Graph PlottersEnter a quadratic equation to find its vertex, axis of symmetry, opening direction, and key graph properties instantly.
Hover to inspect coordinates. Drag the graph to pan the view or use the wheel and zoom controls to explore the function.
ABOUT THE TOOL
The vertex is the highest or lowest point of a quadratic graph. For y = ax² + bx + c, calculate h = −b/(2a), then substitute h into the function to find k.
HOW TO USE IT
The vertex is the turning point of a parabola. When a is positive, it is the minimum point; when a is negative, it is the maximum point. For y = ax² + bx + c, the x-coordinate is h = −b/(2a), and the y-coordinate is k = f(h).
Start by identifying a, b, and c. Substitute them into h = −b/(2a), then evaluate the original quadratic at h to find k. The result (h, k) gives the vertex, while x = h is the axis of symmetry. The graph and result cards keep these values visible as you test examples.
A quadratic can be written in vertex form as y = a(x − h)² + k. The value a controls opening direction and width, h moves the turning point horizontally, and k moves it vertically. From the same equation you can also inspect roots, the y-intercept, range, and discriminant.
The vertex is the point where a quadratic changes direction. For y = ax² + bx + c, the axis of symmetry is x = −b/(2a), and evaluating the function at that x-value gives the vertex. In vertex form y = a(x − h)² + k, the vertex is already visible as (h, k). If a is positive the vertex is a minimum; if a is negative it is a maximum.
For y = x² − 4x + 3, h = −(−4)/(2 · 1) = 2 and k = 2² − 4(2) + 3 = −1, so the vertex is (2, −1). Enter the equation, check the axis x = 2, and compare the roots at x = 1 and x = 3. Seeing the vertex, axis, and roots together explains why the parabola is symmetric and how its minimum relates to the discriminant.
For y = −3(x + 2)² + 5, the vertex is (−2, 5), the axis is x = −2, and the parabola opens downward. The maximum value is 5. Expanding the equation gives −3x² − 12x − 7, so the same vertex can also be found from −b/(2a). Comparing both forms shows why vertex form is the fastest format when the turning point is already known.
The vertex gives the boundary of a quadratic’s range. An upward-opening parabola with vertex (h, k) has y ≥ k, while a downward-opening one has y ≤ k. Roots may or may not exist depending on whether the curve reaches the x-axis. Use the vertex and discriminant together: the vertex shows the extreme height, and the discriminant predicts contact with y = 0.
Explore related tools as you study functions: start with the online graphing calculator to plot a general equation, then use the linear graph calculator for constant rates and the slope calculator for rise over run. When a curve meets the x-axis, check the x intercept calculator; for parabolas, compare the quadratic graph calculator and vertex calculator. Use the cubic graph calculator for polynomial end behavior, the sine graph calculator and cosine graph calculator for periodic models, and the phase shift calculator for horizontal movement. The function transformations calculator compares parent and transformed curves, while the absolute value graph calculator explains V-shaped graphs. For growth and inverse relationships, try the exponential growth graph calculator, reciprocal graph calculator, and log graph calculator. For asymptote and periodic-function practice, use the tangent graph calculator, secant graph calculator, and cosecant graph calculator to compare branches, gaps, and repeating behavior.
COMMON QUESTIONS
For y = ax² + bx + c, use h = −b/(2a), then calculate k = f(h). The vertex is (h, k).
The axis of symmetry is the vertical line x = h that passes through the vertex.
Yes. If a is negative, the parabola opens downward and the vertex is its maximum point.
Yes. An upward-opening parabola with a positive minimum, or a downward-opening parabola with a negative maximum, can miss the x-axis.
Vertex form y = a(x − h)² + k shows the vertex directly as (h, k).
If a is positive, the vertex gives the minimum and the range is y ≥ k. If a is negative, it gives the maximum and the range is y ≤ k.
Yes. Different values of a change width and opening direction while leaving h and k unchanged.