Vertex Calculator

A vertex calculator finds the turning point of a parabola from its equation. Enter the coefficients of y = ax² + bx + c and it returns the vertex, the axis of symmetry and the vertex form. For y = 2x² − 8x + 5 the vertex is (2, −3), the axis of symmetry is x = 2, and the vertex form is y = 2(x − 2)² − 3.

The tool works in both directions. Start from standard form and it converts to vertex form; start from a, h and k in vertex form and it multiplies out to standard form. Along the way it reports the opening direction, the maximum or minimum value, the focus and directrix, the y-intercept, the x-intercepts and a graph of the curve.

What the vertex actually is

Every quadratic graphs as a parabola: a smooth U-shaped or arch-shaped curve. The vertex is the one point where that curve changes direction. On an upward-opening parabola it is the bottom of the U and the y-value there is the smallest the function ever reaches. On a downward-opening parabola it is the top of the arch and the y-value there is the largest. Because a parabola is perfectly symmetric, the vertex is also the point where the curve meets its axis of symmetry, the vertical mirror line that folds one half of the graph exactly onto the other.

That single point answers a surprising number of practical questions. Where does a thrown ball reach its peak? What price maximizes revenue? What is the cheapest production level? Each of those is a quadratic model, and each answer is the vertex.

The vertex formula

For a quadratic written in standard form, the x-coordinate of the vertex is:

The vertex is then the point (h, k), the axis of symmetry is the line x = h, and the equation rewritten in vertex form is:

There is also a shortcut for k that avoids substitution: k = c − b² ÷ (4a). Both routes give the same number, and the calculator uses the substitution version because it is easier to check by hand.

Worked example: y = 2x² − 8x + 5

Take the quadratic y = 2x² − 8x + 5, so a = 2, b = −8 and c = 5. Work through the formula one step at a time:

1. x-coordinate: h = −b ÷ (2a) = 8 ÷ 4 = 2.
2. y-coordinate: k = 2(2)² − 8(2) + 5 = 8 − 16 + 5 = −3.
3. Vertex: (2, −3). Axis of symmetry: x = 2.
4. Vertex form: y = 2(x − 2)² − 3.
5. Direction: a = 2 is positive, so the parabola opens upward and −3 is the minimum value of y.

The rest of the picture follows from the same three coefficients. The y-intercept is (0, 5), because c is the value of y when x = 0. The discriminant is b² − 4ac = 64 − 40 = 24, which is positive, so there are two x-intercepts: x = (8 ± √24) ÷ 4, giving x ≈ 0.775 and x ≈ 3.225. Notice that those two roots sit symmetrically around x = 2, exactly 1.2247 units to each side. The focal length is 1 ÷ (4 × 2) = 0.125, so the focus is at (2, −2.875) and the directrix is the line y = −3.125.

Worked example the other way: y = −3(x + 1)² + 12

Now start in vertex form. Here a = −3, and since the bracket reads (x + 1) rather than (x − 1), the vertex x-coordinate is h = −1 with k = 12. Multiplying out gives b = −2ah = −2(−3)(−1) = −6 and c = a·h² + k = −3(1) + 12 = 9, so the standard form is y = −3x² − 6x + 9.

Check it: h = −b ÷ (2a) = 6 ÷ (−6) = −1, which matches. Because a is negative the parabola opens downward, so 12 is the maximum value of y and it happens at x = −1. The discriminant is 36 + 108 = 144, a perfect square, so the roots are rational: x = −3 and x = 1. The focal length is 1 ÷ (4 × −3) ≈ −0.0833, which places the focus at (−1, 11.9167), just below the vertex, with the directrix at y ≈ 12.0833 just above it. On a downward-opening parabola the focus always sits below the vertex and the directrix above it.

Vertex and axis of symmetry for common quadratics

Each row below was computed with h = −b ÷ (2a) and k = f(h). The last column shows whether the vertex is the low point or the high point of the curve.

Two patterns are worth noticing. The sign inside the bracket is always the opposite of the sign of h, which is where most sign errors come from. And when k = 0, as in the last row, the vertex sits on the x-axis and the parabola touches the axis at exactly one point instead of crossing it.

