Quadratic functions: vertex and extreme value

Warm-up

Three quick questions from earlier lessons.

Discover

A ball's height is $h(t) = -5t^2 + 20t$ meters after $t$ seconds. Its graph is a parabola that opens downward. When does the ball reach its highest point?

After 2 seconds
After 4 seconds

Key idea

Every quadratic $f(x) = ax^2 + bx + c$ has a vertex, its highest or lowest point.

The vertex is at

x = -\dfrac{b}{2a}

. Its height

f\left(-\dfrac{b}{2a}\right)

is the smallest value of

f

if

a > 0

, the largest if

a < 0

.

Completing the square gives the vertex form $f(x) = a(x - h)^2 + k$ , with vertex $(h, k)$ .

Worked example

Find the vertex of $f(x) = 2x^2 - 12x + 5$ .

x = -\dfrac{-12}{2 \cdot 2} = 3

a = 2, b = −12.

f(3) = 18 - 36 + 5 = -13

Evaluate f at the vertex.

Vertex

(3, -13)

, minimum value

-13

a = 2 > 0, so this is the lowest point.

Your turn, guided

Your turn: $f(x) = x^2 + 6x + 11$ .

x = -\dfrac{6}{2} = -3

a = 1, b = 6.

Minimum value:

f(-3) = 2

f(−3) = 9 − 18 + 11.

Practice ★

$f(x) = 5x^2 - 10x + 20$ . What is the $x$ -coordinate of the vertex?
$f(x) = 5x^2 - 40x + 10$ . What is the $x$ -coordinate of the vertex?
$f(x) = 2x^2 - 20x + 19$ . What is the $x$ -coordinate of the vertex?
$f(x) = 5x^2 - 30x + 5$ . What is the $x$ -coordinate of the vertex?

Watch out!

Practice ★★

$f(x) = x^2 + 10x + 21$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 4x + 22$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 10x + 19$ . What is the smallest value of $f(x)$ ?

Draw the graph of $y = -(x - h)^2 + 6$ for $h = 0$ : place the vertex and one more point.
Draw the graph of $y = -(x - h)^2 + 2$ for $h = 2$ : place the vertex and one more point.

Exit ticket

Three questions to check you have got it.

$f(x) = x^2 + 10x + 21$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 4x + 22$ . What is the smallest value of $f(x)$ ?
$f(x) = x^2 + 10x + 19$ . What is the smallest value of $f(x)$ ?

Summary card

Quadratic functions

Vertex at $x = -\dfrac{b}{2a}$ .
Its height $f\left(-\dfrac{b}{2a}\right)$ is the minimum ( $a > 0$ ) or maximum ( $a < 0$ ).
Vertex form: $a(x - h)^2 + k$ , vertex $(h, k)$ .

Watch out: the constant term $c$ is $f(0)$ , not the extreme value.

About this lesson