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.]

If `a > 0`, then the parabola has a minimum point and it opens upwards (U-shaped) eg.

`y = x^2+ 2x − 3`

U-shaped parabola

If `a < 0`, then the parabola has a maximum point and it opens downwards (n-shaped) eg.

`y = -2x^2+ 5x + 3`

n-shaped parabola

Sketching Parabolas

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

`x=-b/(2a)`

(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.

Example 1

Sketch the graph of the function `y = 2x^2− 8x + 6`

Example 2

Sketch the graph of the function `y = -x^2+ x + 6`

Exercise

Sketch the graph `y = -x^2− 4x − 3`

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