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Vertical scaling is a type of transformation that changes the "height" of the graph of a function. It either stretches or compresses the graph along the vertical axis. When you multiply a function by a constant, such as in the example where the function is transformed from \( y=f(x) \) to \( y=3f(x) \), this alters the \( y \)-values.

In this context, "scaling by a factor of 3" means each \( y \)-value of the original function is multiplied by 3:

• If a \( y \)-value is positive, the graph stretches away from the x-axis.
• If a \( y \)-value is zero, it stays the same at zero.
• If a \( y \)-value is negative, it stretches but remains negative, resulting in a larger negative \( y \)-value.

For the point (2, -3) on the graph of \( y = f(x) \), applying the vertical scaling of 3 affects only the \( y \)-coordinate, making it \( 3 \times (-3) = -9 \), resulting in the new point (2, -9).

This transformation does not affect the \( x \)-coordinate, so it remains unchanged.

Coordinate transformations involve altering one or both coordinates of a point to match the requirements of a transformed graph. In the context of functions and their graphs, this simply means changing how input \( x \) and output \( y \) values relate according to a transformation rule.

For example, when you understand that \( y = 3f(x) \) scales the \( y \)-coordinates while leaving the \( x \)-coordinates unchanged, you have a clear strategy for finding new points:

• Keep the original \( x \)-coordinate the same.
• Scale the \( y \)-coordinate according to the transformation factor; here, it's multiplied by 3.

This systematic approach makes it easy to handle coordinate transformations in various transformations, not just vertical scaling. Coordinate transformation is a fundamental concept because it lays the groundwork for understanding more complex graph transformations.

Graph transformations are the backbone of understanding how various changes affect a function's curve. These transformations can be categorized into several types, including translations, reflections, stretching, and compressing.

Vertical scaling, as demonstrated, is a type of stretching or compressing transformation, applied here to the \( y \)-axis. It's essential to observe how:

• A vertical stretch (like multiplying by a factor greater than 1) increases the distance from each point to the x-axis.
• A vertical compression (a factor less than 1 but greater than 0) reduces that distance.

By tracking how transformations alter specific points, like using (2, -3) to reach (2, -9), we gain insight into reshaping a plot graphically without evaluating the function for numerous values. Recognizing graph transformations helps you visualize and predict the effect of algebraic changes on the underlying curve.

Algebraic manipulation is an indispensable skill for transforming functions and solving equations. It involves adjusting the form of a function to make transformations clearer. In the case of the given problem, understanding how \( y = f(x) \) is transformed into \( y = 3f(x) \) through multiplication is crucial.

Working through algebraic manipulation involves several straightforward steps:

• Identify the transformation factor or expression needed to apply a transformation, here \( 3 \).
• Apply this to the function or specific expressions, modifying the function's definition.
• Check your transformation by substituting initial points (e.g., (2, -3)) to see how they change (to (2, -9)).

This practice is crucial for translating a verbal or conceptual transformation into an explicit mathematical operation. By mastering algebraic manipulation, students can tackle function transformations systematically, confirming their understanding of how functional changes impact a graph.