Graphs of quadratic functions: direction, intercepts, symmetry

Warm-up

Three quick questions from earlier lessons.

Discover

$y = -2x^2 + 8x + 3$ . What happens far to the left and right?

$y$ goes down on both sides
$y$ goes up on both sides

Key idea

Read a parabola from $y = ax^2 + bx + c$ .

a > 0

: opens up (minimum);

a < 0

: opens down (maximum); larger

|a|

, narrower.

c

is the $y$ -intercept. The vertex lies on the axis of symmetry, halfway between the

x

-intercepts.

Worked example

$y = x^2 - 6x + 5 = (x - 1)(x - 5)$ .

Opens up

a = 1 > 0.

y

-intercept 5

x = 0.

Vertex at

x = 3

,

y = -4

Halfway between 1 and 5.

Your turn, guided

Your turn: a parabola crosses the $x$ -axis at $-2$ and $6$ . The $x$ -coordinate of its vertex.

2

(−2 + 6)/2.

Practice ★

What is the $y$ -intercept of $y = 2x^2 + 1x + 11$ ?
What is the $y$ -intercept of $y = 4x^2 + 5x + 7$ ?

Does the graph of $y = -6x^2 + 5x + 2$ open up or down?
1. Down, so it has a maximum
2. Up, so it has a minimum
3. It depends on 2

Watch out!

Practice ★★

A parabola crosses the $x$ -axis at $x = -1$ and $x = 3$ . What is the $x$ -coordinate of its vertex?
A parabola crosses the $x$ -axis at $x = -9$ and $x = 5$ . What is the $x$ -coordinate of its vertex?

Boss problem. A ball is kicked from the ground. Its height in feet after $t$ seconds is $h(t) = -16t^2 + 32t = -16t(t - 2)$ .

Exit ticket

Three questions to check you have got it.

What is the $y$ -intercept of $y = 2x^2 + 1x + 11$ ?
What is the $y$ -intercept of $y = 4x^2 + 5x + 7$ ?
What is the $y$ -intercept of $y = 2x^2 + 6x + 15$ ?

Summary card

Graphs of quadratics

Sign of $a$ : up or down; $c$ : $y$ -intercept; vertex midway between the zeros.

Watch out: the $y$ -intercept is $c$ , not $a$ .

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