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Sagot :
To graph the rational function [tex]\( f(x) = \frac{2}{-x-2} \)[/tex], follow these steps:
### Step 1: Determine the Vertical Asymptote
The vertical asymptote occurs where the denominator is zero. Here, the denominator is [tex]\(-x-2\)[/tex]. Setting this equal to zero and solving for [tex]\( x \)[/tex]:
[tex]\[ -x - 2 = 0 \][/tex]
[tex]\[ -x = 2 \][/tex]
[tex]\[ x = -2 \][/tex]
So, the vertical asymptote is at [tex]\( x = -2 \)[/tex].
### Step 2: Determine the Horizontal Asymptote
For large values of [tex]\( x \)[/tex], the rational function [tex]\( f(x) \)[/tex] approaches a horizontal asymptote. As [tex]\( x \)[/tex] approaches either positive or negative infinity, the term [tex]\(-x - 2\)[/tex] grows large in magnitude, and thus [tex]\( f(x) \)[/tex] approaches zero. Hence, the horizontal asymptote is:
[tex]\[ y = 0 \][/tex]
### Step 3: Plot Points Around the Vertical Asymptote
To understand the behavior of the function around the vertical asymptote, we will plot points on either side of [tex]\( x = -2 \)[/tex].
1. Points to the left of [tex]\( x = -2 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = \frac{2}{-(-3) - 2} = \frac{2}{3 - 2} = \frac{2}{1} = 2 \][/tex]
So the point is [tex]\((-3, 2)\)[/tex].
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = \frac{2}{-(-4) - 2} = \frac{2}{4 - 2} = \frac{2}{2} = 1 \][/tex]
So the point is [tex]\((-4, 1)\)[/tex].
2. Points to the right of [tex]\( x = -2 \)[/tex]:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \frac{2}{-(-1) - 2} = \frac{2}{1 - 2} = \frac{2}{-1} = -2 \][/tex]
So the point is [tex]\((-1, -2)\)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{2}{-0 - 2} = \frac{2}{-2} = -1 \][/tex]
So the point is [tex]\((0, -1)\)[/tex].
### Step 4: Draw the Graph
1. Draw the vertical asymptote as a dashed line at [tex]\( x = -2 \)[/tex].
2. Draw the horizontal asymptote as a dashed line at [tex]\( y = 0 \)[/tex].
3. Plot the points [tex]\((-3, 2)\)[/tex] and [tex]\((-4, 1)\)[/tex] to the left of [tex]\( x = -2 \)[/tex].
4. Plot the points [tex]\((-1, -2)\)[/tex] and [tex]\((0, -1)\)[/tex] to the right of [tex]\( x = -2 \)[/tex].
5. Sketch the curve of the function approaching the vertical asymptote [tex]\( x = -2 \)[/tex] from both sides and also approaching the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] goes to positive and negative infinity.
### Final Graph
The final graph will show the behavior of the function with the vertical asymptote at [tex]\( x = -2 \)[/tex], the horizontal asymptote at [tex]\( y = 0 \)[/tex], and the plotted points correctly illustrating the function's shape.
### Step 1: Determine the Vertical Asymptote
The vertical asymptote occurs where the denominator is zero. Here, the denominator is [tex]\(-x-2\)[/tex]. Setting this equal to zero and solving for [tex]\( x \)[/tex]:
[tex]\[ -x - 2 = 0 \][/tex]
[tex]\[ -x = 2 \][/tex]
[tex]\[ x = -2 \][/tex]
So, the vertical asymptote is at [tex]\( x = -2 \)[/tex].
### Step 2: Determine the Horizontal Asymptote
For large values of [tex]\( x \)[/tex], the rational function [tex]\( f(x) \)[/tex] approaches a horizontal asymptote. As [tex]\( x \)[/tex] approaches either positive or negative infinity, the term [tex]\(-x - 2\)[/tex] grows large in magnitude, and thus [tex]\( f(x) \)[/tex] approaches zero. Hence, the horizontal asymptote is:
[tex]\[ y = 0 \][/tex]
### Step 3: Plot Points Around the Vertical Asymptote
To understand the behavior of the function around the vertical asymptote, we will plot points on either side of [tex]\( x = -2 \)[/tex].
1. Points to the left of [tex]\( x = -2 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = \frac{2}{-(-3) - 2} = \frac{2}{3 - 2} = \frac{2}{1} = 2 \][/tex]
So the point is [tex]\((-3, 2)\)[/tex].
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = \frac{2}{-(-4) - 2} = \frac{2}{4 - 2} = \frac{2}{2} = 1 \][/tex]
So the point is [tex]\((-4, 1)\)[/tex].
2. Points to the right of [tex]\( x = -2 \)[/tex]:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \frac{2}{-(-1) - 2} = \frac{2}{1 - 2} = \frac{2}{-1} = -2 \][/tex]
So the point is [tex]\((-1, -2)\)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{2}{-0 - 2} = \frac{2}{-2} = -1 \][/tex]
So the point is [tex]\((0, -1)\)[/tex].
### Step 4: Draw the Graph
1. Draw the vertical asymptote as a dashed line at [tex]\( x = -2 \)[/tex].
2. Draw the horizontal asymptote as a dashed line at [tex]\( y = 0 \)[/tex].
3. Plot the points [tex]\((-3, 2)\)[/tex] and [tex]\((-4, 1)\)[/tex] to the left of [tex]\( x = -2 \)[/tex].
4. Plot the points [tex]\((-1, -2)\)[/tex] and [tex]\((0, -1)\)[/tex] to the right of [tex]\( x = -2 \)[/tex].
5. Sketch the curve of the function approaching the vertical asymptote [tex]\( x = -2 \)[/tex] from both sides and also approaching the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] goes to positive and negative infinity.
### Final Graph
The final graph will show the behavior of the function with the vertical asymptote at [tex]\( x = -2 \)[/tex], the horizontal asymptote at [tex]\( y = 0 \)[/tex], and the plotted points correctly illustrating the function's shape.
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