Questions: Write a formula for the function that results when the toolkit function f(x) = √x is reflected across the x-axis, stretched horizontally by a factor of 2, and shifted left 3 units and down 4 units.

Solution Steps

To transform the function \( f(x) = \sqrt{x} \) as described, we need to apply several transformations in sequence. First, reflecting across the \( x \)-axis involves multiplying the function by \(-1\). Stretching horizontally by a factor of 2 means replacing \( x \) with \( \frac{x}{2} \). Shifting left by 3 units involves replacing \( x \) with \( x + 3 \). Finally, shifting down by 4 units means subtracting 4 from the entire function. Combining these transformations gives us the new function.

The original function is given by

\[ f(x) = \sqrt{x} \]

Reflecting the function across the \( x \)-axis results in

\[ f_{\text{reflected}}(x) = -\sqrt{x} \]

Stretching the function horizontally by a factor of 2 modifies the input, yielding

\[ f_{\text{stretched}}(x) = -\sqrt{\frac{x}{2}} = -\frac{\sqrt{2}}{2} \sqrt{x} \]

Shifting the function left by 3 units results in

\[ f_{\text{shifted\_left}}(x) = -\sqrt{\frac{x + 3}{2}} = -\frac{\sqrt{2}}{2} \sqrt{x + 3} \]

Finally, shifting the function down by 4 units gives us the transformed function:

\[ f_{\text{transformed}}(x) = -\frac{\sqrt{2}}{2} \sqrt{x + 3} - 4 \]

Final Answer