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Let's analyze the given function [tex]\( y = \frac{20 + 8x}{x} \)[/tex] to understand its behavior:
1. Simplification of the function:
[tex]\[ y = \frac{20 + 8x}{x} = \frac{20}{x} + 8 \][/tex]
2. Horizontal asymptote:
To find the horizontal asymptote, we observe the behavior of the function as [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]). When [tex]\( x \)[/tex] becomes very large, the term [tex]\(\frac{20}{x}\)[/tex] becomes very small (approaches 0). Therefore, the function [tex]\( y \)[/tex] approaches:
[tex]\[ y = \frac{20}{x} + 8 \approx 0 + 8 = 8 \][/tex]
Therefore, the horizontal asymptote is [tex]\( y = 8 \)[/tex].
3. Vertical asymptote:
The vertical asymptote occurs where the function is undefined. In this case, the function is undefined when [tex]\( x = 0 \)[/tex], as division by zero is undefined. Hence, there is a vertical asymptote at [tex]\( x = 0 \)[/tex].
4. Interpreting the statements:
- "The horizontal asymptote [tex]\( y = 0 \)[/tex] means the cost approaches zero as the number of people increases."
This statement is incorrect because the horizontal asymptote is not [tex]\( y = 0 \)[/tex], it is [tex]\( y = 8 \)[/tex].
- "The horizontal asymptote [tex]\( y = 8 \)[/tex] means the average cost approaches [tex]\( \$ 8 \)[/tex] as the number of people increases."
This statement is correct. Since the horizontal asymptote is [tex]\( y = 8 \)[/tex], it implies that as the number of people increases, the average cost per person approaches [tex]\( \$ 8 \)[/tex].
- "The vertical asymptote [tex]\( x = 0 \)[/tex] means the cost approaches zero as the number of people increases."
This statement is incorrect. The vertical asymptote at [tex]\( x = 0 \)[/tex] means the function is undefined at that point; it does not imply that the cost approaches zero as the number of people increases.
- "The vertical asymptote [tex]\( x = 8 \)[/tex] means the cost approaches zero as the number of people increases."
This statement is incorrect because there is no vertical asymptote at [tex]\( x = 8 \)[/tex]. The only vertical asymptote is at [tex]\( x = 0 \)[/tex].
Therefore, the true statement about the graph of the function is:
[tex]\[ \text{The horizontal asymptote } y = 8 \text{ means the average cost approaches } \$ 8 \text{ as the number of people increases.} \][/tex]
1. Simplification of the function:
[tex]\[ y = \frac{20 + 8x}{x} = \frac{20}{x} + 8 \][/tex]
2. Horizontal asymptote:
To find the horizontal asymptote, we observe the behavior of the function as [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]). When [tex]\( x \)[/tex] becomes very large, the term [tex]\(\frac{20}{x}\)[/tex] becomes very small (approaches 0). Therefore, the function [tex]\( y \)[/tex] approaches:
[tex]\[ y = \frac{20}{x} + 8 \approx 0 + 8 = 8 \][/tex]
Therefore, the horizontal asymptote is [tex]\( y = 8 \)[/tex].
3. Vertical asymptote:
The vertical asymptote occurs where the function is undefined. In this case, the function is undefined when [tex]\( x = 0 \)[/tex], as division by zero is undefined. Hence, there is a vertical asymptote at [tex]\( x = 0 \)[/tex].
4. Interpreting the statements:
- "The horizontal asymptote [tex]\( y = 0 \)[/tex] means the cost approaches zero as the number of people increases."
This statement is incorrect because the horizontal asymptote is not [tex]\( y = 0 \)[/tex], it is [tex]\( y = 8 \)[/tex].
- "The horizontal asymptote [tex]\( y = 8 \)[/tex] means the average cost approaches [tex]\( \$ 8 \)[/tex] as the number of people increases."
This statement is correct. Since the horizontal asymptote is [tex]\( y = 8 \)[/tex], it implies that as the number of people increases, the average cost per person approaches [tex]\( \$ 8 \)[/tex].
- "The vertical asymptote [tex]\( x = 0 \)[/tex] means the cost approaches zero as the number of people increases."
This statement is incorrect. The vertical asymptote at [tex]\( x = 0 \)[/tex] means the function is undefined at that point; it does not imply that the cost approaches zero as the number of people increases.
- "The vertical asymptote [tex]\( x = 8 \)[/tex] means the cost approaches zero as the number of people increases."
This statement is incorrect because there is no vertical asymptote at [tex]\( x = 8 \)[/tex]. The only vertical asymptote is at [tex]\( x = 0 \)[/tex].
Therefore, the true statement about the graph of the function is:
[tex]\[ \text{The horizontal asymptote } y = 8 \text{ means the average cost approaches } \$ 8 \text{ as the number of people increases.} \][/tex]
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