FunctionsCourse skill

Vertex of a quadratic - from completed-square form

Read the vertex (turning point) of a quadratic in completed-square form.

  • For , where , the vertex is .
  • The square is zero when , so . Read the sign inside the brackets carefully.
  • If , the graph opens upwards and the vertex is a minimum. If , it opens downwards and the vertex is a maximum.
  • Changing changes the shape and direction, but keeps the vertex fixed when and stay the same.

Example 1

: opens upwards

Find the vertex and state whether it is a minimum or maximum.

Quadratic with vertex (1, -2), opening upwards with a minimum.−2−101234−4−3−2−1123456

The vertex is a minimum because .

Example 2

: opens downwards

Find the vertex and state whether it is a minimum or maximum.

Quadratic with vertex (-2, 3), opening downwards with a maximum.−5−4−3−2−101−7−6−5−4−3−2−112345

The vertex is a maximum because .

Remember
In , write : the vertex’s -coordinate is , not .

Your turn

5 practice questions
Question 1

Find the vertex and state whether it is a minimum or maximum.

Check answerQuadratic with vertex (2, 3), opening upwards with a minimum.−1012345−11234567891011

The vertex is a minimum because .

Question 2

Find the vertex and state whether it is a minimum or maximum.

Check answerQuadratic with vertex (-4, 1), opening upwards with a minimum.−7−6−5−4−3−2−101−1123456789

The vertex is a minimum because .

Question 3

Find the vertex and state whether it is a minimum or maximum.

Check answerQuadratic with vertex (3, -5), opening downwards with a maximum.−10123456−15−14−13−12−11−10−9−8−7−6−5−4−3−2−11

The vertex is a maximum because .

Question 4

Find the vertex and state whether it is a minimum or maximum.

Check answerQuadratic with vertex (0, -2), opening upwards with a minimum.−3−2−10123−4−3−2−1123456

The vertex is a minimum because .

Question 5

Find the vertex and state whether it is a minimum or maximum.

Check answerQuadratic with vertex (-1, 4), opening downwards with a maximum.−4−3−2−1012−6−5−4−3−2−1123456

The vertex is a maximum because .

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