![]() Factoring in this way allows us to horizontally stretch first and then shift horizontally. Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. A horizontal reflection: f\left(-t\right)=.This equation combines three transformations into one equation. The point \left(-1,0\right) is transformed next by shifting down 3 units: \left(-1,0\right)\to \left(-1,-3\right).The point \left(0,0\right) is transformed first by shifting left 1 unit: \left(0,0\right)\to \left(-1,0\right).Let us follow one point of the graph of f\left(x\right)=|x|. The transformation of the graph is illustrated below. ![]() The following is an arithmetic sequence as every term is obtained by adding. It is a 'sequence where the differences between every two successive terms are the same' (or) In an arithmetic sequence, 'every term is obtained by adding a fixed number (positive or negative or zero) to its previous term'. Sequences can have formulas that tell us how to find any term in the sequence. An arithmetic sequence is defined in two ways. For example, 2,5,8 follows the pattern 'add 3,' and now we can continue the sequence. Some sequences follow a specific pattern that can be used to extend them indefinitely. The graph of h has transformed f in two ways: f\left(x+1\right) is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in f\left(x+1\right)-3 is a change to the outside of the function, giving a vertical shift down by 3. Sequences are ordered lists of numbers (called 'terms'), like 2,5,8. We know that this graph has a V shape, with the point at the origin. The function f is our toolkit absolute value function. Given f\left(x\right)=|x|, sketch a graph of h\left(x\right)=f\left(x+1\right)-3. Example: Graphing Combined Vertical and Horizontal Shifts
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