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Vertex Form

Vertex form is a way of writing a quadratic equation as y=a(x−h)2+ky = a(x - h)^2 + k, where the point (h,k)(h, k) is the vertex of the parabola. It makes it easy to read off the highest or lowest point of the graph directly from the equation.

Vertex form is a representation of a quadratic function given by y=a(x−h)2+ky = a(x - h)^2 + k, where a≠0a \neq 0. The vertex of the corresponding parabola is located at (h,k)(h, k), and the axis of symmetry is the vertical line x=hx = h. When a>0a > 0, the parabola opens upward and the vertex is a minimum; when a<0a < 0, it opens downward and the vertex is a maximum. The parameter ∣a∣|a| controls how narrow or wide the parabola is.

Key Formula

y=a(x−h)2+ky = a(x - h)^2 + k
Where:
  • aa = controls the direction and width of the parabola (opens up if positive, down if negative)
  • hh = the x-coordinate of the vertex
  • kk = the y-coordinate of the vertex
  • (h,k)(h, k) = the vertex of the parabola

Worked Example

Problem: Convert the quadratic equation y=2x2−12x+22y = 2x^2 - 12x + 22 into vertex form and identify the vertex.
Step 1: Factor the leading coefficient out of the first two terms.
y=2(x2−6x)+22y = 2(x^2 - 6x) + 22
Step 2: Complete the square inside the parentheses. Take half of −6-6, which is −3-3, and square it to get 99. Add and subtract 99 inside the parentheses.
y=2(x2−6x+9−9)+22y = 2(x^2 - 6x + 9 - 9) + 22
Step 3: Rewrite the perfect square trinomial as a squared binomial, and distribute the 22 to the −9-9.
y=2(x−3)2−18+22y = 2(x - 3)^2 - 18 + 22
Step 4: Simplify the constant terms.
y=2(x−3)2+4y = 2(x - 3)^2 + 4
Answer: The vertex form is y=2(x−3)2+4y = 2(x - 3)^2 + 4. The vertex is (3,4)(3, 4), the parabola opens upward, and the axis of symmetry is x=3x = 3.

Visualization

Why It Matters

Vertex form is especially useful when you need to quickly identify the maximum or minimum value of a quadratic function. In physics, for instance, the peak height of a projectile corresponds to the vertex of a parabolic path. It also simplifies graphing, since you can plot the vertex first and then use the value of aa to determine the shape of the curve.

Common Mistakes

Mistake: Reading the vertex as (−h,k)(−h, k) instead of (h,k)(h, k) because the formula has a minus sign: (x−h)(x - h).
Correction: If the equation is y=3(x+2)2+5y = 3(x + 2)^2 + 5, rewrite x+2x + 2 as x−(−2)x - (-2). The vertex is (−2,5)(-2, 5), not (2,5)(2, 5). Always check the sign carefully.
Mistake: Forgetting to distribute the leading coefficient aa when completing the square.
Correction: After adding a value inside the parentheses to complete the square, remember that the value is multiplied by aa. For example, adding 99 inside parentheses preceded by 22 actually adds 1818 to the expression, so you must subtract 1818 outside to compensate.

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