Example 5 from Algebra and Trigonometry, 3.5 Transformation of Functions

Identifying a Horizontal Shift of a Toolkit Function

Figure 8 represents a transformation of the toolkit function $f (x) = x^{2} .$ Relate this new function $g (x)$ to $f (x),$ and then find a formula for $g (x) .$

Figure 8

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Notice that the graph is identical in shape to the $f (x) = x^{2}$ function, but the x-values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so

$g (x) = f (x - 2)$
Notice how we must input the value $x = 2$ to get the output value $y = 0;$ the x-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the $f (x)$ function to write a formula for $g (x)$ by evaluating $f (x - 2) .$

$\begin{matrix} f (x) & = & x^{2} \\ g (x) & = & f (x - 2) \\ g (x) & = & f (x - 2) = {(x - 2)}^{2} \end{matrix}$

How did it go?

Solution

Solution