Quadratic graphs

How the SAT tests this, with worked examples — plus the 2 skills it covers. Free to read, no account needed.

A quadratic graphs as a parabola, and which form the equation is written in decides which feature you get for free.

Vertex form

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

hands you the turning point

(h, k)

directly — minding the sign, since

(x-4)

means

h = 4

. Factored form

y = a(x-r)(x-s)

hands you the roots. Standard form

y = ax^2 + bx + c

hands you the

y

-intercept

(0, c)

The sign of

a

decides the direction: positive opens upward with a minimum at the vertex, negative opens downward with a maximum.

A parabola is symmetric about the vertical line through its vertex, which gives a useful shortcut — the vertex sits exactly halfway between the roots. If the roots are

2

and

8

, the vertex is at

x = 5

So when a question asks for a maximum or minimum value, it is asking for the

y

-coordinate of the vertex, and when it asks where that occurs, it wants the

x

-coordinate.

Worked examples

Give the vertex of

y = -2(x-3)^2 + 7

.
Reading vertex form, the vertex is

(3, 7)

, and since

a = -2 < 0

the parabola opens downward — so

7

is a maximum.

A parabola has roots at

x = 2

and

x = 8

. Where is its axis of symmetry?
The vertex lies halfway between the roots

{x = \frac{2+8}{2} = 5}

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Quadratic graphs

Worked examples

Is quadratic graphs costing you points?

All the skills in Quadratic graphs