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"Vertex / Maximum / Minimum" Explained

Vertex: 3 Best Ways to Master Maximum & Minimum Points

The vertex is the turning point of a parabola, its lowest point (minimum) when it opens upward, or its highest point (maximum) when it opens downward. From standard form, the x-coordinate is x = −b/2a; in vertex form y = a(x−h)² + k, it’s simply (h, k). One of the SAT’s most-tested quadratic concepts.

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Two Forms Effect of a How to Find It On the SAT Mistakes Practice FAQ

DEFINITION

What Is the Vertex of a Parabola?

When you graph a quadratic equation, the U-shaped curve is a parabola, and the vertex is its absolute minimum or maximum point. When the parabola opens upward (a > 0), the vertex is the lowest point a minimum. When it opens downward (a < 0), the vertex is the highest point a maximum.

Axis of symmetry: the vertical line through the vertex, x = -b/2a. It splits the parabola into two mirror-image halves, so the vertex's x-coordinate is always exactly halfway between the two x-intercepts.

Vertex

TWO WAYS TO READ IT

Vertex Form vs. Standard Form

From standard form you calculate the vertex; in vertex form you can read it straight off. Both appear on the SAT.

📐 Standard Form
y = ax2 + bx + c → x = -b/2a

The axis-of-symmetry formula gives the vertex's x-coordinate. Then substitute that x back into the equation to get y.

⭐ Vertex Form
y = a(x - h)2 + k

No calculation needed, the vertex is simply (h, k).

⚠️ Watch the sign of h: in (x - 3), h is +3; in (x + 3), h is -3.

DIRECTION & WIDTH

How the Value of a Affects the Parabola

he coefficient a controls both which way the parabola opens (and so whether the vertex is a min or max) and how wide or narrow it is.

a > 0 Opens upward → vertex is a minimum (lowest point).
a < 0 Opens downward → vertex is a maximum (highest point).
|a| > 1 Parabola is narrower (steeper).
0 < |a| < 1 Parabola is wider (flatter).

STEP BY STEP

How to Find the Vertex

1

Identify your coefficients

From standard form y = ax2 + bx + c, note the values of a and b (keep negative signs).

2

Calculate the x-coordinate

Apply the axis-of-symmetry formula: x = -b/2a.

3

Solve for the y-coordinate

Substitute that x-value back into the original equation to get y. The vertex is (x, y).

Find the vertex of y = 2x2 - 8x + 5.

1. Coefficients: a = 2, b = -8
2. x-coordinate: x = -b/2a = -(-8)/(2·2) = 8/4 = 2
3. y-coordinate: y = 2(22) - 8(2) + 5 = 8 - 16 + 5 = -3

Vertex (2, -3), a minimum, since a = 2 > 0

Find the vertex of y = -3(x + 1)2 + 7.

1. This is vertex form y = a(x - h)2 + k .
2. Match signs: (x + 1) = (x - (-1)) , so h = -1; k = 7.
3. a = -3 < 0 → opens downward → maximum.

Vertex (-1, 7), a maximum

WHERE IT SHOWS UP

The Vertex on the SAT & in the Real World

The vertex answers any “maximum” or “minimum” question, the max/min value is simply the y-coordinate of the vertex. It appears in word problems across business, physics, and engineering contexts.

CONTEXT WHAT THE VERTEX GIVES YOU SAT FREQUENCY
Projectile motion Maximum height of a thrown/launched object (a < 0) 1–2 per test
Business / profit Maximum profit or revenue, or minimum cost 1 per test
"Minimum value of f" The y-coordinate of the vertex when a > 0 1 per test
Graph features Axis of symmetry, vertex coordinates from a graph or equation 1 per test
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AVOID THESE

3 Common Vertex Mistakes

Dropping the Negative in -b/2a

When b is already negative, -b becomes positive. Students write -b/2a as negative anyway.

Fix: -(-8) = +8. Substitute b with its sign in parentheses before simplifying.

Order of Operations on y

When squaring the x-coordinate to find y, students multiply by a before applying the exponent.

Fix: square first, then multiply by a (PEMDAS). 2(22) = 2(4) = 8, not (2·2)2.

Wrong Sign for h in Vertex Form

Reading (x + 3) as h = +3. The form is (x - h), so (x + 3) means h = -3.

Fix: rewrite as (x - (-3)) to see the sign. The vertex x flips the sign inside the parentheses.

Practice Problems

Work each one, then reveal the answer to check yourself.

STANDARD FORM
Find the vertex of y = x2 - 6x + 5.
x = -(-6)/(2·1) = 3; y = 32 - 6(3) + 5 = 9 - 18 + 5 = -4. Vertex (3, -4), a minimum.
VERTEX FORM
State the vertex of y = 2(x - 4)2 - 9 and whether it's a max or min.
h = 4, k = -9 → vertex (4, -9). a = 2 > 0 → minimum.
MAX VALUE
A ball's height is h(t) = -16t2 + 64t + 5. What is its maximum height?
t = -64/(2·-16) = 2 s; h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = 69 (max height).
REASONING
A parabola has vertex (-2, 3) and passes through (0, 7). Is the vertex a max or a min?
At x = 0 the curve (y = 7) is above the vertex (y = 3), so it rises away from the vertex → opens upward → minimum.

TRY THESE

Vertex — FAQ

What is the vertex of a parabola? ×
The vertex is the highest or lowest point of a parabola, depending on the direction the curve opens. If the parabola opens upward (a > 0), the vertex is the minimum; if it opens downward (a < 0), the vertex is the maximum. It always lies on the axis of symmetry, x = −b/2a.
How do I find the maximum or minimum value of a quadratic? +
The maximum or minimum value is simply the y-coordinate of the vertex. Find the vertex's x-coordinate with x = −b/2a, substitute it back to get y, and that y is the max (if a < 0) or min (if a > 0). In vertex form y = a(x − h)² + k, the value is just k.
Can I use a calculator on this part of the SAT? +
Yes — the vertex formula works in both the calculator and non-calculator sections of the SAT, which makes it a highly valuable tool to memorize. On the calculator section you can also graph the quadratic and read the vertex directly from the display.
What's the difference between vertex form and standard form? +
Standard form is y = ax² + bx + c — you find the vertex by calculating x = −b/2a, then solving for y. Vertex form is y = a(x − h)² + k, where the vertex (h, k) can be read directly with no calculation. Converting to vertex form (by completing the square) is a common SAT Advanced Math task.
Can Inlighten's Orlando tutors help with quadratics and the vertex? +
Yes — the vertex formula, vertex vs. standard form, max/min word problems, and the graphing questions that appear across the SAT Advanced Math domain. We diagnose exactly where points are lost — the formula, the algebra, or the word-problem setup — before building a targeted plan. Book a free math assessment to start.

KEEP EXPLORING

Related Algebra Concepts

Quadratic Functions

→

Quadratic Formula

→

Intercepts

→

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