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)=−7^5x+3 Option #2: f(x)=−7^1/5x+3 Option #3: f(x)=1/5⋅7^−x+3 Option #4: f(x)=−5⋅7^x+3

Question

GPT-4o mini · Answer

To transform the function $ f(x) = 7x $ according to the stated operations, we need to apply each transformation step by step. 1. **Horizontal Stretch by a factor of 5**: To horizontally stretch a function by a factor of $ k $, we replace $ x $ with $ \frac{x}{k} $. So, we replace $ x $ with $ \frac{x}{5} $: $$ f(x) = 7\left(\frac{x}{5}\right) = \frac{7x}{5} $$ 2. **Moved up 3 units**: This is done by adding 3 to the function: $$ f(x) = \frac{7x}{5} + 3 $$ 3. **Reflected across the x-axis**: To reflect the function across the x-axis, we multiply the entire function by -1: $$ f(x) = -\left(\frac{7x}{5} + 3\right) = -\frac{7x}{5} - 3 $$ So, the transformed function is: $$ f(x) = -\frac{7x}{5} - 3 $$ Let's analyze the options provided: 1. $ \text{Option #1: } f(x)=-7^{5}x+3 $ → Incorrect 2. $ \text{Option #2: } f(x)=-7^{1/5}x+3 $ → Incorrect 3. $ \text{Option #3: } f(x)=\frac{1}{5}\cdot7^{-x}+3 $ → Incorrect 4. $ \text{Option #4: } f(x)=-5\cdot7^x+3 $ → Incorrect None of the options match the final transformed function $ f(x) = -\frac{7x}{5} - 3 $. It appears that the correct transformation is not represented accurately in the given options.