Graphing Quadratic Equations
Quadratic Equation
A Quadratic Equation in Standard Form
(a, b, and c can have any value, except that a can't be 0.)
Here is an example:
Quadratic Equation
Graphing
You can graph a Quadratic Equation using the Function Grapher, but to really understand what is going on, you can make the graph yourself. Read On!
The Simplest Quadratic
The simplest Quadratic Equation is:
f(x) = x2
And its graph is simple too:
Square function
This is the curve f(x) = x2
It is a parabola.
Now let us see what happens when we introduce the "a" value:
f(x) = ax2
ax^2
Larger values of a squash the curve inwards
Smaller values of a expand it outwards
And negative values of a flip it upside down
Quadratic Graph
Play With It
Now is a good time to play with the
"Quadratic Equation Explorer" so you can
see what different values of a, b and c do.
The "General" Quadratic
Before graphing we rearrange the equation, from this:
f(x) = ax2 + bx + c
To this:
f(x) = a(x-h)2 + k
Where:
h = -b/2a
k = f( h )
In other words, calculate h (=-b/2a), then find k by calculating the whole equation for x=h
First of all ... Why?
Well, the wonderful thing about this new form is that h and k show us the very lowest (or very highest) point, called the vertex:
And also the curve is symmetrical (mirror image) about the axis that passes through x=h, making it easy to graph
quadratic vertex
So ...
h shows us how far left (or right) the curve has been shifted from x=0
k shows us how far up (or down) the curve has been shifted from y=0
Lets see an example of how to do this:
Example: Plot f(x) = 2x2 - 12x + 16
First, let's note down:
a = 2,
b = -12, and
c = 16
Now, what do we know?
a is positive, so it is an "upwards" graph ("U" shaped)
a is 2, so it is a little "squashed" compared to the x2 graph
Next, let's calculate h:
h = -b/2a = -(-12)/(2x2) = 3
And next we can calculate k (using h=3):
k = f(3) = 2(3)2 - 12·3 + 16 = 18-36+16 = -2
So now we can plot the graph (with real understanding!):
2x^2-12x+16
We also know: the vertex is (3,-2), and the axis is x=3
From A Graph to The Equation
What if we have a graph, and want to find an equation?
Example: you have just plotted some interesting data, and it looks Quadratic:
quadratic data
Just knowing those two points we can come up with an equation.
Firstly, we know h and k (at the vertex):
(h, k) = (1,1)
So let's put that into this form of the equation:
f(x) = a(x-h)2 + k
f(x) = a(x-1)2 + 1
Then we calculate "a":
We know (0, 1.5) so: f(0) = 1.5
And we know the function (except for a): f(0) = a(0-1)2 + 1 = 1.5
Simplify: f(0) = a + 1 = 1.5
a = 0.5
And so here is the resulting Quadratic Equation:
f(x) = 0.5(x-1)2 + 1
Note: This may not be the correct equation for the data, but it’s a good model and the best we can come up with.
