How do you find the vertex from vertex form?

In y = a(x − h)² + k the vertex is (h, k). Watch the sign: (x − 3) means h = 3, (x + 3) means h = −3.

How do you find the axis of symmetry from factored form?

Average the two zeros: halfway between them is the axis, and the vertex sits on it.

Graphing Parabolas · Grades 10–10

Graphing Parabolas Worksheets

Read the vertex, axis of symmetry, zeros and maximum or minimum from a graph or an equation, then complete the square and describe transformations of y = x². Grade 10 quadratic relations.

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What it is

A parabola is the graph of a quadratic. In vertex form y = a(x − h)² + k the vertex is (h, k), the axis of symmetry is x = h, and a decides the direction (up if positive, down if negative) and how narrow it is. Factored form y = a(x − r)(x − s) shows the zeros r and s.

Why it matters

A parabola is the path of anything thrown or launched. Its vertex is the highest point and its zeros are where it lands, so reading those features is reading the story the equation tells.

How to do it

Vertex form: read the vertex (h, k), remembering the sign flips inside the bracket.
a > 0 opens up (the vertex is a minimum); a < 0 opens down (a maximum).
Factored form: the zeros are r and s; the axis is halfway between them; substitute it to get the vertex's height.
Completing the square: take half the x-coefficient, square it, add and subtract it, then factor the perfect square.
Transformations of y = x²: a < 0 reflects, |a| ≠ 1 stretches, h shifts right or left, k shifts up or down.

Examples by level

Use the graph or the equation to answer each question.

Beginner
Zeros: x =Answerx = −4 and x = 2
Intermediatey = −(x + 4)². y-intercept =Answer−16
AdvancedA parabola has vertex (−4, −5) and passes through (−2, 3). Write its equation in vertex form.Answery = 2(x + 4)² − 5

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

Reading the vertex of y = (x − 3)² + 2 as (−3, 2): the sign inside the bracket flips.
Forgetting that a negative a makes the parabola open down, so the vertex is a maximum.
Completing the square without balancing: what you add inside must also be subtracted.

Tips

The vertex is always on the axis of symmetry.
Check a vertex-form answer by expanding it back.

For parents

Ask where the ball is highest and where it lands: that's the vertex and the zeros.

For teachers

Pair each equation question with its graph question so students see vertex form, factored form and the picture as one object.

Key vocabulary

vertex: the turning point of a parabola: its highest or lowest point
axis of symmetry: the vertical line through the vertex that splits the parabola in half
zeros: the x-intercepts, where y = 0

Frequently asked questions

How do you find the vertex from vertex form?: In y = a(x − h)² + k the vertex is (h, k). Watch the sign: (x − 3) means h = 3, (x + 3) means h = −3.
How do you find the axis of symmetry from factored form?: Average the two zeros: halfway between them is the axis, and the vertex sits on it.

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