How to write a transformed quadratic function after a horizontal stretch and vertical shift

Answer

For Case A, the expression is $g(x) = f(\frac{1}{2}x) - 3$. For Case B, the expression is $g(x) = f(\frac{1}{2}(x)) - 3$, or more precisely, $g(x) = f(\frac{1}{2}(x+3) - 3)$ depending on whether the translation is applied before or after the stretch; however, as defined, they result in different mappings.

Explanation

1. Analyzing Case A (Horizontal Stretch then Translation) A horizontal stretch by a scale factor of $k=2$ corresponds to the transformation $x \to \frac{x}{2}$. A vertical translation of 3 units downwards is represented by $y \to y - 3$. $g(x) = f\left(\frac{1}{2}x\right) - 3$ This formula shows that we replace the input $x$ with half its value and then shift the entire output function down by three units. ⚠️ This step is required on exams as it correctly identifies the order of operations for transformations.

2. Analyzing Case B (Translation then Horizontal Stretch) A translation of 3 units in the negative $y$-direction is $f(x) - 3$. Applying a horizontal stretch by a factor of 2 to the result of that translation requires replacing $x$ with $\frac{x}{2}$ in the entire expression. $g(x) = f\left(\frac{1}{2}x\right) - 3$ This indicates that even though the verbal order of operations was swapped, because the vertical shift is independent of the horizontal stretch, the resulting algebraic expression remains identical in this specific case.

3. Comparison The following table highlights the nature of these transformations:

The transformations commute because vertical translations and horizontal stretches act on different axes, meaning the vertical displacement does not scale when the horizontal stretch is applied.

Final Answer

The expressions for both cases are: $\boxed{g(x) = f\left(\frac{x}{2}\right) - 3}$

Common Mistakes

• Order of Operations: Students often think that changing the order of transformations always changes the function. While this is true for horizontal and vertical transformations on the same axis, vertical shifts and horizontal stretches are independent.
• Stretch Factor Reciprocal: A common error is writing $f(2x)$ for a horizontal stretch of scale factor 2. Remember that the transformation $x \to \frac{x}{k}$ represents a stretch by factor $k$.