What is vertex form?
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola. The a is the same a as in standard form, so it still controls width and direction.
Watch the sign inside the bracket. y = 3(x − 4)² + 1 has its vertex at (4, 1), while y = 3(x + 4)² + 1 has its vertex at (−4, 1).
You can think of vertex form as the graph of y = x² after three moves. The h slides it left or right, the k lifts or lowers it, and the a stretches or flips it. That is why teachers like this form for graphing: each number has one job.
How do you convert standard form to vertex form?
Complete the square. Factor a out of the x terms, add and subtract the square of half the new x coefficient, then tidy up the constant. Here is y = 2x² − 8x + 3, which becomes y = 2(x − 2)² − 5.
Reading it off: h = 2 and k = −5, so the vertex is (2, −5). The completing the square calculator shows the same steps if you want the method on its own.
Is there a shortcut formula for the vertex?
Yes: h = −b/(2a), then k is the y-value at x = h. This is the quickest way to use a vertex calculator by hand when you only need the point and not the full equation.
For y = 2x² − 8x + 3, a = 2 and b = −8. So h = 8/4 = 2. Then k = 2(2)² − 8(2) + 3 = 8 − 16 + 3 = −5.
How do you find the axis of symmetry?
The axis of symmetry is the vertical line x = h, through the vertex. For y = 2x² − 8x + 3 it is x = 2.
An axis of symmetry calculator does the same h = −b/(2a) step and stops there. Every point left of the line has a mirror image on the right at the same height. That is handy for plotting: once you know (0, 3) is on the curve, (4, 3) is too.
Is the vertex a maximum or a minimum?
It depends on the sign of a. If a is positive, the parabola opens up and the vertex is the lowest point (a minimum). If a is negative, it opens down and the vertex is the highest point (a maximum).
| Equation | Vertex form | Vertex | Type |
|---|---|---|---|
| y = 2x² − 8x + 3 | y = 2(x − 2)² − 5 | (2, −5) | Minimum, opens up |
| y = −x² + 4x + 1 | y = −(x − 2)² + 5 | (2, 5) | Maximum, opens down |
In word problems this is how you find the highest point of a thrown ball or the price that gives the most profit. The vertex gives both parts of the answer: h is where the best value happens, and k is the best value itself.
For y = −x² + 4x + 1, the largest y-value is 5, and it happens at x = 2. No other x can do better, because −(x − 2)² is never positive.
How do you convert vertex form back to standard form?
Expand the square, multiply by a, and add k. For y = 2(x − 2)² − 5, (x − 2)² = x² − 4x + 4, so y = 2x² − 8x + 8 − 5 = 2x² − 8x + 3.
Doing this is also the best way to check a conversion. If you don't land back on the original equation, something went wrong with the constant. Most of the time the x² and x terms come out right and only the last number is off.
What mistakes are common with vertex form?
Getting the sign of h backwards is the classic one. In y = (x + 3)² − 1 the vertex is (−3, −1), not (3, −1), because x + 3 = x − (−3).
The other is forgetting to multiply by a when the subtracted square comes out of the bracket. In the example above, the −4 inside becomes −8 outside, which is why k is −5 and not −1.
How do you sketch a parabola from vertex form?
Plot the vertex, draw the axis of symmetry, and use a to find the next points. From the vertex, step 1 to the right and go up a units, then do the same on the left.
For y = 2(x − 2)² − 5, start at (2, −5). One step right gives (3, −3), since 2(1)² − 5 = −3. Its mirror point is (1, −3). Two steps right gives (4, 3), and its mirror is (0, 3), which is also the y-intercept.
Five points like these are enough for a clean sketch. The graphing calculator draws the full curve if you want to compare.