# Horizontal Stretching Of Functions Common Core Algebra 2 Homework

## Horizontal Stretching of Functions Common Core Algebra 2 Homework

In this article, we will review how to graph functions that have been horizontally stretched or compressed by a factor. This is a topic that is covered in the Common Core Algebra 2 curriculum, and it can help us understand how different functions behave and relate to each other.

## What is horizontal stretching of functions?

A horizontal stretching of a function is a transformation that changes the x-values of the function by multiplying them by a constant factor. For example, if we have a function f(x), then the function g(x) = f(kx) is a horizontal stretching of f(x) by a factor of k. The value of k determines how much the function is stretched or compressed horizontally. If k > 1, then the function is compressed horizontally, meaning that it becomes narrower and steeper. If k

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## How to graph horizontally stretched functions?

To graph a horizontally stretched function, we can use the following steps:

Identify the original function f(x) and the factor k.

Choose some points on the graph of f(x) and write their coordinates as (x, f(x)).

Multiply the x-coordinates by k to get the new x-coordinates for g(x). The y-coordinates remain unchanged.

Plot the new points on the same coordinate plane as f(x) and connect them with a smooth curve.

Label the graph of g(x) = f(kx).

## Examples of horizontal stretching of functions

Let's look at some examples of horizontal stretching of functions using different types of functions and values of k.

### Example 1: Horizontal stretching of a linear function

Suppose we have the linear function f(x) = x + 2 and we want to graph the function g(x) = f(2x). This means that k = 2, so we are compressing the function horizontally by a factor of 2. We can use the following table to find some points on the graphs of f(x) and g(x):

x f(x) g(x) --------------- -4 -2 -2 -2 0 0 0 2 2 2 4 4 4 6 6 We can see that the y-coordinates are the same for both functions, but the x-coordinates for g(x) are half of those for f(x). This means that g(x) is closer to the y-axis than f(x). Here is a graph of both functions:

We can see that g(x) is steeper than f(x), and they have the same y-intercept at (0, 2).

### Example 2: Horizontal stretching of a quadratic function

Suppose we have the quadratic function f(x) = x^2 - 4 and we want to graph the function g(x) = f(0.5x). This means that k = 0.5, so we are stretching the function horizontally by a factor of 0.5. We can use the following table to find some points on the graphs of f(x) and g(x):

x f(x) g(x) --------------- -4 12 -4 -2 0 -3 0 -4 -4 2 0 -3 4 12 -4 We can see that the y-coordinates for g(x) are lower than those for f(x), except at x = 0. The x-coordinates for g(x) are twice as large as those for f(x). This means that g(x) is farther from the y-axis than f(x). Here is a graph of both functions:

We can see that g(x) is flatter than f(x), and they have the same vertex at (0, -4).

### Example 3: Horizontal stretching of a trigonometric function

Suppose we have the trigonometric function f(x) = sin(x) and we want to graph the function g(x) = f(3x). This means that k = 3, so we are compressing the function horizontally by a factor of 3. We can use the following table to find some points on the graphs of f(x) and g(x):

x f(x) g(x) --------------- 0 0 0 pi/6 0.5 0.5 pi/3 0.866 0.866 pi/2 1 1 2pi/3 0.866 -0.866 5pi/6 0.5 -0.5 pi 0 0 We can see that the y-coordinates are the same for both functions, but the x-coordinates for g(x) are one-third of those for f(x). This means that g(x) completes one cycle faster than f(x). Here is a graph of both functions:

We can see that g(x) has a higher frequency than f(x), and they have the same amplitude and phase shift.

## Conclusion

In this article, we learned how to graph functions that have been horizontally stretched or compressed by a factor. We saw how this transformation affects the shape and position of different types of functions, such as linear, quadratic, and trigonometric functions. We also learned how to use a table of values to find the coordinates of the new points on the graph. This is a useful skill that can help us analyze and compare different functions and their properties.