Stretching the Absolute Value Function Vertically

Oct 13, 2025 by magicalceylontours.com 129 views

Introduction

The absolute value function, denoted by $y = |x|$ , is a fundamental concept in mathematics that represents the distance of a number from zero on the number line. It is a piecewise function that can be defined as $y = x$ for $x \geq 0$ and $y = -x$ for $x < 0$ . In this article, we will explore how to stretch and shift the absolute value function vertically by a factor of 2 and up 4 units, respectively.

Stretching the Absolute Value Function Vertically

To stretch the absolute value function vertically by a factor of 2, we need to multiply the entire function by 2. This means that for every value of $y$ , we will multiply it by 2. Mathematically, this can be represented as:

y = 2|x|

This new function represents a vertical stretch of the original absolute value function by a factor of 2. The graph of this function will be a scaled version of the original graph, with all the points doubled in the vertical direction.

Shifting the Absolute Value Function Up 4 Units

To shift the absolute value function up 4 units, we need to add 4 to the entire function. This means that for every value of $y$ , we will add 4 to it. Mathematically, this can be represented as:

y = 2|x| + 4

This new function represents a vertical shift of the original absolute value function up 4 units. The graph of this function will be a translated version of the original graph, with all the points shifted up by 4 units.

Combining Stretching and Shifting

Now that we have learned how to stretch and shift the absolute value function vertically, we can combine these two transformations to get the final function. By stretching the absolute value function vertically by a factor of 2 and shifting it up 4 units, we get:

y = 2|x| + 4

This is the equation of the function that results from stretching $y = |x|$ vertically by a factor of 2 and shifting it up 4 units.

Graphical Representation

To visualize the graph of this function, we can start by graphing the original absolute value function $y = |x|$ . Then, we can stretch it vertically by a factor of 2 to get the graph of $y = 2|x|$ . Finally, we can shift the graph up 4 units to get the final graph of $y = 2|x| + 4$ .

Properties of the Function

The function $y = 2|x| + 4$ has several properties that are worth noting:

Domain: The domain of this function is all real numbers, denoted by $(-\infty, \infty)$ .
Range: The range of this function is all real numbers greater than or equal to 4, denoted by $[4, \infty)$ .
Symmetry: This function is symmetric about the y-axis, since the absolute value function is symmetric about the y-axis.
Asymptotes: This function has no vertical asymptotes, since the absolute value function has no vertical asymptotes.

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions about stretching and shifting the absolute value function.

Q: What is the equation of the function that results from stretching $y = |x|$ vertically by a factor of 3 and shifting it up 2 units?

A: To stretch the absolute value function vertically by a factor of 3, we need to multiply the entire function by 3. This means that for every value of $y$ , we will multiply it by 3. Mathematically, this can be represented as:

y = 3|x|

To shift the absolute value function up 2 units, we need to add 2 to the entire function. This means that for every value of $y$ , we will add 2 to it. Mathematically, this can be represented as:

y = 3|x| + 2

Therefore, the equation of the function that results from stretching $y = |x|$ vertically by a factor of 3 and shifting it up 2 units is:

y = 3|x| + 2

Q: How do I graph the function $y = 2|x| + 5$ ?

A: To graph the function $y = 2|x| + 5$ , you can start by graphing the original absolute value function $y = |x|$ . Then, you can stretch it vertically by a factor of 2 to get the graph of $y = 2|x|$ . Finally, you can shift the graph up 5 units to get the final graph of $y = 2|x| + 5$ .

Q: What is the domain and range of the function $y = 2|x| + 3$ ?

A: The domain of the function $y = 2|x| + 3$ is all real numbers, denoted by $(-\infty, \infty)$ . The range of this function is all real numbers greater than or equal to 3, denoted by $[3, \infty)$ .

Q: Is the function $y = 2|x| + 2$ symmetric about the y-axis?

A: Yes, the function $y = 2|x| + 2$ is symmetric about the y-axis, since the absolute value function is symmetric about the y-axis.

Q: Does the function $y = 2|x| + 1$ have any vertical asymptotes?

A: No, the function $y = 2|x| + 1$ does not have any vertical asymptotes, since the absolute value function does not have any vertical asymptotes.

Q: How do I find the equation of the function that results from stretching $y = |x|$ horizontally by a factor of 2 and shifting it up 3 units?

A: To stretch the absolute value function horizontally by a factor of 2, we need to replace $x$ with $\frac{x}{2}$ . This means that for every value of $x$ , we will divide it by 2. Mathematically, this can be represented as:

y = |x/2|

To shift the absolute value function up 3 units, we need to add 3 to the entire function. This means that for every value of $y$ , we will add 3 to it. Mathematically, this can be represented as:

y = |x/2| + 3

Therefore, the equation of the function that results from stretching $y = |x|$ horizontally by a factor of 2 and shifting it up 3 units is:

y = |x/2| + 3

Conclusion

In conclusion, we have answered some frequently asked questions about stretching and shifting the absolute value function. We have also explored the graphical representation and properties of these functions.