Questions: Write a formula for a function g whose graph is similar to f(x) but shifted to the left 4 units and reflected about the y-axis. f(x) = sqrt(x)

Solution Steps

The original function is given as $f(x)$. The transformations to be applied include shifts, reflections, stretches, and compressions.

Horizontal transformations include shifts and stretches/compressions: In this question , $h=4$, we replace $x$ with $(x - 4)$.

• For horizontal stretching or compression, we replace $x$ with $x/b$ where $b > 1$ indicates compression and $0 < b < 1$ indicates stretching. Reflection across the $y$-axis is a special case with $b = -1$.

Vertical transformations include shifts and stretches/compressions:

• For a shift up by $k$ units, we add $k$ to the function.
• For a shift down by $k$ units, we subtract $k$ from the function.
• For vertical stretching or compression, we multiply the function by $a$ where $a > 1$ indicates stretching and $0 < a < 1$ indicates compression. Reflection across the $x$-axis is a special case with $a = -1$.

The transformed function $g(x)$ can be represented as: $$g(x) = a \cdot f(b(x + h)) + k$$,that is $g(x)=-1\cdot f(1(x - 4)) + 0$. $g(x)=-1\cdot (x-4) + 0$. where $a$ controls vertical stretching/compression and reflection across the $x$-axis, $b$ controls horizontal stretching/compression and reflection across the $y$-axis, $h$ controls horizontal shifts, and $k$ controls vertical shifts.

Final Answer: