Vertex form of a quadratic

Warm-up

Three quick questions from earlier lessons.

Discover

$y = (x - 4)^2 + 1$ . A square is never negative. What is the smallest value of $y$ , and for which $x$ ?

$y = 1$ at $x = 4$
$y = 1$ at $x = -4$

Key idea

Vertex form shows the vertex.

In vertex form

y = a(x - h)^2 + k

the vertex is

(h, k)

. Compared with

y = ax^2

, the graph is shifted

h

right and

k

up. Watch the sign:

(x + 3)^2

means

h = -3

Worked example

$y = 2(x + 3)^2 - 5$ .

h = -3

x + 3 = x − (−3).

k = -5

Constant.

Vertex

(-3, -5)

, a minimum

a = 2 > 0.

Your turn, guided

Your turn: the $y$ -coordinate of the vertex of $y = -(x - 2)^2 + 9$ .

9

k = 9.

Practice ★

What is the $x$ -coordinate of the vertex of $y = 3(x + 1)^2 + 4$ ?
What is the $x$ -coordinate of the vertex of $y = 5(x + 9)^2 + 6$ ?

What is the smallest value of $y = (x - 9)^2 - 3$ ?

Watch out!

Practice ★★

Write a quadratic with vertex $(8, 1)$ and leading coefficient 2, in vertex form.
Write a quadratic with vertex $(4, 1)$ and leading coefficient 4, in vertex form.

Exit ticket

Three questions to check you have got it.

What is the $x$ -coordinate of the vertex of $y = 3(x + 1)^2 + 4$ ?
What is the $x$ -coordinate of the vertex of $y = 5(x + 9)^2 + 6$ ?
What is the $x$ -coordinate of the vertex of $y = 8(x + 5)^2 + 9$ ?

Summary card

Vertex form

$y = a(x - h)^2 + k$ : vertex $(h, k)$ .

Watch out: $(x + h)$ means the vertex is at $-h$ .

End of the lesson

Next lesson: Quadratic functions: vertex and extreme value Back to the chapter

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