Before you read

Answer first — the lesson below explains it.

A projectile follows $f (x) = (x - 4)^{2} + 1$ . What is the vertex (lowest point) of its path?

A
$(4, 1)$
B
$(- 4, 1)$
C
$(4, - 1)$
D
$(1, 4)$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A revenue model is $f (x) = - 3 (x + 2)^{2} + 7$ . What can be said about the maximum?

A
A minimum value of $7$
B
A maximum value of $7$
C
A minimum value of $- 2$
D
A maximum value of $- 2$

Check your answer

Answer:

Theory

The Three Forms of a Quadratic

Every quadratic function can be written in three equivalent forms. Each reveals different information:

Form	Equation	Reveals
Standard	$f(x) = ax^2 + bx + c$	y-intercept ( $c$ ), direction ( $a$ )
Vertex	$f(x) = a(x - h)^2 + k$	Vertex $(h, k)$ , direction ( $a$ )
Factored	$f(x) = a(x - r)(x - s)$	x-intercepts ( $r$ and $s$ )

All three describe the same parabola — just different windows into its properties.

Parabola with vertex (-2, -1) opening upward

Parabola opening downward: a < 0

Quick check

You just read it. Can you apply it?

An engineering model $f (x) = 2 (x - 1) (x - 5)$ crosses the x-axis at which points?

A
$x = 1$ and $x = 5$
B
$x = - 1$ and $x = - 5$
C
$x = 2$ and $x = 10$
D
$x = 1$ and $x = - 5$

Check your answer

Answer:

Factored form: zeros at $x = 1$ and $x = 5$ .

Theory

Key Features of a Parabola

Feature	How to find it
Direction	$a > 0$ : opens up ( $\cup$ , minimum). $a < 0$ : opens down ( $\cap$ , maximum)
Vertex	From vertex form: $(h, k)$ . From standard: $h = -\frac{b}{2a}$ , $k = f(h)$
Axis of symmetry	$x = h$ (vertical line through vertex)
Y-intercept	$f(0) = c$ (in standard form)
X-intercepts	Solve $f(x) = 0$ (factor, formula, or Desmos)
Width	$\|a\|$ large $\to$ narrow. $\|a\|$ small $\to$ wide. $\|a\| = 1$ $\to$ standard width

Example 1

For $f(x) = -2(x - 3)^2 + 8$ , identify: vertex, direction, axis of symmetry, max/min.

Vertex form: $a = -2$ , $h = 3$ , $k = 8$ .

Vertex: $(3, 8)$ .

Direction: $a = -2 < 0$ $\to$ opens down.

Axis of symmetry: $x = 3$ .

Maximum value: $8$ (since it opens down).

Vertex

(3, 8)

, opens down, axis

x = 3

, max

= 8

Example 2

For $f(x) = x^2 - 6x + 5$ , find the x-intercepts and vertex.

X-intercepts: $x^2 - 6x + 5 = 0$ $\to$ $(x-1)(x-5) = 0$ $\to$ $x = 1, 5$ .

Vertex: $h = \frac{1+5}{2} = 3$ (midpoint of intercepts!).

$k = f(3) = 9 - 18 + 5 = -4$ .

Vertex: $(3, -4)$ .

X-intercepts:

(1, 0)

and

(5, 0)

. Vertex:

(3, -4)

Quick check

You just read it. Can you apply it?

A parabolic bridge has equation $y = x^{2} - 10 x + 21$ . What is its vertex?

A
$(5, - 4)$
B
$(5, 4)$
C
$(- 5, - 4)$
D
$(10, 21)$

Check your answer

Answer:

$h = 10/2 = 5$ . $k = 25 - 50 + 21 = - 4$ . Vertex: $(5, - 4)$ .

Theory

The Midpoint Shortcut

The axis of symmetry is always the midpoint of the two x-intercepts:

$h = \frac{r + s}{2}$

This is incredibly useful on the SAT when the factored form is given. No need for $-b/(2a)$ — just average the roots!

Quick check

You just read it. Can you apply it?

If $f (x) = a (x - 3) (x + 1)$ and the y-intercept is $- 6$ , what is $a$ ?

A
$- 2$
B
$2$
C
$- 6$
D
$3$

Check your answer

Answer:

$f (0) = a (- 3) (1) = - 3 a = - 6 \Rightarrow a = 2$ .

Common Mistakes

Common Mistakes to Avoid

Getting the vertex sign wrong from vertex form

$f(x) = (x + 3)^2 - 5$ $\to$ vertex is $(3, -5)$

(x + 3) = (x - (-3))

, so

h = -3

. Vertex:

(-3, -5)

Thinking the vertex is always the y-intercept

The vertex of $x^2 + 4x + 3$ is $(0, 3)$

The y-intercept is

(0, 3)

. The vertex is

(-2, -1)

. They're different!

Quick check

You just read it. Can you apply it?

A parabolic trajectory has x-intercepts at $(- 2, 0)$ and $(6, 0)$ . What is the axis of symmetry?

A
$x = 2$
B
$x = 4$
C
$x = - 2$
D
$x = 3$

Check your answer

Answer:

Axis $= \frac{- 2 + 6}{2} = \frac{4}{2} = 2$ . So $x = 2$ .

Strategy

SAT Strategy Tip

The SAT loves asking about the vertex and what it represents in context. When a quadratic models height, profit, or area, the vertex gives the maximum or minimum value. The x-coordinate tells you when it occurs, the y-coordinate tells you what that value is. Use Desmos to instantly see the vertex.

After reading

Same question as before. What do you say now?

A projectile follows $f (x) = (x - 4)^{2} + 1$ . What is the vertex (lowest point) of its path?

A
$(4, 1)$
B
$(- 4, 1)$
C
$(4, - 1)$
D
$(1, 4)$

Check your answer

Answer:

Vertex form: $h = 4$ , $k = 1$ . Vertex: $(4, 1)$ .

After reading

Same question as before. What do you say now?

A revenue model is $f (x) = - 3 (x + 2)^{2} + 7$ . What can be said about the maximum?

A
A minimum value of $7$
B
A maximum value of $7$
C
A minimum value of $- 2$
D
A maximum value of $- 2$

Check your answer

Answer:

$a = - 3 < 0$ : opens down, so there is a maximum. Vertex: $(- 2, 7)$ . Maximum value: $7$ .

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