A nonlinear function that can be written on the standard
form
is called a quadratic function.
All quadratic functions has a U-shaped graph called a parabola.
The parent quadratic function is
The lowest or the highest point on a parabola is called the
vertex. The vertex has the x-coordinate
The y-coordinate of the vertex is the maximum or minimum value
of the function.
a >
0
parabola opens
up
minimum value
a <
0
parabola opens
down
maximum value
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
The y-intercept of the equation is c.
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
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
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
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
