A quadratic function has the form y = ax^2 + bx + c with a not zero. Its graph is a parabola, a symmetric U shaped curve. Four pieces of information are enough to sketch it accurately.
Which way it opens
If a is positive the parabola opens upward and the vertex is its lowest point. If a is negative it opens downward and the vertex is its highest point. A larger size of a makes the curve narrower.
Vertex and axis of symmetry
The axis of symmetry is the vertical line x = -b / (2a). The vertex sits on it, and substituting that x value back into the equation gives the y coordinate.
Graphing y = x^2 - 4x + 3
a = 1, b = -4, c = 3 a is positive, so it opens upward axis of symmetry: x = 4 / 2 = 2 vertex y value: 4 - 8 + 3 = -1 vertex: (2, -1) y intercept: set x = 0 -> y = 3 x intercepts: set y = 0 x^2 - 4x + 3 = 0 (x - 1)(x - 3) = 0 x = 1 and x = 3
Reading the intercepts
The x intercepts, also called roots or zeros, are where the curve meets the x axis. A parabola can cross twice, touch once at its vertex, or miss the axis entirely. The y intercept is always c, since setting x to zero leaves only that term.
Using symmetry to save work
Once the vertex is known, every point has a mirror image on the other side of the axis. Plotting three points to the right of the vertex gives three more on the left for free. It also gives a fast check: the two x intercepts are always the same distance from the axis of symmetry. Here 1 and 3 are each one unit from x = 2.
Common mistakes
Sign errors in the vertex formula
The formula is -b / (2a). With b = -4 that is 4 / 2 = 2, not -2.
Joining plotted points with straight segments
A parabola is a smooth curve. Straight segments between points, and a sharp corner at the vertex, misrepresent the function.
Check yourself
Find the axis of symmetry of y = x^2 - 6x + 5.
x = 6 / 2 = 3.
Find the vertex of that same function.
At x = 3, y = 9 - 18 + 5 = -4, so the vertex is (3, -4).
Find the x intercepts of y = x^2 - 6x + 5.
(x - 1)(x - 5) = 0, so x = 1 and x = 5. Both are two units from the axis at x = 3.
Does y = -2x^2 + 3 open upward or downward, and where is its vertex?
Downward, since a is negative. With b equal to zero the axis is x = 0, so the vertex is (0, 3), its highest point.
See also
Sources
- Graphing quadratic functions as taught in United States eighth grade and introductory algebra. Diagrams are drawn for this site.