at [tex]\( x = 1 \)[/tex] and the horizontal asymptote at [tex]\( y = 0 \)[/tex].
