Quadratic Vertex Form

\( y=a(x-h)^2+k \)
Algebra GCSE
Question 3 of 20
← All formulas ⟵ Prev Next ⟶

\( Find the maximum value of y = -2(x + 6)^2 - 3 \)

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
\( Maximum = k since a < 0 \)

Explanation

Show / hide — toggle with X

Statement

A quadratic function can also be expressed in vertex form:

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

Here, \((h, k)\) represents the vertex of the parabola, and the parameter \(a\) controls the width and direction. If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards. The axis of symmetry is the vertical line \(x = h\).

Why it’s true

  • Completing the square on a quadratic in standard form \(ax^2 + bx + c\) always produces the vertex form.
  • Shifting the parabola horizontally by \(h\) units and vertically by \(k\) units directly reveals the vertex.
  • The shape factor \(a\) stretches or compresses the parabola.

Recipe (how to use it)

  1. Read off the vertex directly: \((h, k)\).
  2. Identify the axis of symmetry as \(x = h\).
  3. Note whether the parabola opens up (\(a>0\)) or down (\(a<0\)).
  4. Sketch quickly by using symmetry and a few points around the vertex.

Spotting it

If the quadratic is written as \((x - h)^2\) plus or minus some number, it is already in vertex form. This is the easiest form for graphing.

Common pairings

  • Converting from standard form by completing the square.
  • Solving optimisation problems where the maximum or minimum value is obvious from the vertex.
  • Using transformations of the basic graph \(y = x^2\).

Mini examples

  1. Given: \(y = 2(x - 3)^2 + 1\). Vertex: \((3, 1)\).
  2. Given: \(y = -(x + 2)^2 + 4\). Vertex: \((-2, 4)\).

Pitfalls

  • Forgetting that the formula has a minus: \(y = a(x - h)^2 + k\). If you see \(x+2\), then \(h = -2\).
  • Mixing up the sign of \(k\). It is exactly as written, no change.
  • Confusing the value of \(a\): large |a| makes the parabola narrow; small |a| makes it wide.

Exam strategy

  • If you’re asked for the maximum or minimum, just read off \(k\) from vertex form.
  • If you need to sketch, mark the vertex and axis of symmetry first.
  • Always double-check the signs of \(h\) and \(k\).

Summary

The vertex form \(y = a(x-h)^2 + k\) is a powerful way to write quadratics. It gives the vertex and axis of symmetry immediately and is very helpful for graph sketching and optimisation. Mastery of this form allows students to connect algebraic manipulation with graphical understanding.

Worked examples

Show / hide (10) — toggle with E
  1. \( Find the vertex of y = (x - 2)^2 + 5 \)
    1. \( Compare with y = a(x-h)^2 + k \)
    2. \( h = 2, k = 5 \)
    Answer: (2, 5)
  2. \( Find the vertex of y = -3(x + 4)^2 - 2 \)
    1. \( h = -4 (because x+4) \)
    2. \( k = -2 \)
    Answer: (-4, -2)
  3. \( Find the axis of symmetry of y = 2(x - 1)^2 - 7 \)
    1. \( h = 1 \)
    2. \( Axis: x = 1 \)
    Answer: \( x = 1 \)
  4. \( Find the vertex of y = (x + 3)^2 \)
    1. \( h = -3, k = 0 \)
    Answer: (-3, 0)
  5. \( Find the maximum point of y = -(x - 5)^2 + 4 \)
    1. \( h = 5, k = 4 \)
    2. Since a < 0, this is a maximum point
    Answer: (5, 4)
  6. \( Find the vertex of y = 0.5(x - 6)^2 - 3 \)
    1. \( h = 6, k = -3 \)
    Answer: (6, -3)
  7. \( Find the vertex of y = -2(x + 1)^2 + 9 \)
    1. \( h = -1, k = 9 \)
    Answer: (-1, 9)
  8. \( Find the axis of symmetry of y = 4(x - 7)^2 + 2 \)
    1. \( h = 7 \)
    2. \( Axis: x = 7 \)
    Answer: \( x = 7 \)
  9. \( Find the minimum value of y = 3(x - 2)^2 - 5 \)
    1. Vertex (2,-5)
    2. Since a > 0, minimum is -5
    Answer: \( Minimum = -5 at x=2 \)
  10. \( Find the vertex of y = -0.5(x + 2)^2 - 4 \)
    1. \( h = -2, k = -4 \)
    Answer: (-2, -4)