y = |x| is two half-lines meeting at the origin, forming a V.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Moving the vertex
y = |x − h| + k moves the corner to (h, k), with the inside term running backwards as always.
Turning it over
A negative outside flips the V upside down, so the vertex becomes a maximum.
Piecewise generally
Different rules on different stretches. Decide which condition holds before computing.
A V-shape is two lines
y = |x| is y = x for x ≥ 0 and y = −x for x < 0. Every absolute value graph is a piecewise linear function, and rewriting it that way turns an unfamiliar shape into two familiar ones.
The corner is where the inside is zero
For y = |x − 3| + 2 the corner sits at x = 3, where the expression inside changes sign. Finding that point first tells you where to split the function and where the vertex of the V lies.
Transformations apply as usual
y = −2|x − 3| + 5 is the basic V shifted right 3, stretched by 2, reflected downwards and raised 5. The transformation rules from the previous lesson work unchanged on an absolute value graph.
Pieces must agree at the boundary
A piecewise function is continuous only if both formulas give the same value at the join. Checking the boundary is where these problems are actually decided, and a jump there is a real feature, not an error.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Naomi Locate a Vertex
Naomi describes y = |x − 3| + 2.
- Step 1
The (x − 3) is inside, so it shifts right by 3.
Step 4: Your Turn
Practice makes it stick.
The Corner
Problem 1 of 2
y = |x − 5|. What is the x of the vertex?
The Value
Problem 2 of 2
y = |x − 5| + 1. What is y at x = 5?
Vertices and Values
1 of 8
y = |x + 2|. What is the x of the vertex?
2 of 8
y = |x| + 7. What is the minimum value?
3 of 8
y = −|x|. Is the vertex a maximum? 1 for yes, 0 for no.
4 of 8
y = |x − 4|. What is y at x = 1?
5 of 8
f(x) = 2x for x < 3, else 10. What is f(1)?
6 of 8
Same function. What is f(8)?
7 of 8
Match each change to its effect on the V.
Tap a card on the left to start.
8 of 8
y = |x|. How many x values give y = 4?
Step 5: Quick Check
Show what you know.
Question 1 of 1
y = |x − 7| + 3. What is the minimum value?
What You Learned
- y = |x| is a V with its corner at the origin.
- The same transformation rules move and flip it as any other function.
- A piecewise function needs the right condition chosen before anything is computed.