Enter Quadratic Function Values
Use the form for f(x) = ax² + bx + c. The value of a must not be zero.
Formula Used
f(x) = ax² + bx + c
h = -b / 2a and k = f(h)
D = b² - 4ac
x = (-b ± √D) / 2a
p = 1 / 4a, focus is (h, k + p), and directrix is y = k - p.
How to Use This Calculator
- Enter the values of
a,b, andc. - Use a nonzero value for
a. - Add a target y value if you want to solve
f(x) = y. - Set graph limits for the x-axis.
- Choose decimal places for clean output.
- Click the submit button.
- Review results above the form.
- Download the CSV or PDF report.
Example Data Table
| a | b | c | Function | Vertex | Roots | Opening |
|---|---|---|---|---|---|---|
| 1 | -6 | 8 | f(x) = x² - 6x + 8 | (3, -1) | 2, 4 | Upward |
| -2 | 4 | 6 | f(x) = -2x² + 4x + 6 | (1, 8) | -1, 3 | Downward |
| 3 | 2 | 5 | f(x) = 3x² + 2x + 5 | (-0.333, 4.667) | Complex | Upward |
Quadratic Function Feature Guide
Why Features Matter
A quadratic function has a curved graph called a parabola. Its features tell you how the curve behaves. The vertex shows the highest or lowest point. The axis of symmetry shows where the graph balances. The roots show where the graph crosses the x-axis. These points help explain motion, profit, area, height, and many other real problems.
Reading the Coefficients
The value of a controls the opening direction and width. A positive a opens upward. A negative a opens downward. Larger absolute values make the curve narrower. Smaller absolute values make it wider. The value of b moves the vertex sideways. The value of c gives the y-intercept. Together, these values describe the full shape.
Using the Vertex
The vertex is one of the most useful features. It gives the minimum value when the graph opens upward. It gives the maximum value when the graph opens downward. This calculator finds the vertex by using h = -b / 2a. It then places h into the function to find k. The result is shown as an ordered pair.
Understanding Roots
Roots are found with the quadratic formula. The discriminant decides the type of answer. A positive discriminant gives two real roots. A zero discriminant gives one repeated real root. A negative discriminant gives two complex roots. The calculator also solves f(x) equal to a chosen target value.
Graph and Export Options
The graph gives a fast visual check. You can adjust the x range and sample count for better detail. The CSV download is useful for spreadsheets. The PDF download is useful for reports, notes, and assignments. Use both exports when you need to share the calculation with clear evidence.
FAQs
1. What is a quadratic function?
A quadratic function is a second-degree function. Its common form is f(x) = ax² + bx + c, where a is not zero.
2. Why can coefficient a not be zero?
If a is zero, the x² term disappears. The function becomes linear, so quadratic features like a vertex and parabola do not apply.
3. What does the vertex show?
The vertex shows the turning point of the parabola. It is the minimum point for upward graphs and maximum point for downward graphs.
4. What is the discriminant?
The discriminant is b² - 4ac. It tells whether the equation has two real roots, one repeated root, or two complex roots.
5. What is the axis of symmetry?
The axis of symmetry is the vertical line through the vertex. It divides the parabola into two matching sides.
6. What does the range mean?
The range is the set of possible y-values. It starts at the vertex value and extends upward or downward.
7. Can this calculator show complex roots?
Yes. When the discriminant is negative, the calculator shows complex roots using the imaginary unit i.
8. What is the target y option?
The target y option solves ax² + bx + c = target y. It finds x-values where the graph reaches that selected height.