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Learn · Mathematics · Grade 10 · Functions

The parabola: graph of a quadratic function

How a, b and c shape y = ax² + bx + c, the vertex and axis of symmetry, where the graph crosses the axes, and the vertex form a(x − h)² + k.

High school Lesson 19 of 31 Interactive figure Practice

First, the rules

Every quadratic draws a parabola

★ Shape
Example 1
y = ax² + bx + c
a > 0 opens up (a minimum), a < 0 opens down (a maximum). Larger |a| is narrower.
★ Vertex and axis
Examples 1, 2
xv = −b2a, yv = f(xv) = c − b²4a
The axis of symmetry is the vertical line x = xv.
Intercepts
Example 1
y-axis at (0, c). x-axis at the roots of ax² + bx + c = 0 — two, one or none, as Δ says.
★ Vertex form
Example 2
y = a(x − h)² + k
Vertex (h, k) read straight off. Get it by completing the square.
Factored form
Example 3
y = a(x − x₁)(x − x₂)
Roots read straight off; the vertex is halfway between them.
Sign of the function
Example 3
With two roots x₁ < x₂: for a > 0 the function is negative between the roots and positive outside; for a < 0 the opposite.
Maximum / minimum problems
Example 3
The extreme value of a quadratic is yv, reached at x = xv.

See it

Move a, b and c

Watch the vertex (green) slide along, the axis follow it, the y-intercept stay at c, and the x-intercepts appear and disappear as Δ changes sign.

Step by step

Worked examples

1

Sketch from the standard form

vertex, intercepts

Sketch y = x² − 4x − 5: direction, vertex, axis, intercepts.

Ask yourself: Four facts make the sketch: which way it opens, where the vertex is, where it meets each axis.

Show the solution
  1. a = 1 > 0: opens upward.
  2. Vertex: xv = −−42 = 2, yv = 4 − 8 − 5 = −9. Axis x = 2.
  3. y-intercept (0, −5). x-intercepts: x² − 4x − 5 = (x − 5)(x + 1) = 0, at (−1, 0) and (5, 0) — symmetric about x = 2. ✓

Answer up; vertex (2, −9); axis x = 2; (0, −5), (−1, 0), (5, 0)

2

Completing the square

vertex form

Write y = 2x² + 12x + 7 in vertex form and state its minimum value.

Ask yourself: Factor the a out of the x-terms first, then make a perfect square inside.

Show the solution
  1. y = 2(x² + 6x) + 7.
  2. Half of 6 is 3, 3² = 9: y = 2(x² + 6x + 9 − 9) + 7 = 2(x + 3)² − 18 + 7.
  3. y = 2(x + 3)² − 11: vertex (−3, −11), and since a > 0 the minimum value is −11. Check with the formula: xv = −12 / 4 = −3. ✓

Answer y = 2(x + 3)² − 11; minimum −11

3

A maximum area

optimisation

A farmer has 40 m of fence to enclose a rectangular field against a straight wall (no fence needed on the wall side). What dimensions give the largest area, and what is that area?

Ask yourself: Write the area as a function of one variable; the maximum of a downward parabola is its vertex.

Show the solution
  1. Let x be the side perpendicular to the wall. Fence: 2x + w = 40 ⇒ w = 40 − 2x.
  2. Area A(x) = x(40 − 2x) = −2x² + 40x — a parabola opening down (a = −2).
  3. Vertex: xv = −402 · (−2) = 10, so w = 20 and A = 10 · 20 = 200.
  4. Factored form −2x(x − 20) shows the roots 0 and 20; the vertex sits halfway, at 10. ✓

Answer 10 m by 20 m, area 200 m²

Practice: the parabola

0 / 0

Vertex coordinates, direction, axis of symmetry and intercepts from the equation. Your best score is kept in this browser only.

About this lesson

Quadratic functions and their graphs: the parabola y = ax² + bx + c, the sign of a (opens up or down), the vertex at x = −b / 2a, the axis of symmetry, the y-intercept c, the x-intercepts from the roots, the vertex form a(x − h)² + k, maximum and minimum values, and the sign of the function between and outside the roots. Drag the sliders and watch the parabola move, then practise.

Part of Learn mathematics · Grade 10 · Functions. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.