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Benny Ferraro
Benny Ferraro

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.


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