The graph of y = ax^2 + bx + c

A nonlinear function that can be written on the standard form

$$ax^{2}+bx+c,\: \: where\: \: a\neq 0$$

is called a quadratic function.

All quadratic functions has a U-shaped graph called a parabola. The parent quadratic function is

$$y=x^{2}$$

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The lowest or the highest point on a parabola is called the vertex. The vertex has the x-coordinate

$$x=-\frac{b}{2a}$$

The y-coordinate of the vertex is the maximum or minimum value of the function.

a > 0                   parabola opens up                    minimum value

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a < 0                    parabola opens down              maximum value

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A rule of thumb reminds us that when we have a positive symbol before x2 we get a happy expression on the graph and a negative symbol renders a sad expression.

The vertical line that passes through the vertex and divides the parabola in two is called the axis of symmetry. The axis of symmetry has the equation

$$x=-\frac{b}{2a}$$

The y-intercept of the equation is c.

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When you want to graph a quadratic function you begin by making a table of values for some values of your function and then plot those values in a coordinate plane and draw a smooth curve through the points.


Example

Graph

$$x=x^{2}+2x+1$$

Make a table of value for some values of x. Use both positive and negative values!

x y = x2 + 2x + 1
-3 4
-2 1
-1 0
0 1
1 4
2 9
3 16

Graph the points and draw a smooth line through the points and extend it in both directions

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Notice that we have a minimum point which was indicated by a positive a value (a = 1). The vertex has the coordinates (-1, 0)  which is what you will get if you use the formula for the x-coordinate of the vertex

$$x=-\frac{b}{2a}=-\frac{2}{2\cdot 1}=-1$$

and that the line has an y-intercept of (0, 1) which could have been determined from the c-value which is 1.

If you have an absolute value of a that is greater than 1 the parabola will be narrower than the parental quadratic function. And the opposite that if you have a absolute value of a that is less than 1 then the parabola will be wider than the parental quadratic function.

Here you can get a visual of your quadratic equations


Video lesson

Graph the function

$$x^{2} - 3x - 10$$