Question

Consider the logarithmic function, f(x)=logx.


Write an expression for f(x)
after it undergoes the following transformations:

shifted 3
units left
shifted up 5
units
stretched vertically by a factor of 2

reflected across the x
axis
(2 points)

Answer 1

To apply the specified transformations to the function \( f(x) = \log x \), we will do them step-by-step.

1. **Shifted 3 units left**: To shift a function \( f(x) \) to the left by \( h \) units, we replace \( x \) with \( x + h \). For a 3-unit left shift, this gives us \( f(x + 3) = \log(x + 3) \).

2. **Shifted up 5 units**: To shift a function up by \( k \) units, we add \( k \) to the entire function. Therefore, we take our previous result and add 5:
\[
g(x) = \log(x + 3) + 5
\]

3. **Stretched vertically by a factor of 2**: To stretch a function vertically by a factor of \( a \), we multiply the function by \( a \). Here, we multiply by 2:
\[
h(x) = 2 \left( \log(x + 3) + 5 \right) = 2 \log(x + 3) + 10
\]

4. **Reflected across the x-axis**: To reflect a function across the x-axis, we multiply the entire function by -1. Thus, we have:
\[
f(x) = -\left(2 \log(x + 3) + 10\right) = -2 \log(x + 3) - 10
\]

Putting all transformations together, the final expression for \( f(x) \) after all transformations is:
\[
f(x) = -2 \log(x + 3) - 10
\]