Read the vertex straight off the equation
The vertex form of a quadratic writes the equation so that its most useful point is visible with no work at all:
y = a(x - h)² + k
- (h, k) is the vertex, the parabola’s turning point: the lowest point when the curve opens up, the highest when it opens down.
- x = h is the axis of symmetry, the vertical mirror line through the vertex. The grapher above draws it as the dashed purple line, labeled with its equation.
- a sets the shape: positive opens up, negative opens down, and |a| larger than 1 makes the curve narrower than y = x².
What a, h and k each are
| Letter | Where it is | How to read it | What it does |
|---|---|---|---|
| a | In front of the bracket. | Its own sign and size, no trick. | Sets the direction and the width. Positive opens up, negative opens down, and a bigger size makes the parabola narrower. |
| h | Inside the bracket, subtracted. | The opposite of the sign you see. (x + 5)² means h = -5. | The x-coordinate of the vertex, the axis of symmetry x = h, and the horizontal shift of y = x². |
| k | Outside the bracket, added. | Exactly as written. No sign flip. | The y-coordinate of the vertex, the vertical shift, and the minimum value when a > 0 or the maximum when a < 0. |
Only h carries a trap, and it is the one the next section is about. a and k mean exactly what they look like.
The grapher starts at y = (x - 2)² - 1. Read the equation against the picture: the vertex marker sits at (2, -1), the dashed axis reads x = 2, and the Expanded (standard form) box shows the very same parabola written as y = x² - 4x + 3. Drag the vertex anywhere (it snaps to grid points, or select it and use the arrow keys) and both equations rewrite themselves instantly. If plotting a point like (2, -1) still takes a second thought, warm up on the coordinate plane first.
The sign trap: (x - 2)² means h = +2
Vertex form subtracts h. So the sign you see inside the parentheses is the opposite of the vertex’s x-coordinate:
| You see | Rewrite as (x - h)² | h | Vertex is on the… |
|---|---|---|---|
| (x - 2)² - 1 | already matches, h = 2 | +2 | right of the y-axis |
| (x + 3)² + 2 | (x - (-3))² + 2 | -3 | left of the y-axis |
The safe habit: force whatever is in the parentheses into the shape x - h and read h from that. x + 3 only fits the pattern as x - (-3), so h = -3. There is no such trap for k: it sits outside the parentheses with its true sign, so + 2 really does mean the vertex is 2 units up.
The widget prints this exact warning under its equation boxes because it is the error graders see most: on a test, “the vertex of y = (x + 3)² + 2” answered as (3, 2) instead of (-3, 2).
Reading transformations against the y = x² ghost
Every parabola in vertex form is the basic curve y = x², picked up, moved, and reshaped. Keep the show the y = x² ghost checkbox ticked: the dashed grey ghost stays put while your parabola moves, so each parameter’s job is visible on its own.
| Parameter | What it does to y = x² | Watch for |
|---|---|---|
| h | shifts the graph horizontally (right if h > 0, left if h < 0) | the dashed axis x = h travels with it |
| k | shifts the graph vertically (up if k > 0, down if k < 0) | the vertex rises or sinks, shape unchanged |
| a | stretches or flips: negative opens down, |a| > 1 narrows, 0 < |a| < 1 widens | compare arm steepness against the ghost |
Standard form to vertex form (and back)
The two boxes in the panel are the same parabola in two costumes. Vertex form to standard form is just multiplying out; standard form to vertex form is completing the square, or a two-line shortcut. One important constant: a is identical in both forms. Converting only reshuffles h and k into b and c and back.
Vertex form to standard form: expand
Start with the grapher’s default and multiply out the square:
y = (x - 2)² - 1
First the square: (x - 2)² = (x - 2)(x - 2) = x² - 2x - 2x + 4 = x² - 4x + 4.
Then attach the k: y = x² - 4x + 4 - 1, so
y = x² - 4x + 3
which is exactly what the Expanded (standard form) box shows.
Standard form to vertex form: complete the square
Now run the same example backwards. Start from
y = x² - 4x + 3
- Take the coefficient of x, which is -4. Halve it: -2. Square that: 4.
- Add and subtract the 4 so nothing changes: y = (x² - 4x + 4) - 4 + 3.
- The parentheses are now a perfect square: y = (x - 2)² - 1.
Back to where we started, and the vertex (2, -1) is visible again.
The vertex shortcut from standard form
Completing the square works every time, but if all you need is the vertex of y = ax² + bx + c, there is a faster route:
h = -b / (2a), then k = f(h)
On the same example, a = 1 and b = -4, so h = -(-4) / (2 × 1) = 4/2 = 2. Substitute back: k = f(2) = 2² - 4 × 2 + 3 = 4 - 8 + 3 = -1. Vertex (2, -1), same a, so the vertex form is y = (x - 2)² - 1. Done in two lines.
