Vertex form of a quadratic
Vertex form writes a quadratic so that its most important point, the vertex, can be read straight off the equation. In y = a(x − h)² + k the vertex is (h, k), the axis of symmetry is x = h, and a says how wide the parabola is and which way it opens.
The vertex form equation
- (h, k) is the vertex: the lowest point if the parabola opens upward, the highest if it opens downward.
- x = h is the axis of symmetry, the vertical line through the vertex.
- a controls the shape. If a > 0 the parabola opens up; if a < 0 it opens down. If |a| > 1 it is narrower than y = x²; if |a| < 1 it is wider.
The form works because (x − h)² is never negative and equals zero only at x = h. So when a > 0, the smallest value of y is k, reached exactly at x = h.
Watch the sign of h
The vertex form has a minus sign built in: (x − h). So the h in the vertex is the number that makes the bracket zero, which is the opposite sign of what appears in the bracket.
| Equation | Vertex |
|---|---|
| y = (x − 3)² + 2 | (3, 2) |
| y = (x + 3)² + 2 | (−3, 2), because x + 3 = x − (−3) |
| y = 2(x − 1)² − 5 | (1, −5) |
| y = −x² + 4 | (0, 4) |
The k needs no such care: it is added on the outside, so it is the y-coordinate as written.
Converting standard form to vertex form
A quadratic in standard form, y = ax² + bx + c, can be rewritten in vertex form by completing the square. Take y = 2x² − 8x + 3.
- Factor a out of the x terms: y = 2(x² − 4x) + 3.
- Take half the x-coefficient inside the bracket, −4/2 = −2, and square it: 4. Add and subtract it inside: y = 2(x² − 4x + 4 − 4) + 3.
- The first three terms are a perfect square: y = 2((x − 2)² − 4) + 3.
- Expand the outer bracket: y = 2(x − 2)² − 8 + 3 = 2(x − 2)² − 5.
The vertex is (2, −5). As a shortcut, the vertex of ax² + bx + c is always at x = −b/(2a); substitute that x into the equation to get k. Here −(−8)/(2·2) = 2, and y(2) = 8 − 16 + 3 = −5.
Converting vertex form to standard form
Going the other way only needs expanding. y = 3(x + 1)² − 7 becomes 3(x² + 2x + 1) − 7 = 3x² + 6x − 4. Standard form makes the y-intercept obvious (here −4), while vertex form makes the vertex obvious. Factored form, a(x − r)(x − s), shows the roots r and s. The same parabola can be written all three ways.
Finding the roots from vertex form
Set y = 0 and solve: 0 = a(x − h)² + k gives (x − h)² = −k/a, so
If −k/a is negative there are no real roots: the parabola never reaches the x-axis. That happens when the vertex is above the axis and the parabola opens up, or below it and opens down. For y = 2(x − 2)² − 5, the roots are x = 2 ± √(5/2) ≈ 0.419 and 3.581.
See it with sliders
In the calculator, type y = a(x - h)^2 + k and accept the three sliders it offers. Drag h and the parabola slides sideways; drag k and it moves up and down; drag a through zero and it flattens into a line and flips over. Selecting the curve marks its vertex and roots, so you can check the formulas above as you go.
Practice
A new problem every time. Answer in any form you like: fractions, decimals, √ and π all work, and equivalent equations count as right.
Find the vertex of the parabola y = −2(x + 4)² − 5.
Show the answer
Answer: (−4, −5)
- Vertex form is y = a(x − h)² + k, and the vertex is (h, k).
- x + 4 is x − (−4), so h = −4. The sign flips.
- The constant at the end is k = −5.
- The vertex is (−4, −5).
Frequently asked questions
What is vertex form?
How do I find the vertex from vertex form?
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What does a do in vertex form?
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Keep going
Every figure on this page opens in ihateDesmos, where you can drag sliders and watch the graph respond.