This calculator finds the vertex of any parabola in a few seconds. Enter your equation in standard form, y = ax² + bx + c, or in vertex form, y = a(x − h)² + k, and it returns the exact coordinates of the vertex along with the steps used to get there. It also tells you whether that point is a maximum or a minimum, based on the direction the parabola opens.
What Is the Vertex of a Parabola?
The vertex is the single point where a parabola turns. It is the sharpest bend on the curve, and it always sits on the parabola's axis of symmetry. If the parabola opens upward, the vertex is the lowest point on the graph. If it opens downward, the vertex is the highest point. Every parabola has exactly one vertex, no more and no less.
What Is the Formula for the Vertex of a Parabola?
For a parabola written in standard form, y = ax² + bx + c, the x-coordinate of the vertex comes from x = -b/2a. Once you have that value, you substitute it back into the original equation to get the y-coordinate. Written as a coordinate pair, the vertex formula is (h, k) = (-b/2a, c - b²/4a). You do not need to memorize both versions. If you know x, plugging it back in to solve for y works just as well. Here is the quick reference:
What You Need | Formula |
X-coordinate of the vertex | x = -b/2a |
Y-coordinate of the vertex | y = c - b²/4a |
Vertex as an ordered pair | (h, k) = (-b/2a, c - b²/4a) |
How Do You Find the X-Coordinate of the Vertex?
Start by identifying a and b from your equation. Take y = 2x² - 8x + 5 as an example, where a = 2 and b = -8. Plug these into x = -b/2a: x = -(-8)/(2×2) = 8/4 = 2. That x-value, 2, is also the equation of the parabola's axis of symmetry, so this one calculation gives you two answers.
How Do You Find the Y-Coordinate of the Vertex?
With x = 2 from the previous step, substitute it back into y = 2x² - 8x + 5: y = 2(2)² - 8(2) + 5 = 8 - 16 + 5 = -3. The vertex of this parabola is (2, -3). That y-value is also the minimum the function can reach, since a is positive here.
How Does This Vertex Calculator Work?
Behind the input boxes, the calculator takes whatever coefficients you type in, runs them through the vertex formula, and returns both the coordinates and the direction the parabola opens. You can enter your equation in standard form or vertex form, whichever one you already have, and the tool adjusts the calculation accordingly. Rather than just handing you a final answer, it lays out each step so you can check your own work against it or catch where you went wrong.
How Do You Find the Vertex From Standard Form vs Vertex Form?
Which method you use depends on which form your equation is already in. From standard form, you apply x = -b/2a and substitute to find y, exactly as shown above. From vertex form, y = a(x − h)² + k, there is no calculation needed at all. The vertex is simply (h, k), read straight off the equation. The one place people slip up is the sign inside the parentheses. Because the formula already has a minus sign built in, an equation like y = 3(x - 5)² + 2 gives a vertex of (5, 2), not (-5, 2).
Aspect | Standard Form | Vertex Form |
Equation | y = ax² + bx + c | y = a(x − h)² + k |
Vertex found by | Calculating x = -b/2a, then substituting to get y | Reading (h, k) directly off the equation |
Best for | Finding the y-intercept and using the quadratic formula | Graphing quickly, since the vertex is already visible |
Common mistake | Forgetting to substitute x back in to find y | Missing the sign flip on h inside the parentheses |
How Do You Find the Vertex From y = ax² + bx + c?
Try a second example: y = -x² + 6x - 5. Here a = -1 and b = 6, so x = -6/(2×-1) = 3. Substituting x = 3 back in: y = -(3)² + 6(3) - 5 = -9 + 18 - 5 = 4. The vertex is (3, 4). Because a is negative, this point is a maximum, not a minimum.
How Do You Find the Vertex From y = a(x−h)² + k?
If your equation is y = 2(x + 4)² - 7, notice the plus sign inside the parentheses. Rewriting it as y = 2(x - (-4))² - 7 shows that h = -4, so the vertex is (-4, -7). Compare that with y = 2(x - 4)² - 7, where h = 4 and the vertex is (4, -7). Same numbers, opposite sign, and an entirely different point on the graph.
Worked Example: Finding the Vertex of a Parabola Step by Step
Say you throw a ball into the air and its height in feet is modeled by h(t) = -16t² + 64t + 5, where t is time in seconds. To find the ball's maximum height, you are really just looking for the vertex. Here a = -16 and b = 64. The x-coordinate, in this case time, is t = -64/(2×-16) = 2. Substituting t = 2 back into the equation gives h(2) = -16(2)² + 64(2) + 5 = -64 + 128 + 5 = 69. The vertex is (2, 69), which means the ball reaches its maximum height of 69 feet exactly 2 seconds after it is thrown. That is the vertex doing real work, not just sitting on a graph.
