Horizontal Stretching Of Functions Common Core Algebra 2 Homework
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How to Understand Horizontal Stretching of Functions in Common Core Algebra 2
One of the topics that you may encounter in your common core algebra 2 homework is horizontal stretching of functions. This is a type of transformation that changes the shape of a function's graph by stretching or compressing it horizontally. In this article, we will explain what horizontal stretching of functions means, how to identify it, and how to apply it to different types of functions.
What is Horizontal Stretching of Functions
Horizontal stretching of functions is a transformation that affects the x-values of a function's graph. It occurs when a constant factor, called the stretch factor, is multiplied by the input variable x before applying the function rule. For example, if f(x) is a function, then g(x) = f(kx) is a horizontally stretched version of f(x), where k is the stretch factor.
The value of k determines how much the graph of f(x) is stretched or compressed horizontally. If k > 1, then the graph of g(x) is compressed by a factor of k. This means that the x-values are closer together than in f(x), and the graph appears narrower. If 0 < k < 1, then the graph of g(x) is stretched by a factor of 1/k. This means that the x-values are farther apart than in f(x), and the graph appears wider. If k = 1, then there is no horizontal stretching or compression, and g(x) = f(x).
How to Identify Horizontal Stretching of Functions
To identify horizontal stretching of functions, you need to compare the graphs of f(x) and g(x) = f(kx) and look for changes in the x-values. You can also use a table of values or an equation to find the value of k and determine whether it is greater than 1, less than 1, or equal to 1.
For example, suppose you have the function f(x) = x^2 and its horizontally stretched version g(x) = f(2x) = (2x)^2 = 4x^2. You can compare their graphs as shown below:
You can see that the graph of g(x) is narrower than the graph of f(x), which means that it is compressed horizontally by a factor of 2. You can also use a table of values to compare the x-values for each function:
You can see that for every value of x in f(x), there is a corresponding value of x/2 in g(x). This means that g(x) has half the x-values as f(x), which confirms that it is compressed horizontally by a factor of 2. You can also use an equation to find the value of k by solving for x in terms of g(x):
g(x) = f(kx)
4x^2 = (kx)^2
kx = Â2x
k = Â2
You can see that k = 2 or -2, which are both greater than 1, which confirms that g(x) is compressed horizontally by a factor of 2.
How to Apply Horizontal Stretching of Functions to Different Types of Functions
You can apply horizontal stretching of functions to different types of functions by using the same general rule: g(x) = f(kx), where k is the stretch factor. 061ffe29dd