Question

question 13 (1 point)
the function $y = \log - (x - 5) + 3$ is

a) translated up 1 unit

b) reflected in the x-axis

c) reflected in the y-axis

d) compressed vertically by a factor of 3

Explanation:

Step1: Recall Transformations

For a logarithmic function \( y = \log_b(x) \), transformations:

• Vertical reflection: \( y = -\log_b(x) \) (reflect over x - axis).
• Horizontal reflection: \( y=\log_b(-x) \) (reflect over y - axis).
• Vertical translation: \( y=\log_b(x)+k \) (up if \( k>0 \), down if \( k < 0 \)).
• Vertical compression/stretch: \( y = a\log_b(x) \) (stretch if \( |a|>1 \), compress if \( 0<|a|<1 \)).

Step2: Analyze Given Function

The function is \( y=-\log(x - 5)+3 \) (assuming base 10, but base doesn't affect reflection/translation analysis here).

• Option a: The vertical translation is +3 (up 3 units), not 1. Eliminate a.
• Option b: The negative sign in front of \( \log(x - 5) \) means \( y = -\log(x - 5)+3 \) is a vertical reflection (over x - axis) of \( y=\log(x - 5)+3 \). This matches the form of vertical reflection (\( y = -f(x)+k \) where \( f(x)=\log(x - 5) \)).
• Option c: For reflection over y - axis, we need \( \log(-(x - 5))=\log(-x + 5) \), but our function has \( \log(x - 5) \) with a negative sign in front of the log, not inside the log argument. Eliminate c.
• Option d: The coefficient of the log term is - 1 (not related to vertical compression by factor 3, since vertical compression would be a coefficient between 0 and 1 or greater than 1, and here the coefficient is - 1 which is a reflection, not compression). Eliminate d.

Answer:

b) reflected in the x - axis