Consider the quadratic function $f(x) = A(x-h)^2 + k$. This is
studied extensively with its parabolic graph in most texts.
We will examine mapping diagrams for an alternative visualization
of the key features of $f $.
In considering the quadratic functions in
this form, we can read the instructions for the function as a
composition of commands:
- Subtract $h$ to $x$ to form $x-h$.
- Square the result of step 1 to form $(x-h)^2$.
- Multiply the result of step 2 by the magnification factor $A$
to form $A(x-h)^2$.
- Add $k$ to the result of step 3 to form the quadratic function
$f(x) = A(x-h)^2 + k$
Note that for Example QF.0, $x^2-2x-3 = (x-1)^2 -4$ so $A=1$,
$h=1$ and $k = -4$.
You should review
Example QF.0 to explore for yourself how the numbers
$A$, $h$, and $k$ are realized in the features of the
mapping diagram.
Before further discussion we'll examine some simple and important
examples.
Example
QF.1.1 :
$q(x) = x^2$. " Square x" : $x^2$ The core
quadratic function.
Example
QF.1.3 :
$q(x) = (x-2)^2$. "Added value before $(...)^2$ ": $x - 2$
Now that you've looked a some simple
examples here are four more [important] examples for the quadratic
function $q(x) = Ax^2 $.
These examples illustrate the effect of the quadratic
coefficient, $A$. Compare the mapping diagram with the graph.
You can consider the effect of other
adding "values after $x^2$ "in the exercises or with the dynamic Example QF.DV.0.
"Adding value before $(...)^2$ " is worth a few more examples
treated as compositions of $g(x) =x - h$ before $f(x) = x^2$
From this construction and understanding
of the core quadratic function $q(x) = x^2$, the following
fundamental observations should make sense in the mapping
diagrams:
You can use the next dynamic example to
see the effects of the quadratic coefficient and adding constants
before and after $x^2$ simultaneously on a mapping diagram and a
graph.