4. The Graph of the Quadratic Function
In general, the graph of a quadratic equation
`y = ax^2+ bx + c`
is a parabola.
[You can also see a more detailed description of parabolas in the Plane Analytic Geometry section.]
Shape of the parabola
If `a > 0`, then the parabola has a minimum point and it opens upwards (U-shaped) eg.
`y = x^2+ 2x − 3`
If `a < 0`, then the parabola has a maximum point and it opens downwards (n-shaped) eg.
`y = -2x^2+ 5x + 3`
In order to sketch the graph of the quadratic equation, we follow these steps :
(a) Check if `a > 0` or `a < 0` to decide if it is U-shaped or n-shaped.
(b) The Vertex: The x-coordinate of the minimum point (or maximum point) is given by
(which can be shown using completing the square method, which we met earlier).
We substitute this x-value into our quadratic function (the y expression). Then we will have the (x, y) coordinates of the minimum (or maximum) point. This is called the vertex of the parabola.
(c) The coordinates of the y-intercept (substitute `x = 0`). This is always easy to find!
(d) The coordinates of the x-intercepts (substitute `y = 0` and solve the quadratic equation), as long as they are easy to find.
Sketch the graph of the function `y = 2x^2− 8x + 6`
Sketch the graph of the function `y = -x^2+ x + 6`
Sketch the graph `y = -x^2− 4x − 3`