Horizontal Stretch - Properties, Graph, & Examples (2024)

You may already have encountered graphs that look alike but share different widths. These functions may have been horizontally stretched using a base function. Horizontal stretches are among the most applied transformation techniques when graphing functions, so it’s best to understand its definition.

Horizontal stretches happen when a base graphis widened along the x-axis and away from the y-axis.

Learning how we can stretch graphs horizontally can help us understand the family of functions’ graphs. We can also learn how to speed up graphing new functions based on the scale factors applied.

Make sure to review your knowledge of parent functions, translations, and vertical stretches before moving to the next section.

Understanding the common parent functions we might encounter.
Refresh your knowledge of vertical and horizontal transformations.
Review how we apply vertical stretches.

These linked articles may help you refresh your knowledge, and when you’re ready, let’s go ahead and learn more about horizontal stretching.

What is a horizontal stretch?

We can apply horizontal stretch to a function by multiplying its input values by a scale factor, a, where 0 < 1/a < 1. What does this mean for functions such as f(x)? When 1/a is multiplied to x, f(x)’s graph stretches horizontally by a scale factor of a.

Let’s go ahead and look at how f(x) = x² will be affected by a scale factor of 1/2 and 1/3.

As we have expected, the graph stretches by a factor of 2 and 3. Notice how the y-values remain the same? This is true for all horizontal stretches. The graph only stretches away from the y-axis when we horizontally stretch a graph.

Horizontal stretch on other functions will exhibit similar properties. Let’s say we have f(x) = |x|, if this function’s graph is to be stretched horizontally to attain g(x), the new function’s expression can be expressed as |1/3 ∙ x| = |x/3|.

Why don’t we compare some values from both functions?

x	-2	-1	0	1	2
f(x) = \|x\|	2	1	0	1	2

We also expect all its x-coordinates to increase by 3. When we do this, the output values will remain the same.

x	-6	-3	0	3	6
g(x) = \|x/3\|	2	1	0	1	2

What does this mean for the coordinates of the new function’s graph? If the function f(x) passes through (m, n) and is stretched horizontally by a factor of 1/a, the new function will pass through (am, n).

How to horizontally stretch a function?

Now that we’ve learned about horizontal stretches and how they affect a base function, it’s time that we apply these on functions’ graphs. Before we start stretching functions horizontally by a certain factor, remember these pointers to stretch graphs faster horizontally:

Only stretch the base of the graph horizontally so that the y-coordinates would remain in the same position.
Since the y-coordinates will remain the same, the y-intercept stays the same as well.
Make sure to double-check critical points on the graph, such as its intercepts, maximum points, and more.
See if the coordinate points are scaled correctly as well.

Let’s use the table of values shown in the previous section to graph both y = |x| and y = |x/3| to visualize the effects of a horizontal stretch on a graph.

As we have discussed, we’re expecting the graphs to stretch along the base with the values for y remaining constant.

To attain y = |x/3|, we stretch the parent function y = |x| by a factor of 3. The graph shown above confirms this, and we can apply the same process when horizontally stretching the graphs of other functions.

Ready to graph more functions and apply horizontal stretches? Let’s summarize what we have learned so far first before we try out more questions.

Summary of horizontal stretch definition and properties

Here are some important pointers to remember when answering problems and graphing functions that involve horizontal stretches:

We can only horizontally stretch a graph by a factor of 1/a when the input value is also increased by a.
When f(x) is stretched horizontally to f(ax), multiply the x-coordinates by a.
Retain the y-intercepts’ position.
The resulting function will have the same range but may have a different domain.
Given a point (m, n), it becomes (am, n) when stretched horizontally.

Let’s always go back to these five-pointers when in doubt. Being able to master the technique of horizontally stretching graphs can help us graph functions faster and understand their behavior.

When the function is stretched horizontally, we multiply the input values (for this case, it’s the values of x) by the given scale factor’s denominator.

This means that (-2, -4) will be transformed to (4∙ -2, -4) = (-8, -4). We apply the same process for the rest of the values and have h(x)’s table of values as shown below.

x	-12	-8	-4	0	4	8	12
h(x)	-6	-4	-2	0	2	4	6

Example 5

Observe the functions shown below. What is the relationship between f(x) and g(x)?

Solution

Just by looking at the graph, we can see that g(x) results from f(x) being stretched horizontally. Why don’t we inspect some points from both graphs?

This is what happens to the points:

(-3, 9) → (-9, 9)
(-2, 4) → (-6, 4)
(-1, 1) → (-3, 1)
(0, 0) → (0, 0)
(1, 1) → (3, 1)
(2, 4) → (6, 4)
(3, 9) → (9, 9)

Notice how for each case: the x-coordinates of g(x) are all three times larger than f(x). This means that the scale factor used to stretch f(x) is 1/3.

The function g(x) is the result of f(x) being horizontally stretched by a scale factor of 1/3.

Example 6

The image below shows the graph of f(x). Graph g(x) using the fact that it is the result of f(x) being stretched horizontally by a factor of 1/2. Make sure to include the new critical points for g(x).

Solution

Let’s go ahead and find the new critical points of g(x) first. Since f(x) will be stretched horizontally, we’ll multiply the x-coordinates by 2. Hence, we have the following critical points:

(-3, 0) → (-6, 0)
(0, 3) → (0, 3)
(3, 0) → (6, 0)

Plot these points and stretch the graph of f(x) by 3. Make sure that the y-coordinates and y-intercept remain the same.

This is how the resulting graph should look like. Since g(x) is the result of f(x) being stretched horizontally, we stretch the graph of f(x) by a scale of 3.

Example 7

Describe the transformations done on the following functions shown below.

a.f(x) = x² → g(x) = x²/9

b. m(x) = √x → n(x) = 3√(x/4)
c. p(x) = 3x – 1→ q(x) = 3x/4 – 2

Solution

If you might have noticed that some may not only exhibit horizontal stretches, you are right!

Keep in mind that we sometimes have to apply different transformation techniques for us to obtain a particular function’s expression.

First, let’s start with f(x) and g(x):

To check the scale factor applied in the input value of f(x) to attain g(x), let’s express g(x) as a perfect square: g(x) = (x/3)² = (1/3 · x)².

Expressing g(x) in terms of f(x), we have g(x) = f(x/3). Hence, g(x) is the result of f(x) being horizontally stretched by a scale factor of 1/3.

Moving on to m(x) and n(x):

We can see two scale factors applied to n(x): 3 on the output value and 1/4 for the input value. Applying what we know on vertical and horizontal stretches, we have n(x) = 3·m(1/4 · x). Meaning, n(x) is the result of m(x) being vertically stretched by a scale factor of 3 and horizontally stretched by a scale factor of 1/4.

Lastly, let’s observe the translations done on p(x).

q(x) = 3/4 x – 1 – 1

= 3(x/4) – 1 – 1

= p(x/4) – 1

From this, we can see that q(x) results from p(x) being stretched horizontally by a scale factor of 1/4 and translated one unit downward.

Example 8

What are the transformations done on f(x) so that it results to g(x) = 2|x/3| – 1? Use the graph of f(x) shown below to guide you. Apply the transformations to graph g(x).