What are Horizontal Stretches and Shrinks?
Horizontal stretches and shrinks, respectively, horizontally pull the base graph, or push it together, while leaving the y-intercept unchanged to anchor the graph.
For the base function f (x) and a constant k, where k > 0 and k ≠ 1, the function given by
g(x) = f (kx),
can be sketched by horizontally shrinking f (x) by a factor of 1/k if k > 1
by horizontally stretching f (x) by a factor of 1/k
if 0 < k < 1.
A horizontal stretch or shrink by a factor of 1/k means that the point (x, y) on the graph of f(x) is transformed to the point (x/k, y) on the graph of g(x).
Examples of Horizontal Stretches and Shrinks
Consider the following base functions,
(1) f (x) = x2 - 3,
(2) g(x) = cos (x).
The graphical representation of function (1), f (x), is a parabola. What do you suppose the graph of
y1(x) = f (4x)
looks like? Using the definition of f (x), we can write y1(x) as,
y1(x) = f (4x) = (4x)2 - 3 = 16x2-3.
Based on the definition of horizontal shrink, the graph of y1(x) should look like the graph of
f (x), shrunk horizontally by a factor of 1/4. Take a look at the graphs of f (x) and y1(x).
Function (2), g (x), is a cosine function. What would the graph of
y2(x) = g(2/3x)
look like? Using our knowledge of horizontal stretches, the graph of y2(x) should look like the base graph g(x) stretched horizontally by a factor of 3/2. To check this, we can write y2(x) as,
y2(x) = g(2/3x) = cos (2/3x),
construct a table of values, and plot the graph of the new function. As you can see, the graph of y2(x) is in fact the base graph g(x) stretched horizontally by a factor of 3/2.
In the next section, we will explore reflections.