Understanding the Vertical Stretch of Functions: Exploring the Impact of a Scaling Factor Greater Than 1

y=af(x) a>1

In this scenario, we have a function y = af(x), where a is a positive constant greater than 1

In this scenario, we have a function y = af(x), where a is a positive constant greater than 1.

1. First, let’s understand the role of a in the equation. Since a is greater than 1, it acts as a scaling factor for the function. This means that it will stretch the graph vertically compared to the original function f(x). If a were less than 1, it would compress the graph vertically.

2. Next, let’s consider the impact on the y-values. If we have a point (x, y) on the graph of f(x), when we multiply the y-coordinate by a, we get (x, ay). This means that the y-values of the function af(x) are stretched vertically by a factor of a.

3. Let’s think about specific examples to illustrate this concept. Suppose we have the function f(x) = x^2. If we introduce a scaling factor a = 2, our new function becomes y = 2x^2. This means that for any given x, the corresponding y-value will be twice as large compared to the original function f(x).

4. Similarly, if we have the function f(x) = sin(x), and we multiply it by a scaling factor a = 3, our new function will be y = 3sin(x). This implies that the y-values of the new function will be stretched vertically by a factor of 3 compared to the original function.

5. Finally, keep in mind that the shape of the graph remains the same, only the vertical size changes. The x-values and the overall pattern of the function are not affected by multiplying by a constant.

Overall, when a constant a greater than 1 is multiplied with a given function f(x), it will cause the graph to stretch vertically, increasing the y-values compared to the original function. This knowledge will enable you to analyze and understand the behavior of functions with scaling factors.

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