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To graph the function [tex]\( k(x) = \frac{3x - 15}{x + 4} \)[/tex], we need to consider several key points such as the vertical and horizontal asymptotes, intercepts, and behavior at various points.
1. Determine the Vertical Asymptote:
A vertical asymptote occurs where the denominator is zero, as the function becomes undefined.
[tex]\[ x + 4 = 0 \implies x = -4 \][/tex]
Thus, there is a vertical asymptote at [tex]\( x = -4 \)[/tex].
2. Determine the Horizontal Asymptote:
For the horizontal asymptote, we analyze the end behavior of the function as [tex]\( x \)[/tex] approaches [tex]\(\pm \infty\)[/tex].
The degrees of the numerator and the denominator are the same (both are 1), so the horizontal asymptote is determined by the ratio of their leading coefficients.
[tex]\[ \frac{3x}{x} = 3 \][/tex]
Therefore, the horizontal asymptote is [tex]\( y = 3 \)[/tex].
3. Find the Intercepts:
- x-intercept: Set [tex]\( k(x) = 0 \)[/tex]:
[tex]\[ \frac{3x - 15}{x + 4} = 0 \implies 3x - 15 = 0 \implies x = 5 \][/tex]
So, the x-intercept is [tex]\( (5, 0) \)[/tex].
- y-intercept: Set [tex]\( x = 0 \)[/tex]:
[tex]\[ k(0) = \frac{3(0) - 15}{0 + 4} = \frac{-15}{4} = -3.75 \][/tex]
Thus, the y-intercept is [tex]\( (0, -3.75) \)[/tex].
4. Choose Additional Points for Plotting:
To provide clarity around the vertical asymptote, choose a few values left and right of [tex]\( x = -4 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ k(-5) = \frac{3(-5) - 15}{-5 + 4} = \frac{-15 - 15}{-1} = \frac{-30}{-1} = 30 \][/tex]
For [tex]\( x = -3 \)[/tex]:
[tex]\[ k(-3) = \frac{3(-3) - 15}{-3 + 4} = \frac{-9 - 15}{1} = \frac{-24}{1} = -24 \][/tex]
For [tex]\( x = -4.1 \)[/tex]:
[tex]\[ k(-4.1) = \frac{3(-4.1) - 15}{-4.1 + 4} = \frac{-12.3 - 15}{-0.1} = \frac{-27.3}{-0.1} = 273 \][/tex]
For [tex]\( x = -3.9 \)[/tex]:
[tex]\[ k(-3.9) = \frac{3(-3.9) - 15}{-3.9 + 4} = \frac{-11.7 - 15}{0.1} = \frac{-26.7}{0.1} = -267 \][/tex]
These points help to show the rapid growth or decay as [tex]\( x \)[/tex] approaches the vertical asymptote.
### Summary
- Vertical Asymptote: [tex]\( x = -4 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 3 \)[/tex]
- x-intercept: [tex]\( (5, 0) \)[/tex]
- y-intercept: [tex]\( (0, -3.75) \)[/tex]
- Additional Points: [tex]\(( -5, 30 )\)[/tex], [tex]\(( -3, -24 )\)[/tex], [tex]\(( -4.1, 273 )\)[/tex], [tex]\(( -3.9, -267 )\)[/tex]
### Graphing Steps:
1. Plot the vertical asymptote [tex]\( x = -4 \)[/tex] with a dashed line.
2. Plot the horizontal asymptote [tex]\( y = 3 \)[/tex] with a dashed line.
3. Plot the intercepts [tex]\((5, 0)\)[/tex] and [tex]\((0, -3.75)\)[/tex].
4. Plot the additional points [tex]\(( -5, 30 )\)[/tex], [tex]\(( -3, -24 )\)[/tex], [tex]\(( -4.1, 273 )\)[/tex], [tex]\(( -3.9, -267 )\)[/tex].
5. Sketch the curve by connecting these points and ensuring the behavior near asymptotes is accurately represented. Draw the function approaching the vertical asymptote at [tex]\( x = -4 \)[/tex] from both sides and getting closer to the horizontal asymptote [tex]\( y = 3 \)[/tex] as [tex]\( x \)[/tex] moves towards [tex]\(\pm \infty\)[/tex].
