Skill: quadratic-graphing

Graphing Quadratics

Every parabola is the parent y = x² stretched, flipped, or shifted. Vertex form y = a(x − h)² + k reads the whole graph at a glance: vertex (h, k), axis x = h, and the direction from the sign of a.

1 Understand

The core idea in plain language.

What it is. Vertex form y = a(x − h)² + k puts the vertex (h, k) — the turning point — on display. The axis of symmetry is the vertical line x = h. If a > 0 the parabola opens up (vertex is a minimum); if a < 0 it opens down (vertex is a maximum). |a| > 1 narrows it, |a| < 1 widens it.

Why it matters. The vertex is the answer to every optimization question — max height, min cost, best price. Reading it straight from the equation beats plotting points.

Where it is used. Projectile max height · profit maximization · satellite dishes and headlight reflectors (parabolic shapes).

2 See It

Diagrams that make the idea visual.

Vertex form reads the graph

y = 2(x − 1)² + 3: the vertex is (1, 3), the axis is x = 1, a = 2 > 0 so it opens up and is narrower than the parent. Every number in the form means something.

y = 2(x − 1)² + 3: vertex (1, 3), axis x = 1, opens up, narrower.

Transformations of the parent

Start from y = x². The h inside the square shifts left/right (opposite intuition), the k outside shifts up/down, a stretches or compresses, and a negative a reflects over the x-axis.

Parent y = x² (dashed) vs y = 2(x − 1)² + 3: narrower, shifted right 1, up 3.

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Vertex: y = x² − 4x + 3

h = −b/2a = 4/2 = 2.
k = f(2) = 4 − 8 + 3 = −1. Vertex (2, −1), axis x = 2.

Vertex (2, −1); axis of symmetry x = 2

Check: a > 0 so it opens up — (2, −1) is the minimum point.

Example 2 Vertex: y = 2x² − 4x + 5

h = −(−4)/(2·2) = 1.
k = 2(1) − 4(1) + 5 = 3. Vertex (1, 3), axis x = 1.

Vertex (1, 3); axis x = 1; opens up (minimum y = 3)

Check: f(1) = 3 and nearby values are larger (checks).

Example 3 Read: y = −(x + 2)² − 1

Vertex form with a = −1, h = −2 (x + 2 = x − (−2)), k = −1.
Opens down → vertex (−2, −1) is the maximum.

Vertex (−2, −1); axis x = −2; opens down (maximum y = −1)

Check: Expanded: −(x² + 4x + 4) − 1 = −x² − 4x − 5 (checks).

Example 4 Convert: y = x² + 6x + 5 to vertex form

h = −6/2 = −3; k = 9 − 18 + 5 = −4.
Vertex form: y = (x + 3)² − 4.

y = (x + 3)² − 4; vertex (−3, −4)

Check: (x + 3)² − 4 = x² + 6x + 9 − 4 = x² + 6x + 5 (checks).

4 Common Mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

Mistake 1: sign of h

Wrong

y = (x − 3)² has vertex (−3, 0).

Right

(3, 0) — vertex form is (x − h)², so (x − 3) means h = +3.

Rule: the sign flips: (x − 3)² shifts RIGHT 3.

Mistake 2: axis = h, not (h, k)

Wrong

The axis of symmetry is (2, −1).

Right

x = 2 — the axis is a vertical LINE, not a point.

Rule: axis: x = h. Vertex: (h, k).

Mistake 3: max/min mix-up

Wrong

y = −x² + 4 has a minimum at (0, 4).

Right

Maximum — a < 0 opens down, so the vertex is the top.

Rule: opens up → minimum; opens down → maximum.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Find the vertex and axis of y = x² + 6x + 5.

Answer

Vertex (−3, −4), axis x = −3.

2. Find the vertex and axis of y = −x² + 2x + 1. Does it open up or down?

Answer

Vertex (1, 2), axis x = 1, opens down (maximum).

3. Find the vertex of y = 3x² − 12x + 7.

Answer

Vertex (2, −5).

Key Points to Remember

Vertex form y = a(x − h)² + k: vertex (h, k), axis x = h.
(x − h)² with MINUS means shift right; (x + h)² shifts left.
a > 0 opens up (vertex = minimum); a < 0 opens down (vertex = maximum).
|a| > 1 narrows vs parent; |a| < 1 widens; negative a reflects.
From standard form: h = −b/2a, k = f(h).
y-intercept is always (0, c).
The axis is a line (x = h), not a point.

Practice problems Try the tool See it on FiboMap