3.4 Quadratic Functions
Wednesday, November 10, 2010
3:29 PM
We know by now that there are many different types of functions, one special type is a quadratic function.
Definition: Quadratic Function
General Form:
The graph of a quadratic function is called a parabola.
Domain of quadratic functions are all real numbers.
They have symmetry vertical symmetry (about the y-axis) although we will see that the axis of symmetry can be
changed.
Definition: Vertex form
Graph the two functions:
What are the lowest points on each function?
If you move one to the right of the lowest point, how does the change in the function value change?
There is a name for the point
It is called the _________________
When will the vertex be the minimum point on the graph?
When will it be the maximum?
Finding Vertex form of a Parabola
Let's do another one.
Notes Page 1
Let's do another one.
So now we have vertex form, and this is helpful especially for graphing.
Graphing a Parabola
-Start with the vertex.
-Now identify the Axis of Symmetry
-Choose a couple of points to either side, and plot them
-Use symmetry to graph points on the side not chosen
-Connect the dots
Reading information from the graph
-Increasing and decreasing.
-Domain and Range.
-Maximum and Minimum.
Deriving a way to find the Vertex of a Quadratic in standard form.
Start with
So, the x-coordinate of the vertex is
Then, to get the other coordinate, evaluate the function at the point you find for x.
Notes Page 2
So the vertex is
What is the vertex of the Quadratic given by:
Now can we go the other way? If given a vertex and a point, can we find the formula?
So find the equation of the Quadratic with the vertex
We want something of the form
and x-intercept 4.
-What is h? What is k? How can we use the x-intercept?
-How could we get a? (
Okay, now for a Word problem:
A dairy farm has a barn that is 150ft long and 75ft wide. The owner has 240ft of fencing and plans to
use all of it in the construction of two identical adjacent outdoor pens, with part of the long side of the
barn as one side of the pens, and a common fence between the two. The owner wants the pens to be
as large as possible.
a) Construct a model for the combined area of both pens in the form of a function
the domain of
b) Find the value of x that produces the maximum combined area.
c) Find the dimensions and the area of each pen.
Notes Page 3
and state
Starting with a picture is helpful.
Now what formulas are needed?
Notes Page 4