Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To graph the rational function [tex]\( f(x) = \frac{4}{-x + 4} \)[/tex], follow these steps:
### Step 1: Identify the form of the Function
The function [tex]\( f(x) = \frac{4}{-x + 4} \)[/tex] is a rational function. Rational functions typically have vertical and horizontal asymptotes.
### Step 2: Determine the Vertical Asymptote
The vertical asymptote occurs where the denominator is zero because the function is undefined at that point. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
So, there is a vertical asymptote at [tex]\( x = 4 \)[/tex].
### Step 3: Determine the Horizontal Asymptote
To determine the horizontal asymptote for a rational function of the form [tex]\( f(x) = \frac{a}{bx + c} \)[/tex], you take the limit as [tex]\( x \)[/tex] approaches infinity or negative infinity. For this function, as [tex]\( x \to \pm \infty \)[/tex], the term [tex]\(-x\)[/tex] dominates the constant 4 in the denominator:
[tex]\[ \lim_{x \to \pm \infty} f(x) = \lim_{x \to \pm \infty} \frac{4}{-x + 4} \approx \lim_{x \to \pm \infty} \frac{4}{-x} = 0 \][/tex]
So, there is a horizontal asymptote at [tex]\( y = 0 \)[/tex].
### Step 4: Plot Key Points
To help sketch the graph, calculate a few important points:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{4}{-0 + 4} = \frac{4}{4} = 1 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \frac{4}{-2 + 4} = \frac{4}{2} = 2 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{4}{-5 + 4} = \frac{4}{-1} = -4 \][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = \frac{4}{-8 + 4} = \frac{4}{-4} = -1 \][/tex]
- For [tex]\( x < 4 \)[/tex] but close to 4, say [tex]\( x = 3.9 \)[/tex]:
[tex]\[ f(3.9) = \frac{4}{-3.9 + 4} = \frac{4}{0.1} = 40 \][/tex]
- For [tex]\( x > 4 \)[/tex] but close to 4, say [tex]\( x = 4.1 \)[/tex]:
[tex]\[ f(4.1) = \frac{4}{-4.1 + 4} = \frac{4}{-0.1} = -40 \][/tex]
### Step 5: Sketch the Graph
1. Draw the vertical asymptote as a dashed line at [tex]\( x = 4 \)[/tex].
2. Draw the horizontal asymptote as a dashed line at [tex]\( y = 0 \)[/tex].
3. Plot the points calculated above (e.g., (0,1), (2,2), (5,-4), (8,-1), (3.9, 40), and (4.1, -40)).
4. Sketch the general shape of the graph, noting the behavior near the asymptotes:
- As [tex]\( x \)[/tex] approaches 4 from the left, the function values become very large positive (e.g., at [tex]\( x = 3.9 \)[/tex], [tex]\( f(x) = 40 \)[/tex]).
- As [tex]\( x \)[/tex] approaches 4 from the right, the function values become very large negative (e.g., at [tex]\( x = 4.1 \)[/tex], [tex]\( f(x) = -40 \)[/tex]).
- As [tex]\( x \to \pm \infty \)[/tex], [tex]\( f(x) \)[/tex] approaches the horizontal asymptote [tex]\( y = 0 \)[/tex].
By plotting these points and considering the asymptotic behavior, you get a visualization of the function's graph. The resulting graph should reflect all these characteristics, illustrating the rational function's behavior across its domain.
### Step 1: Identify the form of the Function
The function [tex]\( f(x) = \frac{4}{-x + 4} \)[/tex] is a rational function. Rational functions typically have vertical and horizontal asymptotes.
### Step 2: Determine the Vertical Asymptote
The vertical asymptote occurs where the denominator is zero because the function is undefined at that point. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -x + 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
So, there is a vertical asymptote at [tex]\( x = 4 \)[/tex].
### Step 3: Determine the Horizontal Asymptote
To determine the horizontal asymptote for a rational function of the form [tex]\( f(x) = \frac{a}{bx + c} \)[/tex], you take the limit as [tex]\( x \)[/tex] approaches infinity or negative infinity. For this function, as [tex]\( x \to \pm \infty \)[/tex], the term [tex]\(-x\)[/tex] dominates the constant 4 in the denominator:
[tex]\[ \lim_{x \to \pm \infty} f(x) = \lim_{x \to \pm \infty} \frac{4}{-x + 4} \approx \lim_{x \to \pm \infty} \frac{4}{-x} = 0 \][/tex]
So, there is a horizontal asymptote at [tex]\( y = 0 \)[/tex].
### Step 4: Plot Key Points
To help sketch the graph, calculate a few important points:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{4}{-0 + 4} = \frac{4}{4} = 1 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \frac{4}{-2 + 4} = \frac{4}{2} = 2 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = \frac{4}{-5 + 4} = \frac{4}{-1} = -4 \][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[ f(8) = \frac{4}{-8 + 4} = \frac{4}{-4} = -1 \][/tex]
- For [tex]\( x < 4 \)[/tex] but close to 4, say [tex]\( x = 3.9 \)[/tex]:
[tex]\[ f(3.9) = \frac{4}{-3.9 + 4} = \frac{4}{0.1} = 40 \][/tex]
- For [tex]\( x > 4 \)[/tex] but close to 4, say [tex]\( x = 4.1 \)[/tex]:
[tex]\[ f(4.1) = \frac{4}{-4.1 + 4} = \frac{4}{-0.1} = -40 \][/tex]
### Step 5: Sketch the Graph
1. Draw the vertical asymptote as a dashed line at [tex]\( x = 4 \)[/tex].
2. Draw the horizontal asymptote as a dashed line at [tex]\( y = 0 \)[/tex].
3. Plot the points calculated above (e.g., (0,1), (2,2), (5,-4), (8,-1), (3.9, 40), and (4.1, -40)).
4. Sketch the general shape of the graph, noting the behavior near the asymptotes:
- As [tex]\( x \)[/tex] approaches 4 from the left, the function values become very large positive (e.g., at [tex]\( x = 3.9 \)[/tex], [tex]\( f(x) = 40 \)[/tex]).
- As [tex]\( x \)[/tex] approaches 4 from the right, the function values become very large negative (e.g., at [tex]\( x = 4.1 \)[/tex], [tex]\( f(x) = -40 \)[/tex]).
- As [tex]\( x \to \pm \infty \)[/tex], [tex]\( f(x) \)[/tex] approaches the horizontal asymptote [tex]\( y = 0 \)[/tex].
By plotting these points and considering the asymptotic behavior, you get a visualization of the function's graph. The resulting graph should reflect all these characteristics, illustrating the rational function's behavior across its domain.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
