To transform the function \( f(x) = 7x \) according to the specified operations, we will apply each transformation step by step.
-
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, \( k = 5 \): \[ f(x) = 7\left(\frac{x}{5}\right) = \frac{7x}{5} \]
-
Reflection Across the X-axis: Reflecting a function across the x-axis involves multiplying the entire function by \(-1\): \[ f(x) = -\frac{7x}{5} \]
-
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?