Is the Vertex a Maximum or a Minimum?
This comes down entirely to the sign of a. If a is positive, the parabola opens upward like a bowl, and the vertex is the lowest point, the minimum. If a is negative, the parabola opens downward like a dome, and the vertex is the highest point, the maximum. This is the same a you already typed into the calculator above, so once you know the vertex, you already know which one you are looking at.
Sign of a | Parabola Opens | Vertex Is |
a > 0 | Upward | A minimum (the lowest point) |
a < 0 | Downward | A maximum (the highest point) |
What Is the Axis of Symmetry of a Parabola?
The axis of symmetry is the vertical line that runs through the vertex and splits the parabola into two mirror-image halves. Its equation is x = -b/2a, the same x-value you calculate for the vertex. There is no separate formula to learn here. Once you know the vertex, you already know the axis of symmetry too, since they share the same x-coordinate.
How Does the Vertex Relate to Other Parabola Properties?
The vertex does not exist in isolation. It is the reference point for several other properties of a parabola. The focus and directrix both sit an equal distance from the vertex, on opposite sides. The roots, or x-intercepts, are spaced symmetrically around the vertex's x-value, so their average always equals -b/2a. And the discriminant, which tells you how many times the parabola crosses the x-axis, is built into the same formula used to find the vertex's y-coordinate. The table below sums this up, and the sections after it cover each property in more detail.
Property | How It Relates to the Vertex |
Focus | Sits at a fixed distance, p = 1/4a, from the vertex, along the axis of symmetry |
Directrix | Sits the same distance from the vertex as the focus, on the opposite side |
Roots (x-intercepts) | Spaced symmetrically around the vertex's x-value, so their average equals -b/2a |
Discriminant | The vertex's y-coordinate can also be written as -D/4a |
How Do You Find the Focus and Directrix of a Parabola?
Once you have the vertex, finding the focus and directrix is a short extra step. The focal length, p = 1/4a, tells you how far the focus sits from the vertex, and the directrix sits the same distance on the opposite side. Both lie on the axis of symmetry you already found. For the full calculation with your own numbers, use the focus and directrix calculator.
How Do You Solve a Quadratic Equation Using the Quadratic Formula?
The vertex's x-value, -b/2a, is actually the first half of the quadratic formula. That is not a coincidence. The two roots of a quadratic equation sit symmetrically on either side of the vertex, which is why averaging them gives you back -b/2a. If you need to solve a full equation for its roots rather than just its turning point, the quadratic formula calculator handles that.
How Do You Convert Standard Form to Vertex Form by Completing the Square?
Completing the square is the algebra the vertex formula is built on. In short, you rewrite ax² + bx + c as a(x − h)² + k by adjusting the equation until it forms a perfect square. Working through this by hand is useful for understanding where the formula comes from, but if you just need the converted equation, the completing-the-square calculator does it instantly.
How Do You Find the Discriminant of a Quadratic Equation?
The discriminant, b² - 4ac, answers a different question from the vertex: how many times does the parabola cross the x-axis? It is worth knowing that the vertex's y-coordinate can also be written as -D/4a, so if you already have the discriminant, you are one step from the vertex too. For the full discriminant and what it means for your equation, try the discriminant calculator.
How Do You Graph a Parabola Step by Step?
With the vertex, the axis of symmetry, and the direction of opening in hand, you already have most of what you need to sketch a parabola by hand. Plot the vertex, plot the y-intercept and its mirror point across the axis of symmetry, then connect the points with a smooth curve. For a full graph with more plotted points, use the parabola graphing calculator.
How Do You Find the Vertex of a Parabola on a TI-84 or TI-83 Calculator?
If you are working from a graphing calculator instead of by hand, the process is a little different. Enter your equation next to Y1 in the Y= menu, graph it, then press 2nd and Trace to open the Calculate menu. From there, choose Maximum if the parabola opens downward, or Minimum if it opens upward. Picking Zero by mistake is a common slip since it sits right above Maximum and Minimum, but Zero finds a root, not the vertex. For a full keystroke by keystroke walkthrough, see the graphing calculator guide.
How Do You Find a Parabola's Equation From Its Vertex and Another Point?
Sometimes the problem runs in reverse. You are given the vertex and one other point on the curve, and you need to find the full equation. Start with y = a(x − h)² + k, plug in the vertex for h and k and the other point for x and y, then solve for a. With a vertex of (1, 3) and a point at (5, 0), substituting gives 0 = a(5 - 1)² + 3, which works out to a = -3/16. For the complete calculation with your own numbers, use the vertex and point calculator.