**What is a Horizontal Translation? **
Horizontally translating a graph is equivalent to shifting the base graph left or right in the direction of the *x*-axis. A graph is translated *k* units horizontally by moving each point on the graph *k* units horizontally.
**Definition**
For the base function *f* (*x*) and a constant *k*, the function given by
*g*(*x*) = *f* (*x - k*),
can be sketched by shifting *f* (*x*) *k* units horizontally. |
The value of *k* determines the direction of the shift. Specifically,
if *k *> 0, the base graph shifts *k* units to the right, and
if *k* < 0, the base graph shifts *k *units to the left. |
**Examples of Horizontal Translations**
Consider the following base functions,
(1) *f* (*x*) = 2*x*^{2},
(2) *g*(*x*) = 5√*x*.
The graphical representation of function (1), *f* (*x*), is a parabola. What do you suppose the graph of
*y*_{1}(*x*) = *f* (*x* -3)
looks like? Using the definition of *f* (*x*), we can write *y*_{1}(*x*) as,
*y*_{1} (*x*) = *f* (*x*-3) = 2(*x* -3)^{2} = 2(*x*^{2} - 6*x* + 9) = 2*x*^{2} -12*x* + 18.
However, this expansion is not necessary if you understand graphical transformations. Based on the definition of horizontal shift, the graph of *y*_{1} (*x*) should look like the graph of *f* (*x*), shifted 3 units to the right. Take a look at the graphs of *f* (*x*) and *y*_{1}(*x*).
Function (2), *g* (*x*), is a square root function. What would the graph of
*y*_{2}(*x*) = *g* (*x* + 2)
look like? Using our knowledge of horizontal shifts, the graph of *y*_{2} (*x*)should look like the base graph *g* (*x*) shifted 2 units to the left. We can write *y*_{2}(*x*) as,
Take a look at the graphs of *g*(*x*) and *y*_{2}(*x*).
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**In the next section, we will explore vertical stretches and shrinks.**
Vertical Stretches and Shrinks |