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.

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.

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

Horizontal Stretches and Shrinks