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.
Using the formula for area for a parallelogram, A=BH, where A is area, B is Base, and H is height, you follow that. So in this case, you would want to find one of the area's then multiply it by 4. So for the area you get 20.52 meters. If you multiply it by 4, you get 82.08 meters. Have a nice day!
