Question

Describe the shape of the function whose equation is given.

y=3log(x−2)−4


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(2 points)
Responses

an increasing curve with a horizontal asymptote
an increasing curve with a horizontal asymptote

an increasing curve with a vertical asymptote
an increasing curve with a vertical asymptote

a decreasing curve with a horizontal asymptote
a decreasing curve with a horizontal asymptote

a decreasing curve with a vertical asymptote

Answer 1

To analyze the shape of the function \( y = 3\log(x-2) - 4 \), we can consider its components:

1. **Logarithmic Function:** The base shape of the function is \( \log(x) \), which is a logarithmic function. The logarithmic function has a vertical asymptote at \( x = 0 \). However, in this case, we have \( \log(x - 2) \), which means the input to the logarithm shifts to the right by 2, resulting in a vertical asymptote at \( x = 2 \).

2. **Transformations:**
- The coefficient of 3 in front of the logarithm stretches the graph vertically but does not affect the horizontal asymptote.
- The subtraction of 4 shifts the entire graph down by 4 units.

3. **Behavior near the Asymptote:**
- As \( x \) approaches 2 from the right (i.e., \( x \to 2^+ \)), \( \log(x - 2) \) approaches \( -\infty \), so \( y \) will also approach \( -\infty \).
- As \( x \) increases beyond 2, \( y \) will increase because the logarithmic function is increasing.

From this analysis, we can conclude that the function \( y = 3\log(x-2) - 4 \) has:

- A vertical asymptote at \( x = 2 \),
- An increasing curve for \( x > 2 \) (since logarithmic functions are increasing).

Thus, the correct description of the function is:

**an increasing curve with a vertical asymptote.**