The term 'af(x)' refers to the transformation of a function f(x) by a constant factor a. This transformation scales the function vertically, either expanding or contracting the graph of the function depending on the value of a.
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The constant factor a in the expression af(x) determines the direction and magnitude of the vertical scaling of the function f(x).
If a is positive, the graph of f(x) is stretched vertically by a factor of a.
If a is negative, the graph of f(x) is reflected about the x-axis and then stretched vertically by a factor of |a|.
When a = 1, the graph of f(x) is unchanged, as the function is simply multiplied by 1.
Vertical scaling can be combined with other transformations, such as translations, to create more complex transformations of functions.
Review Questions
Explain how the constant factor a affects the graph of the function f(x) in the expression af(x).
The constant factor a in the expression af(x) determines the direction and magnitude of the vertical scaling of the function f(x). If a is positive, the graph of f(x) is stretched vertically by a factor of a. If a is negative, the graph of f(x) is reflected about the x-axis and then stretched vertically by a factor of |a|. When a = 1, the graph of f(x) is unchanged, as the function is simply multiplied by 1.
Describe how vertical scaling can be combined with other transformations to create more complex transformations of functions.
Vertical scaling can be combined with other transformations, such as translations, to create more complex transformations of functions. For example, the expression a(f(x) + b) would first vertically scale the function f(x) by a factor of a, and then shift the resulting graph up or down by a constant b. This combination of vertical scaling and vertical shifting allows for a wide range of transformations to be applied to the original function.
Analyze the effect of the constant factor a on the graph of the function f(x) in the context of the transformation of functions.
$$\text{The constant factor a in the expression af(x) plays a crucial role in the transformation of functions. When a is positive, the graph of f(x) is stretched vertically by a factor of a, resulting in a wider or taller graph depending on the original function. When a is negative, the graph of f(x) is reflected about the x-axis and then stretched vertically by a factor of |a|. This transformation can significantly alter the shape and behavior of the original function, which is an important consideration when working with transformations of functions.}$$
Related terms
Vertical Scaling: The process of stretching or shrinking a function's graph in the vertical direction by multiplying the function by a constant factor.
The process of moving a function's graph up or down by adding a constant to the function.
Transformation of Functions: The process of modifying the graph of a function by applying various transformations, such as translations, reflections, and scaling.