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.

The value of k determines the direction of the shift. Specifically,

Examples of Horizontal Translations

Consider the following base functions,

(1) f (x) = 2x²,

(2) g(x) = 5√x.

The graphical representation of function (1), f (x), is a parabola. What do you suppose the graph of

y₁(x) = f (x -3)

looks like? Using the definition of f (x), we can write y₁(x) as,

y₁ (x) = f (x-3) = 2(x -3)² = 2(x² - 6x + 9) = 2x² -12x + 18.

However, this expansion is not necessary if you understand graphical transformations. Based on the definition of horizontal shift, the graph of y₁ (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 y₁(x).

Function (2), g (x), is a square root function. What would the graph of

y₂(x) = g (x + 2)

look like? Using our knowledge of horizontal shifts, the graph of y₂ (x)should look like the base graph g (x) shifted 2 units to the left. We can write y₂(x) as,

Take a look at the graphs of g(x) and y₂(x).

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In the next section, we will explore vertical stretches and shrinks.

Vertical Stretches and Shrinks