Which of the following describes the shape of the function family for y=−4log3(x−1)+5.1

Question

?(1 point)
Responses

waves that go up and down at regular intervals
waves that go up and down at regular intervals

a curve with a vertical asymptote and decreasing growth/decay
a curve with a vertical asymptote and decreasing growth/decay

repeating S-shaped curves that are separated by vertical asymptotes
repeating S-shaped curves that are separated by vertical asymptotes

a V-shaped curve

Answer 1

The function \( y = -4\log_3(x-1) + 5.1 \) describes a logarithmic function that has been transformed.

1. The term \( \log_3(x-1) \) indicates a logarithmic function where the argument \( x-1 \) shifts the graph to the right by 1 unit.
2. The negative coefficient (-4) indicates that the graph is reflected over the x-axis and vertically stretched.
3. The +5.1 shifts the entire graph up by 5.1 units.
4. Additionally, logarithmic functions have vertical asymptotes. In this case, there is a vertical asymptote at \( x = 1 \) (since the function goes to negative infinity as \( x \) approaches 1 from the right).

Given these observations, the best description of the function's shape is:

a curve with a vertical asymptote and decreasing growth/decay.