For example, in the toolkit functions, we introduced the absolute value function f\left(x\right)=|x|. If the positive constant is a fraction less than 1, the graph will appear to stretch horizontally.Sometimes, we come across a function that requires more than one formula in order to obtain the given output.
If the constant is a positive number greater than 1, the graph will appear to stretch vertically.
Multiplying a function by a constant other than 1, a ⋅ f ( x ), produces a dilation. When applying multiple transformations, apply reflections first. Multiplying the values in the domain by −1 before applying the function, f ( − x ), reflects the graph about the y-axis. Multiplying a function by a negative constant, − f ( x ), reflects its graph in the x-axis. If a positive constant is subtracted from the value in the domain before the function is applied, f ( x − h ), the graph will shift right. If a positive constant is added to the value in the domain before the function is applied, f ( x + h ), the graph will shift to the left. The basic shape of the graph will remain the same. If a positive constant is subtracted from a function, f ( x ) − k, the graph will shift down. If a positive constant is added to a function, f ( x ) + k, the graph will shift up. Often a geometric understanding of a problem will lead to a more elegant solution. This skill will be useful as we progress in our study of mathematics. Identifying transformations allows us to quickly sketch the graph of functions. Begin by evaluating for some values of the independent variable x. For example, consider the functions g ( x ) = x 2 − 3 and h ( x ) = x 2 + 3. If we add a negative constant, the graph will shift down. If we add a positive constant to each y-coordinate, the graph will shift up. This occurs when a constant is added to any function. 
is a rigid transformation that shifts a graph up or down relative to the original graph. changes the size and/or shape of the graph.Ī vertical translation A rigid transformation that shifts a graph up or down. A non-rigid transformation A set of operations that change the size and/or shape of a graph in a coordinate plane.
changes the location of the function in a coordinate plane, but leaves the size and shape of the graph unchanged. A rigid transformation A set of operations that change the location of a graph in a coordinate plane but leave the size and shape unchanged. When the graph of a function is changed in appearance and/or location we call it a transformation.
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