Sure, let's start by graphing the rational function [tex]\( f(x) = \frac{-x+5}{-2x+1} \)[/tex].
### Step 1: Vertical Asymptote
To find the vertical asymptote, set the denominator equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[ -2x + 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex],
[tex]\[ -2x = -1 \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
So, the vertical asymptote is [tex]\( x = \frac{1}{2} \)[/tex].
### Step 2: Horizontal Asymptote
To find the horizontal asymptote, consider the degrees and leading coefficients of the numerator and the denominator. In this function, both the numerator and the denominator are linear (degree 1).
The horizontal asymptote is given by the ratio of the leading coefficients:
[tex]\[ \lim_{{x \to \infty}} \frac{-x+5}{-2x+1} = \frac{-1}{-2} = \frac{1}{2} \][/tex]
So, the horizontal asymptote is [tex]\( y = \frac{1}{2} \)[/tex].
### Step 3: X-intercept
To find the x-intercept, set the numerator equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[ -x + 5 = 0 \][/tex]
Solving for [tex]\( x \)[/tex],
[tex]\[ x = 5 \][/tex]
So, the x-intercept is at [tex]\( (5, 0) \)[/tex].
### Step 4: Y-intercept
To find the y-intercept, evaluate the function at [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = \frac{-0 + 5}{-2(0) + 1} = \frac{5}{1} = 5 \][/tex]
So, the y-intercept is at [tex]\( (0, 5) \)[/tex].
### Summary
Now we have all the key points and asymptotes needed to graph the rational function:
- Vertical asymptote: [tex]\( x = \frac{1}{2} \)[/tex]
- Horizontal asymptote: [tex]\( y = \frac{1}{2} \)[/tex]
- X-intercept: [tex]\( (5, 0) \)[/tex]
- Y-intercept: [tex]\( (0, 5) \)[/tex]
### Sketching the Graph
1. Draw a dashed vertical line at [tex]\( x = \frac{1}{2} \)[/tex].
2. Draw a dashed horizontal line at [tex]\( y = \frac{1}{2} \)[/tex].
3. Plot the x-intercept [tex]\( (5, 0) \)[/tex] and y-intercept [tex]\( (0, 5) \)[/tex].
4. Sketch the curve, approaching the asymptotes and passing through the intercepts.
With these elements, you should be able to accurately graph the function [tex]\( f(x) = \frac{-x+5}{-2x+1} \)[/tex].
