Use a computer or a graphing calculator Let f(x)=x^2-3 x . Using the same axes, draw the graphs

step 1 In this figure, the right curve is a graph of y equals f x. The blue curve is a graph of y equal

Use a computer or a graphing calculator Let f(x)=x^2-3 x ....

Key Concepts

Function Transformation

Function transformation involves modifying the graph of a function in various ways such as translating, reflecting, stretching, or compressing. These changes are applied either to the input (x-values) or the output (y-values) of the function, and allow for an understanding of how the graph shifts or changes shape in response to algebraic manipulations.

Horizontal Translation

A horizontal translation shifts a graph left or right. This occurs when a constant is added or subtracted inside the function's argument. For example, replacing x with x - h translates the original graph h units to the right, enabling a clear visual comparison between the original and shifted graphs.

Vertical Translation

A vertical translation shifts a graph up or down. This is achieved by adding or subtracting a constant from the entire function. For example, f(x) - k shifts the entire graph downward by k units. This transformation is useful when comparing how continuous shifts affect the function’s output over a given domain.

Horizontal Scaling

Horizontal scaling (or horizontal stretching/compression) involves multiplying the input variable by a constant, which changes the rate at which the function's graph progresses along the x-axis. Specifically, replacing x with a constant times x will compress the graph if the constant is greater than 1 or stretch it if the constant is between 0 and 1. This transformation alters the spread of the graph without affecting its y-intercept.

Domain of a Function

The domain of a function is the set of input values for which the function is defined. Specifying the domain, such as [-2, 5] in this context, restricts the section of the graph that is considered, which is important for analysis and visualization, especially when graphing or using technology to render the function.

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