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Vertical scaling is an important concept in function transformations. When you scale a function vertically, you multiply its output values (y-values) by a constant factor. This doesn't affect the x-values. Instead, it stretches or compresses the graph up and down.
For example, if you have the function \( y = f(x) \) and you apply a vertical scaling by a factor of 2, the modified function is \( y = 2f(x) \).
This means every output value from \( f(x) \) is now twice as large. So if there is a point (1, 3) on the graph \( y = f(x) \), it transforms to (1, 6) on the graph \( y = 2f(x) \).
This is because the original y-value 3 is multiplied by 2, giving you 6. This understanding helps solve problems involving vertical scaling, like finding the corresponding points after a transformation.

Graph transformations involve changing the position or shape of a graph. These changes can occur in various ways, including translations, reflections, and scaling.
- **Translations** move the graph horizontally or vertically without changing its shape. For example, adding a constant to a function, like \( y = f(x) + c \), shifts the graph up by \( c \) units.
- **Reflections** flip the graph across a line, like the x-axis or y-axis. An example is \( y = -f(x) \), which reflects the graph across the x-axis.
- **Scaling** changes the size of the graph. Vertical scaling, which we've seen, multiplies the y-values. Horizontal scaling multiplies the x-values.
Knowing these transformations helps you understand how a graph of a function changes in response to different operations. In our exercise, vertical scaling was used to find the new point on the graph \( y = 2f(x) \) given the point on \( y = f(x) \).

Function evaluation is the process of finding the output of a function for a specific input. To evaluate a function, substitute the input value into the function and calculate the result.
For instance, if you have \( f(x) = x^2 \) and want to evaluate it at \( x = 3 \), you substitute 3 for x: \( f(3) = 3^2 \). This yields 9.
In our exercise, the evaluation process was slightly different because we were dealing with a transformed function. Given the original point on the function \( y = f(x) \), we needed to evaluate the new function \( y = 2f(x) \).
For the point (1, 3) on y = f(x), the corresponding point on y = 2f(x) is evaluated by multiplying the original y-value by 2:
- *Original*: (1, 3)
- *Transformed*: 2 * 3 = 6
The new point is (1, 6).