Algebra I

Graphing Quadratics

Lesson

The graph of a quadratic $y = ax^2 + bx + c$ is a parabola. Two key features to lock in: the vertex (turning point) and the axis of symmetry (the vertical line through the vertex).

Vertex form

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

The vertex is $(h, k)$ . The sign of $a$ tells you the direction:

$a > 0$ : parabola opens UP, vertex is the minimum.
$a < 0$ : parabola opens DOWN, vertex is the maximum.

Standard form

y = ax^2 + bx + c

Axis of symmetry: $x = -\frac{b}{2a}$ .
Vertex: x is the same; plug back in to get y.
y-intercept: $c$ (the constant).

Worked example 1 — vertex form

y = (x - 3)^2 + 5

Vertex: $(3, 5)$ . Opens up (since $a = 1 > 0$ ), so minimum y is 5.

Worked example 2 — standard form

y = x^2 - 6x + 5

Axis: $x = -(-6)/2 = 3$ . y at $x = 3$ : $9 - 18 + 5 = -4$ . Vertex: $(3, -4)$ .

Worked example 3 — y-intercept

y = 2x^2 - 3x + 7

y-intercept is the constant $c$ , so $7$ .

Watch the sign in vertex form

$y = (x - 3)^2$ has vertex $(3, 0)$ , not $(-3, 0)$ . The form is $(x - h)$ , so $h$ is the OPPOSITE sign of what you see.

How to type your answer

For a vertex point: 3,-4. For a single value (axis, min, max, intercept): just the number.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Vertex of } y = (x - 3)^2 + 5

Problem 2

\text{Vertex of } y = (x + 2)^2 - 1

Problem 3

\text{Axis of symmetry of } y = (x - 4)^2 + 7 \text{ (give just the } x \text{ value)}

Problem 4

\text{y-intercept of } y = x^2 - 3

Practice

Standard problems matching the lesson.

Problem 5

\text{Vertex of } y = (x - 1)^2 + 4

Problem 6

\text{Vertex of } y = (x + 5)^2 - 3

Problem 7

\text{Vertex of } y = -(x - 2)^2 + 8

Problem 8

\text{Axis of symmetry of } y = 2(x - 7)^2 - 5 \text{ (} x \text{ value)}

Problem 9

\text{y-intercept of } y = x^2 + 4x + 3

Problem 10

\text{y-intercept of } y = 2x^2 - 3x + 5

Problem 11

\text{Axis of symmetry of } y = x^2 - 6x + 5

Problem 12

\text{Axis of symmetry of } y = x^2 + 8x - 1

Problem 13

\text{Vertex of } y = x^2 - 4x + 1

Problem 14

\text{Vertex of } y = x^2 + 6x + 5

Problem 15

\text{Maximum } y \text{ of } y = -(x - 3)^2 + 10

Problem 16

\text{Minimum } y \text{ of } y = (x + 1)^2 - 7

Problem 17

Does y = -x^2 + 4 open up (1) or down (0)?

Problem 18

h(t) = -(t-2)^2 + 16. Max height?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

\text{Vertex of } y = 3(x - 2)^2 + 1

Problem 20

\text{Axis of symmetry of } y = 2x^2 + 8x - 3

Problem 21

\text{Vertex of } y = x^2 - 10x + 21

Problem 22

\text{Vertex of } y = -(x + 1)^2 + 3

Problem 23

\text{Minimum value of } y = 2(x - 3)^2 + 5

Problem 24

h(t) = -5(t-2)^2 + 20. Max height?

Problem 25

\text{y-intercept of } y = (x - 2)^2 - 4

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