Where the vertex meets the x-axis: the discriminant

The vertex tells you how the parabola sits relative to the x-axis before you solve anything. If the parabola opens upward and k is above zero, there is no way for the curve to reach the axis. The discriminant b² − 4ac states the same fact algebraically:

Read the middle two rows together. Both parabolas open upward, but one has its vertex on the axis (k = 0, one repeated root) and the other has its vertex two units above it (k = 2, no real roots at all). The vertex and the discriminant always tell the same story.

Focus, directrix and the width of the curve

A parabola can also be defined geometrically: it is the set of all points that are the same distance from a fixed point, the focus, as from a fixed line, the directrix. The focus sits inside the curve at (h, k + 1 ÷ 4a) and the directrix runs parallel to the x-axis at y = k − 1 ÷ 4a. The distance between them, 2 divided by the absolute value of 4a, is what makes a satellite dish or a headlight reflector work: rays arriving parallel to the axis all bounce to the focus.

The table below fixes the vertex at the origin and varies only a. The last column is the latus rectum, the width of the parabola measured across the focus, which equals 1 divided by the absolute value of a.

Doubling a halves the focal length and halves the width: the curve tightens around its axis. A negative a mirrors the whole picture, putting the focus below the vertex and the directrix above it.

Completing the square, step by step

Completing the square is the algebra behind the shortcut, and most courses want to see it at least once. Using y = 2x² − 8x + 5 again: factor 2 out of the first two terms to get y = 2(x² − 4x) + 5. Half of −4 is −2 and its square is 4, so add and subtract 4 inside the bracket: y = 2(x² − 4x + 4 − 4) + 5. The first three terms form a perfect square, giving y = 2(x − 2)² − 8 + 5, which simplifies to y = 2(x − 2)² − 3. The same vertex, (2, −3), arrived at the long way. Note the −8: the 4 you subtracted was inside a bracket multiplied by 2, so it leaves the bracket doubled.

How to use this vertex calculator

1. Pick your form. Choose standard form if your equation looks like y = ax² + bx + c, or vertex form if it already looks like y = a(x − h)² + k.
2. Rearrange first. Move every term to one side so the equation reads y = something, ordered by descending power of x. The quadratic 5 + 2x² = 7x becomes y = 2x² − 7x + 5.
3. Enter the coefficients with their signs. A missing term means that coefficient is 0, so y = x² − 9 has b = 0 and c = −9.
4. Read the vertex. The headline shows the ordered pair (h, k) and whether it is a maximum or a minimum.
5. Check the extras. The breakdown card lists the axis of symmetry, focus, directrix, intercepts and discriminant, and the step-by-step card shows the substitution so you can copy the working.

In vertex form, watch the sign of h. The equation y = 3(x + 4)² − 1 has h = −4, not 4, because the standard bracket is (x − h) and (x + 4) equals (x − (−4)).

Who this calculator is for

• Algebra 1 and Algebra 2 students converting between standard and vertex form and sketching parabolas.
• Precalculus students working with conic sections, where the focus, directrix and latus rectum matter as much as the vertex.
• Physics students reading the peak height and the time of the peak off a projectile equation.
• Business and economics students maximizing revenue or minimizing average cost from a quadratic model.
• Teachers and tutors generating clean example problems with known vertices and intercepts.
• Anyone checking homework who wants a second opinion on a sign or a fraction before turning in the assignment.

Key terms explained

• Parabola: the U-shaped or arch-shaped graph of a quadratic function.
• Vertex: the turning point (h, k), the minimum when a is positive and the maximum when a is negative.
• Axis of symmetry: the vertical line x = h that splits the parabola into mirror halves.
• Standard form: y = ax² + bx + c, the form in which the y-intercept c is easiest to read.
• Vertex form: y = a(x − h)² + k, the form in which the turning point is easiest to read.
• Discriminant: b² − 4ac, which counts the x-intercepts: two if positive, one if zero, none if negative.
• Focus and directrix: the fixed point and fixed line that define the parabola by equal distances.
• Latus rectum: the chord through the focus parallel to the directrix, with length equal to 1 divided by the absolute value of a.

