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
Before further discussion we'll examine some simple and important
$q(x) = x^2$. " Square x" : $x^2$ The core
$q(x) = (x-2)^2$. "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
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
Dynamic Visualization for Vertex Form of
Quadratic Functions: Graphs and Mapping Diagrams