Question

If f(x)=7x is horizontally stretched by a factor of 5, moved up 3 units, and reflected across the x-axis, which of the following is the new expression of f(x) ? Option #1: f(x)=−75x+3 Option #2: f(x)=−715x+3 Option #3: f(x)=15⋅7−x+3 Option #4: f(x)=−5⋅7x+3

Answer 1

To transform the function \( f(x) = 7x \) according to the specified operations, we will apply each transformation step by step.

1. **Horizontal Stretch by a Factor of 5**: When a function is stretched horizontally by a factor of \( k \), we replace \( x \) with \( \frac{x}{k} \). Here, \( k = 5 \):
\[
f(x) = 7\left(\frac{x}{5}\right) = \frac{7x}{5}
\]

2. **Reflection Across the X-axis**: Reflecting a function across the x-axis involves multiplying the entire function by \(-1\):
\[
f(x) = -\frac{7x}{5}
\]

3. **Moving Up 3 Units**: Moving a function up by \( c \) units means we add \( c \) to the function's value. Here, \( c = 3 \):
\[
f(x) = -\frac{7x}{5} + 3
\]

Now let's simplify this expression:
\[
f(x) = -\frac{7}{5}x + 3
\]

Next, we can check which of the given options matches this expression:

- **Option #1**: \( f(x) = -75x + 3 \) (incorrect)
- **Option #2**: \( f(x) = -\frac{7}{15}x + 3 \) (incorrect)
- **Option #3**: \( f(x) = 15 \cdot 7^{-x} + 3 \) (incorrect)
- **Option #4**: \( f(x) = -5\cdot 7x + 3 \) (incorrect)

None of the options given accurately reflect the transformation. The correct transformed function is:
\[
f(x) = -\frac{7}{5}x + 3
\]

It seems there may have been an oversight in the provided options, as none match accurately with our derived expression. Could there be a mistake in the options provided?