Cogito
Integrated Math 2 · Chapter 1 · Lesson 2
Vertex Form and Transformations
Reading the vertex straight off the equation.
12 problems · about 21 minutes · F-BF.B.3, F-IF.C.8
What this lesson teaches
The student converts to vertex form and describes a parabola as a transformation of y = x².
- Vertex form is y = a(x − h)² + k, with vertex (h, k).
- The sign of h appears reversed because the form subtracts it.
- Completing the square converts standard form to vertex form.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy = (x − 6)² + 1. What is the vertex x-coordinate?
Answer 6
Why 6.
Why does y = (x − 3)² have its vertex at x = 3?
Answer The bracket is zero there, which is its smallest possible value.
Why The squared term is smallest when the bracket vanishes.
y = (x − 5)² + 2. Vertex x-coordinate?
Answer 5
Why The sign reverses.
y = (x + 4)² − 1. Vertex x-coordinate?
Answer -4
Why x + 4 is x − (−4).
y = (x − 2)² + 7. What is the minimum value?
Answer 7
Why The squared part is at least zero.
Build It Up
The same ideas with more to keep track of.
3 problemsTo complete the square on x² − 8x, what is added?
Answer 16
Why (−4)².
To complete the square on x² + 10x, what is added?
Answer 25
Why 5².
y = −(x − 1)² + 9. What is the maximum value?
Answer 9
Why The k value, since a is negative.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the completing-the-square steps in order.
Answer 1. Halve the coefficient of x. 2. Square that result. 3. Add and subtract it inside the expression. 4. Write the perfect square and collect the constants.
Why Halving comes before squaring.
y = (x − 7)² − 3. Vertex y-coordinate?
Answer -3
Why The k value.
The Vertex: y = (x − 3)² − 4. What is the x-coordinate of the vertex?
Answer 3
Why 3.
The Minimum: Same equation. What is the minimum value of y?
Answer -4
Why −4.