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.
Definition
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
or
by vertically shrinking f (x) by a factor of k
if 0 < k < 1.

Remember that xintercepts 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) = x^{2}  2,
(2) g(x) = sin (x).
The graphical representation of function (1), f (x), is a parabola. What do you suppose the graph of
y_{1}(x) = 1/2f (x)
looks like? Using the definition of f (x), we can write y_{1}(x) as,
y_{1} (x) = 1/2f (x) = 1/2 ( x^{2}  2) = 1/2 x^{2}  1.
Based on the definition of vertical shrink, the graph of y_{1}(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 y_{1}(x).
Notice that the xintercepts have not moved.
Function (2), g (x), is a sine function. What would the graph of
y_{2}(x) = 6g (x)
look like? Using our knowledge of vertical stretches, the graph of y_{2}(x)should look like the base graph g(x) vertically stretched by a factor of 6. To check this, we can write y_{2}(x) as,
y_{2}(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 y_{2}(x) is in fact the base graph g(x) stretched vertically by a factor of 6.
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In the next section, we will explore horizontal stretches and shrinks.
Horizontal Stretches and Shrinks 