Two real-world vertex problems

Projectile height. A ball leaves a 5-foot-high platform at 64 feet per second straight up. In United States customary units, gravity gives the model y = −16x² + 64x + 5, with x in seconds and y in feet. The vertex is at h = −64 ÷ (2 × −16) = 2 seconds, and k = −16(4) + 128 + 5 = 69 feet. So the ball peaks 69 feet up, two seconds after release. The positive root of the same equation, x ≈ 4.077 seconds, is when it lands.

Maximizing revenue. A vendor finds that selling x units at a price that falls as volume rises produces revenue R = −5x² + 200x dollars. The vertex is h = −200 ÷ (2 × −5) = 20 units, and k = −5(400) + 200(20) = $2,000. Selling more than 20 units actually lowers total revenue, because the price concession outruns the extra volume. The two x-intercepts, 0 and 40, mark where revenue falls to zero.

What changes the result the most

• The sign of a: flips the whole curve and turns a minimum into a maximum. Nothing else affects direction.
• The size of a: a larger absolute value narrows the parabola and pulls the focus closer to the vertex, but it also moves h, since h depends on a.
• The coefficient b: shifts the vertex horizontally. Doubling b doubles the distance of the axis of symmetry from the y-axis.
• The constant c: slides the whole curve vertically. It changes k by exactly the same amount and never moves h.
• Rounding: using a rounded value of h to compute k compounds the error, so always substitute the exact fraction.

Tips for getting it right

• Keep h as a fraction while you substitute. For y = x² − 5x + 6, using 5/2 gives exactly k = −1/4; using 2.5 works too, but rounding to 2.5 in a messier problem will not.
• Sanity-check with symmetry: the vertex x-value is always the average of the two x-intercepts when they exist. For roots 2 and 3 the average is 2.5, matching the vertex.
• Verify by substitution. Put h back into the original equation and confirm you get k.
• Sketch the direction first. Knowing the parabola opens downward tells you the answer must be a maximum before you compute anything.
• Expand your vertex form back out. If it does not return the original a, b and c, a sign slipped somewhere.

Limitations and assumptions

• This page handles vertical parabolas of the form y = ax² + bx + c only. A sideways parabola x = ay² + by + c has its axis of symmetry horizontal and needs the same formulas with the roles of x and y swapped.
• The coefficient a must not be zero. With a = 0 the graph is a straight line with no vertex.
• When the discriminant is negative the calculator reports that no x-intercepts exist rather than printing complex roots. For complex roots in a + bi form, use the quadratic formula calculator.
• Results are rounded for display to four decimal places, so an exact value such as one third shows as 0.3333.
• Very large coefficients can push intermediate values past the limits of floating-point arithmetic; keep coefficients within a normal homework range for exact-looking results.

How this compares to related calculators

This page answers "where is the turning point of this parabola?" Neighboring questions have their own tools: use the Quadratic Formula Calculator when you need the roots of ax² + bx + c = 0, including complex ones; use the Factoring Calculator when the quadratic factors neatly and you want the factored form; use the Slope Calculator and the Point-Slope Form Calculator for straight lines rather than curves; and use the Midpoint Calculator or the Distance Calculator when you are measuring between two plotted points. For arithmetic on the numbers you get out, the Scientific Calculator and the Fraction Calculator are the quickest follow-ups.

Sources

Everything on this page is deterministic mathematics that follows from the definition of a quadratic function, so no external data source applies. The identities h = −b ÷ (2a), k = f(h), the vertex form y = a(x − h)² + k, the discriminant b² − 4ac and the focal length 1 ÷ (4a) are all derived by completing the square on y = ax² + bx + c, and each worked example above was computed directly from those identities. The one place a physical constant appears is the projectile example, where the coefficient −16 is half the standard acceleration of gravity expressed in United States customary units (32.174 feet per second squared, the standard value adopted for the foot and pound as defined in NIST Special Publication 811, Guide for the Use of the International System of Units), rounded to 32 as textbooks do.