Question

state the transformations, in proper order, that are needed to turn f(x)=- 8\log_{10}x+6 into f(x)=4\log_{10}x+1.
\bigcirc a)
reflection about the x-axis, vertical compression by a factor of \frac{1}{2}, vertical translation 5 units down.
\bigcirc b)
vertical translation 5 units down, vertical compression by a factor of \frac{1}{2}, reflection about the x-axis.
\bigcirc c) reflection about the x-axis, vertical translation 5 units down, vertical compression by a factor of \frac{1}{2}.
\bigcirc d) reflection about the y-axis, vertical stretch by a factor of 2, vertical translation 5 units up.
question 6 (1 point)

Explanation:

Step1: Analyze the coefficient of the log term

The original function is \( f(x) = -8\log_{10}x + 6 \), and the target is \( f(x) = 4\log_{10}x + 1 \). First, we deal with the sign of the coefficient. A reflection about the \( x \)-axis changes \( -8\log_{10}x \) to \( 8\log_{10}x \) (since reflecting \( y = -8\log_{10}x \) over the \( x \)-axis gives \( y = 8\log_{10}x \)).

Step2: Analyze the vertical compression

Next, we need to go from \( 8\log_{10}x \) to \( 4\log_{10}x \). A vertical compression by a factor of \( \frac{1}{2} \) (because \( 8\times\frac{1}{2}=4 \)) will transform \( 8\log_{10}x \) to \( 4\log_{10}x \).

Step3: Analyze the vertical translation

Now, we deal with the constant term. The original constant term after the first two transformations (reflection and compression) is \( 6 \), and we need to get to \( 1 \). The difference is \( 1 - 6=- 5 \), so a vertical translation of \( 5 \) units down (since we subtract \( 5 \) from the constant term) will change \( 6 \) to \( 1 \). So the order is: reflection about the \( x \)-axis, vertical compression by a factor of \( \frac{1}{2} \), vertical translation \( 5 \) units down. This matches option a.

Answer:

a) reflection about the \( x \)-axis, vertical compression by a factor of \( \frac{1}{2} \), vertical translation 5 units down.