What are Vertical Stretches and Shrinks?
While translations move the x and y intercepts of a base graph, stretches and shrinks effectively pull the base graph outward or compress the base graph inward, changing the overall dimensions of the base graph without altering its shape. When a graph is stretched or shrunk vertically, the x -intercepts act as anchors and do not change under the transformation.
For the base function f (x) and a constant k > 0, the function given by
g(x) = k f (x),
can be sketched by vertically stretching f (x) by a factor of k if k > 1
by vertically shrinking f (x) by a factor of k
if 0 < k < 1.
Remember that x-intercepts do not move under vertical stretches and shrinks. In other words, if f (x) = 0 for some value of x, then k f (x) = 0 for the same value of x. Also, a vertical stretch/shrink by a factor of k means that the point (x, y) on the graph of f (x) is transformed to the point (x, ky) on the graph of g(x).
Examples of Vertical Stretches and Shrinks
Consider the following base functions,
(1) f (x) = x2 - 2,
(2) g(x) = sin (x).
The graphical representation of function (1), f (x), is a parabola. What do you suppose the graph of
y1(x) = 1/2f (x)
looks like? Using the definition of f (x), we can write y1(x) as,
y1 (x) = 1/2f (x) = 1/2 ( x2 - 2) = 1/2 x2 - 1.
Based on the definition of vertical shrink, the graph of y1(x) should look like the graph of f (x), vertically shrunk by a factor of 1/2. Take a look at the graphs of f (x) and y1(x).
Notice that the x-intercepts have not moved.
Function (2), g (x), is a sine function. What would the graph of
y2(x) = 6g (x)
look like? Using our knowledge of vertical stretches, the graph of y2(x)should look like the base graph g(x) vertically stretched by a factor of 6. To check this, we can write y2(x) as,
y2(x) = 6g(x) = 6 sin (x),
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 vertically by a factor of 6.
In the next section, we will explore horizontal stretches and shrinks.
Horizontal Stretches and Shrinks