Understanding the Behavior of the Equation y = f(bx) When b > 1: Exploring the Effects of Scaling and Transformation on the Function

y=f(bx), b>1

In the equation y = f(bx), where b > 1, we have a function f being applied to the value bx

In the equation y = f(bx), where b > 1, we have a function f being applied to the value bx. Here’s a detailed explanation:

1. The variable x represents the input value of the function. It could be any real number.
2. The variable b is a positive constant greater than 1. This means that b acts as a scaling factor or multiplier for the x-values.
3. When b > 1, it implies that the x-values will be stretched or expanded horizontally. This is because as x increases, bx increases faster due to the larger value of b.
4. The function f represents the specific mathematical operations or transformations that are applied to bx. It could be any mathematical function or expression. Examples include linear functions, quadratic functions, logarithmic functions, trigonometric functions, etc.
5. The output of the function, y, represents the result after applying the function f to the value of bx.

To understand the behavior of the equation y = f(bx) when b > 1, we can consider some examples:

Example 1: Let’s say f(x) = x^2 and b = 2.
In this case, the equation becomes y = (2x)^2 = 4x^2.
The function f(x) = x^2 represents a quadratic function. When b = 2, the x-values are being doubled before being squared. This means that the graph of y = 4x^2 will be narrower than the graph of y = x^2. The parabola will open upwards, becoming steeper as x increases.

Example 2: Let’s say f(x) = sin(x) and b = 3.
In this case, the equation becomes y = sin(3x).
The function f(x) = sin(x) represents a trigonometric function. When b = 3, the x-values are being stretched horizontally by a factor of 3. The graph of y = sin(3x) will have three times as many complete oscillations as the graph of y = sin(x) over the same x-interval. The period of the sine function will be decreased.

These examples illustrate how the value of b in the equation y = f(bx) affects the behavior of the function and its graph. The specific nature of the function f will determine the exact transformation applied to the x-values.

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