By following these steps, you'll get a comprehensive graph of the function [tex]\( k(x) = \frac{3x - 15}{x + 4} \)[/tex].
1. Determine the Vertical Asymptote:
A vertical asymptote occurs where the denominator is zero, as the function becomes undefined.
[tex]\[ x + 4 = 0 \implies x = -4 \][/tex]
Thus, there is a vertical asymptote at [tex]\( x = -4 \)[/tex].
2. Determine the Horizontal Asymptote:
For the horizontal asymptote, we analyze the end behavior of the function as [tex]\( x \)[/tex] approaches [tex]\(\pm \infty\)[/tex].
The degrees of the numerator and the denominator are the same (both are 1), so the horizontal asymptote is determined by the ratio of their leading coefficients.
[tex]\[ \frac{3x}{x} = 3 \][/tex]
Therefore, the horizontal asymptote is [tex]\( y = 3 \)[/tex].
3. Find the Intercepts:
- x-intercept: Set [tex]\( k(x) = 0 \)[/tex]:
[tex]\[ \frac{3x - 15}{x + 4} = 0 \implies 3x - 15 = 0 \implies x = 5 \][/tex]
So, the x-intercept is [tex]\( (5, 0) \)[/tex].
- y-intercept: Set [tex]\( x = 0 \)[/tex]:
[tex]\[ k(0) = \frac{3(0) - 15}{0 + 4} = \frac{-15}{4} = -3.75 \][/tex]
Thus, the y-intercept is [tex]\( (0, -3.75) \)[/tex].
4. Choose Additional Points for Plotting:
To provide clarity around the vertical asymptote, choose a few values left and right of [tex]\( x = -4 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ k(-5) = \frac{3(-5) - 15}{-5 + 4} = \frac{-15 - 15}{-1} = \frac{-30}{-1} = 30 \][/tex]
For [tex]\( x = -3 \)[/tex]:
[tex]\[ k(-3) = \frac{3(-3) - 15}{-3 + 4} = \frac{-9 - 15}{1} = \frac{-24}{1} = -24 \][/tex]
For [tex]\( x = -4.1 \)[/tex]:
[tex]\[ k(-4.1) = \frac{3(-4.1) - 15}{-4.1 + 4} = \frac{-12.3 - 15}{-0.1} = \frac{-27.3}{-0.1} = 273 \][/tex]
For [tex]\( x = -3.9 \)[/tex]:
[tex]\[ k(-3.9) = \frac{3(-3.9) - 15}{-3.9 + 4} = \frac{-11.7 - 15}{0.1} = \frac{-26.7}{0.1} = -267 \][/tex]
These points help to show the rapid growth or decay as [tex]\( x \)[/tex] approaches the vertical asymptote.
### Summary
- Vertical Asymptote: [tex]\( x = -4 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 3 \)[/tex]
- x-intercept: [tex]\( (5, 0) \)[/tex]
- y-intercept: [tex]\( (0, -3.75) \)[/tex]
- Additional Points: [tex]\(( -5, 30 )\)[/tex], [tex]\(( -3, -24 )\)[/tex], [tex]\(( -4.1, 273 )\)[/tex], [tex]\(( -3.9, -267 )\)[/tex]
### Graphing Steps:
1. Plot the vertical asymptote [tex]\( x = -4 \)[/tex] with a dashed line.
2. Plot the horizontal asymptote [tex]\( y = 3 \)[/tex] with a dashed line.
3. Plot the intercepts [tex]\((5, 0)\)[/tex] and [tex]\((0, -3.75)\)[/tex].
4. Plot the additional points [tex]\(( -5, 30 )\)[/tex], [tex]\(( -3, -24 )\)[/tex], [tex]\(( -4.1, 273 )\)[/tex], [tex]\(( -3.9, -267 )\)[/tex].
5. Sketch the curve by connecting these points and ensuring the behavior near asymptotes is accurately represented. Draw the function approaching the vertical asymptote at [tex]\( x = -4 \)[/tex] from both sides and getting closer to the horizontal asymptote [tex]\( y = 3 \)[/tex] as [tex]\( x \)[/tex] moves towards [tex]\(\pm \infty\)[/tex].
By following these steps, you'll get a comprehensive graph of the function [tex]\( k(x) = \frac{3x - 15}{x + 4} \)[/tex].
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