How to Graph a Quadratic Function
A visual walkthrough on sketching parabolas. Discover how to identify the vertex, axis of symmetry, intercepts, and orientation step by step.
1. Key Features of a Parabola
The standard form of a quadratic function is:
Every quadratic function produces a symmetric curve called a parabola. Here is what each constant controls:
- Sign of a: If
a > 0, the parabola opens upwards (curving upward with a minimum vertex). Ifa < 0, it opens downwards (with a maximum point). - Vertex: The turning point of the curve.
- Axis of Symmetry: The central vertical line running straight through the vertex.
- y-intercept: The point where the curve crosses the y-axis, which is simply
(0, c).
2. The 4-Step Graphing Method
- Find the Vertex: Calculate
x = -b / (2a). Then computef(x)to get the y-coordinate. Your vertex is(x, y). - Draw the Axis of Symmetry: A dashed vertical line at
x = -b / (2a). - Find the y-intercept: Plug in
x = 0to get(0, c). Then use symmetry to plot the mirror point across the axis. - Find the x-intercepts (Roots): Set
ax² + bx + c = 0and solve via factoring or the quadratic formula.
3. Worked Example: f(x) = x² - 4x + 3
(2, -1).x = 2.(0, 3). Reflected across x = 2 gives (4, 3).x = 1 and x = 3.Plot f(x) = x² - 4x + 3 Live
Try entering x^2 - 4x + 3 into the graph plotter below. Notice the vertex at (2, -1) and intercepts at x = 1 and 3!
Frequently Asked Questions
What shape does a quadratic function make?
A quadratic function forms a U-shaped curve called a parabola. If 'a' is positive, it opens upwards. If 'a' is negative, it opens downwards.
How do you find the vertex of a parabola?
The x-coordinate of the vertex is found using the formula x = -b / (2a). Then, plug this x-value back into the original function to find the y-coordinate.
What is the axis of symmetry?
The axis of symmetry is the vertical line that divides the parabola exactly in half. It passes through the vertex and its equation is x = -b / (2a).
How do you find x-intercepts of a quadratic?
You can find x-intercepts (also called roots or zeros) by setting y = 0 and solving for x. You can solve by factoring, completing the square, or using the quadratic formula.
Does a quadratic equation ax² + bx + c = 0 always have two roots?
Yes. By the Fundamental Theorem of Algebra, every degree 2 polynomial has exactly two roots in the complex number system: either two distinct real roots (b² - 4ac > 0), one repeated real root of multiplicity 2 (b² - 4ac = 0), or two complex conjugate roots (b² - 4ac < 0).
How do you know if a parabola opens upwards or downwards?
Look at the leading coefficient 'a' in y = ax² + bx + c. If a > 0 (positive), the parabola opens upwards like a 'U' and the vertex is a minimum point. If a < 0 (negative), the parabola opens downwards and the vertex is a maximum point.
How do you convert standard form to vertex form?
You can either complete the square or find the vertex (h, k) using h = -b / (2a) and k = f(h). Then substitute these values into vertex form y = a(x - h)² + k, retaining the original leading coefficient 'a'.