How this calculation works
Compute the turning point with decimal arithmetic, rewrite the quadratic in vertex form and classify its global minimum or maximum from the sign of a. The SVG shows the quadratic, vertex and vertical symmetry line, and the table verifies matching values at equal offsets from the axis.
h = −b/(2a); k = c − b²/(4a); vertex (h,k), axis x=hA worked example
For y=x²−6x+5, h=3 and k=−4. The vertex is (3,−4), the axis is x=3 and the completed-square form is y=(x−3)²−4. The parabola opens upward, so −4 is its global minimum.
Good to know
Vertical real parabolas only; a cannot be zero. Coefficients have magnitude at most 10¹² and |a| must be at least 10⁻¹². Decimal calculations are rounded at 60 significant digits, displayed coordinates are rounded further, and SVG coordinates use floating-point numbers. The auto-centered graph uses independent axis scales and is not a root solver.
Reference: OpenStax College Algebra: quadratic functions (opens in a new tab) ↗
Locate the turning point before drawing the curve
A vertical quadratic y=ax²+bx+c has a unique turning point whenever a is nonzero. Its horizontal coordinate is h=−b/(2a). Substituting that value into the quadratic gives k, the vertical coordinate. The vertex is the ordered pair (h,k), and its two entries have different meanings: h identifies where the turn occurs, while k identifies the function value there.
The vertical line x=h is the axis of symmetry. Points the same horizontal distance to either side of this line have the same height. The graph marks the turning point and the axis together, while the table checks that symmetry numerically. This makes it possible to connect the coefficient calculation with an actual picture of the function.
The piecewise function grapher can restrict a quadratic to a stated interval and combine it with other formulas, while retaining open or closed endpoints. The vertex calculation alone does not define those domain restrictions. The local maximum and minimum calculator instead scans a general bounded expression for numerical stationary candidates. Its finite search is different from the analytic quadratic vertex formula.
Work through the original quadratic example
For y=x²−6x+5, the coefficients are a=1, b=−6 and c=5. The horizontal vertex calculation is h=−(−6)/(2×1)=3. Evaluating the function at x=3 gives 9−18+5=−4, so the vertex is (3,−4). Because a is positive, the curve opens upward and −4 is the global minimum over all real x.
Complete the square to write the same function as y=(x−3)²−4. At x=2 and x=4, the squared offset from three is one, giving y=−3 in both cases. At x=1 and x=5, the squared offset is four, giving y=0 in both cases. The symmetry table retains those pairs rather than relying solely on how the curve appears on screen.
Use the sign of a to classify the extremum
A positive a multiplies every squared offset by a positive number. Since a square cannot be negative, the vertex height k is the lowest possible function value. The range is y≥k. With a negative a, every nonzero squared offset lowers the function value, so the vertex is the global maximum and the range is y≤k.
The sign of b does not by itself tell you whether the curve opens upward or downward, and c does not generally give the vertex height. The coefficient c is the y-intercept because the terms containing x vanish at x=0. It equals the vertex height only in special cases, such as b=0. Keeping the intercept and turning point separate avoids a common mistake when sketching quadratics.
Understand why the completed-square form gives symmetry
Every nondegenerate quadratic can be written as y=a(x−h)²+k. Replacing x by h+t gives y=a·t²+k. Replacing it by h−t produces the same value, because (−t)²=t². That short argument explains both the symmetry line and the paired rows in the table. It also shows why the vertex occurs at an offset of zero.
The calculator obtains k using c−b²/(4a), which is algebraically equal to ah²+bh+c. Decimal arithmetic keeps more working precision than direct floating-point substitution when large terms nearly cancel. It still has a finite precision limit; copied values and graph coordinates are rounded representations rather than unlimited symbolic exact numbers.
Read the automatic graph as a coordinate illustration
The graph chooses a symmetric window around the vertex so that the turning point remains visible even when h is far from zero. It draws the quadratic in its vertex form, which also helps avoid subtracting large nearly equal terms at each plotted point. The x and y axes are scaled independently to make the visible curve useful across a wide range of coefficients.
Independent axis scales mean that apparent width and angles in the image are not equal-scale geometric measurements. The same coefficient can look wider or narrower after the window changes. Analytic results such as the vertex and opening direction remain independent of that visual scaling. The selected window may omit the y-axis or some intercepts, and the tool does not claim that an omitted crossing is absent.
Use the quadratic equation solver for the separate task of solving ax²+bx+c=0. A vertex below the horizontal axis on an upward-opening curve suggests two real roots, but obtaining those roots is a different calculation.
Recognize a line before asking for a parabola vertex
If a=0, the expression becomes y=bx+c. With nonzero b it is a line, and with b=0 it is constant. Neither has the unique turning point described by the parabola formulas. The calculator therefore asks for a nonzero a rather than dividing by zero or returning a fabricated vertex.
Very small nonzero a can place the vertex far from the origin. The supported coefficient limits and minimum magnitude of a keep the automatic graph finite and interpretable. Within those bounds, export the SVG alongside the copied coefficients and vertex values so a later reader can distinguish the entered function from the chosen view of it.
A few questions, answered
What is the difference between the vertex and the axis?
The vertex is one point (h,k). The symmetry axis is the entire vertical line x=h that passes through that point.
Can a be zero?
No. A=0 produces a line or constant function rather than a parabola with a unique turning point.
Is c the vertex height?
Usually not. c is the vertical intercept at x=0. The vertex height is k=c−b²/(4a).
Why does the graph not always include x=0?
The graph is centered automatically around the vertex. Its finite window can omit the origin or other intercepts.
Does this tool find complex roots?
No. It analyzes the real vertex, symmetry axis and graph. Use the quadratic equation solver for roots.