How to Understand and Apply Horizontal Stretching of Functions in Common Core Algebra 2
Horizontal stretching of functions is a type of transformation that changes the shape of a function’s graph by stretching or compressing it horizontally. It is also known as horizontal dilation or horizontal scaling. In this article, we will explain what horizontal stretching of functions is, how to identify it, how to perform it, and how to use it in common core algebra 2 homework problems.
What is Horizontal Stretching of Functions?
A function is a rule that assigns an output value to each input value. A function can be represented by an equation, a table, a graph, or a verbal description. A graph of a function shows the relationship between the input values (x-coordinates) and the output values (y-coordinates).
A transformation is a change in the position, shape, or size of a figure. A transformation can be applied to a function’s graph to create a new graph that represents a new function. There are four types of transformations: translations, reflections, rotations, and dilations.
A dilation is a transformation that changes the size of a figure by multiplying all its dimensions by a constant factor. A dilation can be either an enlargement (when the factor is greater than 1) or a reduction (when the factor is between 0 and 1).
A horizontal stretching of functions is a type of dilation that changes the size of a function’s graph by multiplying all its x-coordinates by a constant factor. A horizontal stretching of functions can be either a horizontal expansion (when the factor is greater than 1) or a horizontal compression (when the factor is between 0 and 1).
How to Identify Horizontal Stretching of Functions?
To identify horizontal stretching of functions, we need to compare the original function’s graph with the transformed function’s graph and look for changes in the x-coordinates. If all the x-coordinates are multiplied by the same factor, then we have a horizontal stretching of functions.
For example, consider the graphs of f(x) = x^2 and g(x) = (1/2)x^2 below:
We can see that g(x) is obtained from f(x) by multiplying all the x-coordinates by 1/2. This means that g(x) is a horizontal compression of f(x) by a factor of 1/2.
How to Perform Horizontal Stretching of Functions?
To perform horizontal stretching of functions, we need to apply the following rule:
If y = f(x), then y = f(bx) is a horizontal stretching of f(x) by a factor of 1/b.
This rule means that we need to replace x with bx in the original function’s equation, where b is the constant factor that determines the degree of stretching. Note that if b > 1, then we have a horizontal compression; if b < 1, then we have a horizontal expansion; if b = 1, then we have no change.
For example, to perform a horizontal expansion of f(x) = x^2 by a factor of 3, we need to replace x with (1/3)x in the equation:
y = f((1/3)x) = ((1/3)x)^2 = (1/9)x^2
This means that y = (1/9)x^2 is a horizontal expansion of f(x) = x^2 by a factor of 3.
How to Use Horizontal Stretching of Functions in Common Core Algebra 2 Homework?
Horizontal stretching of functions is an important concept in common core algebra 2 because it helps us understand how changing the input values affects the output values and the shape of the graph. Horizontal stretching of functions can also help us model real-world situations that involve