Everything you can read off vertex form
The vertex is the headline, but it is not the only thing sitting in the equation. Here is the whole list, worked through on y = 2(x + 5)² - 4:
| What you want | Read it from | On y = 2(x + 5)² - 4 |
|---|---|---|
| The vertex | (h, k) | (-5, -4) |
| The axis of symmetry | x = h | x = -5 |
| Which way it opens | the sign of a | a = 2, so it opens upward |
| The minimum value | k | -4, reached at x = -5 |
| The y-intercept | (0, ah² + k) | (0, 46) |
| The x-intercepts | x = h ± √(-k/a) | x = -5 ± √2, about -6.41 and -3.59 |
Two of those rows are worth pausing on.
k is not the y-intercept. k is how high the vertex is. The y-intercept is where the curve crosses the y-axis, at x = 0, which is ah² + k. They agree only when h = 0, that is when the vertex is already on the y-axis. Above, the vertex is 4 below the axis and the curve crosses the y-axis at 46.
The x-intercepts come from rearranging, not from a new formula. Set y = 0:
a(x - h)² + k = 0 → (x - h)² = -k/a → x = h ± √(-k/a)
That square root is the whole story of how many crossings there are. If -k/a is positive there are two, if it is zero the vertex sits on the x-axis and there is one, and if it is negative there are none. It is the discriminant test in vertex form’s own letters, and it is often quicker: you can see at a glance whether k and a have opposite signs.
How to graph a parabola from vertex form
Vertex form hands you the starting point, so graphing is four steps and no equation solving:
- Plot the vertex (h, k) and draw the axis of symmetry x = h through it.
- Point the curve with the sign of a: up if positive, down if negative.
- Step out from the vertex. One unit sideways changes y by a, two units by 4a, three by 9a, because the bracket is squared. With a = 1 that is the familiar 1, 3, 5 pattern of gaps; with a = 2 it is 2, 6, 10.
- Mirror everything across the axis and sweep a smooth curve through the points. The y-intercept (0, ah² + k) comes free, so plot that too.
Nine worked examples
| Vertex form | Standard form | Vertex | Opens | y-intercept | x-intercepts |
|---|---|---|---|---|---|
| y = (x - 2)² - 1 | y = x² - 4x + 3 | (2, -1) | up | (0, 3) | 1, 3 |
| y = (x + 3)² + 2 | y = x² + 6x + 11 | (-3, 2) | up | (0, 11) | none |
| y = -(x + 1)² | y = -x² - 2x - 1 | (-1, 0) | down | (0, -1) | -1 |
| y = 2(x + 5)² - 4 | y = 2x² + 20x + 46 | (-5, -4) | up | (0, 46) | ≈ -6.41, -3.59 |
| y = -2(x - 3)² + 8 | y = -2x² + 12x - 10 | (3, 8) | down | (0, -10) | 1, 5 |
| y = 0.5x² - 2 | y = 0.5x² - 2 | (0, -2) | up | (0, -2) = k | -2, 2 |
| y = x² + 3 | y = x² + 3 | (0, 3) | up | (0, 3) = k | none |
| y = 3(x - 1)² | y = 3x² - 6x + 3 | (1, 0) | up | (0, 3) | 1 |
| y = -8(x + 4)² + 128 | y = -8x² - 64x | (-4, 128) | down | (0, 0) | -8, 0 |
The last row is a pasted homework question: p(x) = -8x² - 64x written in vertex form is -8(x + 4)² + 128, so a = -8, h = -4 and k = 128. Read as transformations, that is y = x² reflected across the x-axis, stretched by a factor of 8, shifted 4 left and 128 up.
Note the two rows with h = 0, marked in the y-intercept column. Those are the only ones where k really is the y-intercept, because their vertex is already on the y-axis.
Sideways and conic forms
Swap x and y and the same equation describes a parabola lying on its side:
x = a(y - k)² + h
The vertex is still (h, k), and a still sets the width, but now positive a opens it to the right and negative to the left. It is a parabola and it is not a function of x: a vertical line through it crosses twice. Conic-section courses write parabolas a third way again, (x - h)² = 4p(y - k), where p is the distance from the vertex to the focus, which is a different parameter from the a used here.
Which form should you use?
Neither form is “better”; each makes a different question easy.
| You want | Reach for | Because |
|---|---|---|
| The vertex, or the maximum/minimum value | Vertex form | (h, k) is written in the equation |
| The transformations of y = x² | Vertex form | h, k, and a are the shift and stretch |
| The y-intercept | Standard form | set x = 0 and y = c immediately; vertex form needs ah² + k |
| The quadratic formula, or factoring for roots | Standard form | both need a, b, and c |
In y = (x - 2)² - 1 the vertex is instant but the y-intercept takes a step of arithmetic; in y = x² - 4x + 3 the y-intercept (0, 3) is instant but the vertex takes the shortcut above. For how a, b, and c each bend the curve in standard form, and where the roots sit on the graph, see the parabola lesson.
The grapher above also works as a vertex form calculator: place the vertex, set a, and read both forms of the equation instantly.
It’s free to embed on your own site or LMS. Next, explore the standard-form coefficients and roots in the parabola lesson, or rebuild your point plotting instincts on the coordinate plane.