Graphing a shifted and stretched absolute value function

TL;DR

Learn how to graph a transformed absolute value function by shifting it left, stretching it vertically, and shifting it up.

Transcript

• [Instructor] So we're asked to graph f of x is equal to two times the absolute value of x plus three, plus two. And what they've already graphed for us, this right over here, this is the graph of y is equal to the absolute value of x. So let's do this through a series of transformations. So the next thing I wanna graph, let's see if we can graph ... Read More

Key Insights

• 👈 Replacing x with x + 3 shifts the graph of an absolute value function to the left by the amount being subtracted.
• 🫥 A negative value within the absolute value sign results in a downward-sloping line.
• 🧑‍🏭 Multiplying the absolute value function by a factor stretches it vertically, changing the slopes.
• 😀 Adding a constant to the y-values shifts the entire function vertically.
• 📈 Graphing a transformed absolute value function involves combining multiple transformations.
• 🪈 Different transformation order can be used to achieve the same final graph.
• 🍃 Shifting left, stretching vertically, and shifting up are common transformations in graphing absolute value functions.

Questions & Answers

Q: How does replacing x with x + 3 shift the absolute value function?

Replacing x with x + 3 shifts the graph of an absolute value function three units to the left. The value being subtracted determines the amount of horizontal shift.

Q: What happens if the value inside the absolute value sign is negative?

If the value inside the absolute value sign is negative, the graph will have a slope of -1. This results in a downward-sloping line.

Q: How does multiplying by 2 affect the absolute value function graph?

Multiplying an absolute value function by 2 stretches the graph vertically, making it steeper. The slopes of the graph are multiplied by the same factor.

Q: What is the purpose of adding 2 to the y-values?

Adding 2 to the y-values of an absolute value function shifts the graph vertically two units up. This creates an upward translation.

Summary & Key Takeaways

• The graph of y = |x| represents the absolute value of x.

• To graph y = |x + 3|, shift the graph of y = |x| three units to the left.

• When graphing y = 2|x + 3|, multiply the slopes by 2 and create a steeper graph.

• Add 2 to the y-values of y = 2|x + 3| to shift the graph two units up and obtain the final function, f(x).