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)

Transcript text: 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

Solution Steps

Step 1: Identify the Original Function

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

Step 2: Apply Horizontal Transformations

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 $b = -1$.

Step 3: Apply Vertical Transformations

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 $a = -1$.

Step 4: Combine the Transformations

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:

The transformed function $g(x)$ is obtained by applying the specified transformations to the original function $f(x)$. The specific form of $g(x)$ depends on the function $f(x)$ and the parameters -1, 1, 4, and 0.

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