Vertex Form

Accueil » Vertex Form
Vertex Form: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Vertex form is the layout y = a(x − h)² + k for a quadratic, which shows the turning point (h, k) and the opening direction without any calculation.

Algebra 1Functions and GraphsA-SSE.B.3b

What is Vertex Form?

The vertex form \(y = a(x – h)^2 + k\) places the turning point right in view: it is \((h, k)\), and the mirror line is \(x = h\). The number \(a\) is the same stretch factor you meet in the other layout.

A positive \(a\) gives a bowl, a negative \(a\) gives an arch, and a bigger absolute value squeezes the curve narrower. Watch the sign trap: \(y = (x + 2)^2\) has \(h = -2\), since \(x + 2\) means \(x – (-2)\).

How to spot it

  • A quadratic shows a squared parenthesis with a number added or subtracted outside it.
  • A turning point is supplied and an equation through one more point is wanted.
  • A question on vertex form slides a basic parabola right, left, up or down.
  • The task asks for the greatest or least value, which is the number k.

The method

  1. Read h and k from the equation, remembering that \((x + 2)\) means \(h = -2\).
  2. Report the turning point \((h, k)\) and the mirror line \(x = h\); the sign of a gives the orientation.
  3. To build an equation, insert h and k, then substitute a second point and solve for a.
  4. To reach the other layout, expand the square and collect like terms.
vertex (h, k) = (3, 8)(1, 0)(5, 0)x = 3y = −2(x − 3)² + 8a = −2: opens down
y = −2(x − 3)² + 8 has its vertex at (3, 8) and opens downward.

Vertex Form: solved example

Problem

A curve has its turning point at \((-2, 5)\) and goes through \((0, -3)\). Write its equation in this layout.

Solution

Begin with \(y = a(x + 2)^2 + 5\). Insert \(x = 0\) and \(y = -3\): \(-3 = a(2)^2 + 5\).

That gives \(4a = -8\), so \(a = -2\), and the equation reads \(y = -2(x + 2)^2 + 5\).

Try it

Question 1

Name the turning point of \(y = 3(x – 4)^2 – 7\).

Show the answer

\(h = 4\) and \(k = -7\), so \((4, -7)\).

Question 2

Name the turning point of \(y = (x + 1)^2 + 6\).

Show the answer

\(x + 1 = x – (-1)\), so \(h = -1\) and the point is \((-1, 6)\).

Vertex Form vs. Standard Form of a Quadratic

The expanded layout \(ax^2 + bx + c\) shows the y-intercept c immediately and hides the turning point. This layout does the reverse. Neither is superior; they describe the same curve, and expanding or completing the square converts between them.

Standard Form of a Quadratic

Keep going

Lessons

Worksheets and quizzes

Related words

Whole dictionary: A–Z · Algebra 1 · Functions and Graphs

Voter.. page