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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.
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