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Sagot :
To graph the rational function
[tex]\[ f(x) = \frac{x + 5}{-2x - 1} \][/tex]
we will start by determining the vertical and horizontal asymptotes, and then plot some points for more precision in drawing the graph. Here is a step-by-step breakdown:
### Step 1: Determine Vertical Asymptote
A vertical asymptote occurs where the denominator of the rational function is zero and the numerator is not zero. In this case, the denominator is [tex]\(-2x - 1\)[/tex].
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -2x - 1 = 0 \][/tex]
[tex]\[ -2x = 1 \][/tex]
[tex]\[ x = -\frac{1}{2} \][/tex]
So, there is a vertical asymptote at [tex]\( x = -\frac{1}{2} \)[/tex].
### Step 2: Determine Horizontal Asymptote
The horizontal asymptote of a rational function is determined by the degrees of the numerator and the denominator. Both the numerator and denominator here are linear (degree 1), so the horizontal asymptote is the ratio of the leading coefficients.
Leading coefficient of [tex]\( x \)[/tex] in the numerator: 1
Leading coefficient of [tex]\( x \)[/tex] in the denominator: -2
The horizontal asymptote is:
[tex]\[ y = \frac{1}{-2} = -\frac{1}{2} \][/tex]
### Step 3: Plot Points
Let's plot points to provide a clearer picture of the function's behavior around the asymptotes:
#### Point for [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = \frac{-2 + 5}{-2(-2) - 1} = \frac{3}{4 - 1} = \frac{3}{3} = 1 \][/tex]
Coordinate: [tex]\( (-2, 1) \)[/tex]
#### Point for [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \frac{-1 + 5}{-2(-1) - 1} = \frac{4}{2 - 1} = \frac{4}{1} = 4 \][/tex]
Coordinate: [tex]\( (-1, 4) \)[/tex]
#### Point for [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{0 + 5}{-2(0) - 1} = \frac{5}{-1} = -5 \][/tex]
Coordinate: [tex]\( (0, -5) \)[/tex]
#### Point for [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \frac{1 + 5}{-2(1) - 1} = \frac{6}{-2 - 1} = \frac{6}{-3} = -2 \][/tex]
Coordinate: [tex]\( (1, -2) \)[/tex]
### Step 4: Draw the Graph
1. Vertical Asymptote: Draw a dashed vertical line at [tex]\( x = -\frac{1}{2} \)[/tex].
2. Horizontal Asymptote: Draw a dashed horizontal line at [tex]\( y = -\frac{1}{2} \)[/tex].
3. Points to Plot:
- [tex]\( (-2, 1) \)[/tex]
- [tex]\( (-1, 4) \)[/tex]
- [tex]\( (0, -5) \)[/tex]
- [tex]\( (1, -2) \)[/tex]
By plotting these points and asymptotes, you can sketch the curve of the function, ensuring it approaches the asymptotes appropriately. The function will likely have two branches divided by the vertical asymptote at [tex]\( x = -\frac{1}{2} \)[/tex], displaying typical rational function behavior.
Once these elements are plotted, you should be able to visualize and sketch the graph of [tex]\( f(x) = \frac{x + 5}{-2x - 1} \)[/tex] accurately.
[tex]\[ f(x) = \frac{x + 5}{-2x - 1} \][/tex]
we will start by determining the vertical and horizontal asymptotes, and then plot some points for more precision in drawing the graph. Here is a step-by-step breakdown:
### Step 1: Determine Vertical Asymptote
A vertical asymptote occurs where the denominator of the rational function is zero and the numerator is not zero. In this case, the denominator is [tex]\(-2x - 1\)[/tex].
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -2x - 1 = 0 \][/tex]
[tex]\[ -2x = 1 \][/tex]
[tex]\[ x = -\frac{1}{2} \][/tex]
So, there is a vertical asymptote at [tex]\( x = -\frac{1}{2} \)[/tex].
### Step 2: Determine Horizontal Asymptote
The horizontal asymptote of a rational function is determined by the degrees of the numerator and the denominator. Both the numerator and denominator here are linear (degree 1), so the horizontal asymptote is the ratio of the leading coefficients.
Leading coefficient of [tex]\( x \)[/tex] in the numerator: 1
Leading coefficient of [tex]\( x \)[/tex] in the denominator: -2
The horizontal asymptote is:
[tex]\[ y = \frac{1}{-2} = -\frac{1}{2} \][/tex]
### Step 3: Plot Points
Let's plot points to provide a clearer picture of the function's behavior around the asymptotes:
#### Point for [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = \frac{-2 + 5}{-2(-2) - 1} = \frac{3}{4 - 1} = \frac{3}{3} = 1 \][/tex]
Coordinate: [tex]\( (-2, 1) \)[/tex]
#### Point for [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \frac{-1 + 5}{-2(-1) - 1} = \frac{4}{2 - 1} = \frac{4}{1} = 4 \][/tex]
Coordinate: [tex]\( (-1, 4) \)[/tex]
#### Point for [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \frac{0 + 5}{-2(0) - 1} = \frac{5}{-1} = -5 \][/tex]
Coordinate: [tex]\( (0, -5) \)[/tex]
#### Point for [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \frac{1 + 5}{-2(1) - 1} = \frac{6}{-2 - 1} = \frac{6}{-3} = -2 \][/tex]
Coordinate: [tex]\( (1, -2) \)[/tex]
### Step 4: Draw the Graph
1. Vertical Asymptote: Draw a dashed vertical line at [tex]\( x = -\frac{1}{2} \)[/tex].
2. Horizontal Asymptote: Draw a dashed horizontal line at [tex]\( y = -\frac{1}{2} \)[/tex].
3. Points to Plot:
- [tex]\( (-2, 1) \)[/tex]
- [tex]\( (-1, 4) \)[/tex]
- [tex]\( (0, -5) \)[/tex]
- [tex]\( (1, -2) \)[/tex]
By plotting these points and asymptotes, you can sketch the curve of the function, ensuring it approaches the asymptotes appropriately. The function will likely have two branches divided by the vertical asymptote at [tex]\( x = -\frac{1}{2} \)[/tex], displaying typical rational function behavior.
Once these elements are plotted, you should be able to visualize and sketch the graph of [tex]\( f(x) = \frac{x + 5}{-2x - 1} \)[/tex] accurately.
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