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
- Extreme Value of $f$: The extreme value for $f$ will occur
at $x=h$. This value will be $f(h) = A(h-h)^2 +k = k$.
- When $A \gt 0$ the value at $x=h$, that is, $k$
will be the smallest (minimum) value for $f$.
- When $A < 0$ the value at
$x=h$, that is, $k$ will be the largest
(maximum) value for $f$.
- Magnification, The effect of $A$: The value of
$A$ can be seen by looking at the change in the value of $f$
from $x=h$ to $x= h+1$. $f(h) = k$ while $f(h+1) = A +k$. Though
this change is not a constant for all changes in $x$, it
indicates the magnification that will effect all other changes
for $f$ from the core function of $x^2$. It will be greater for
greater values of $|A|$, and will determine whether it is
an increase of decrease by whether $A \gt 0$ or $A < 0$. On
the graph this is usually described as controlling the bowl
shape of the parabolic graph of $f$.
- Symmetry of $f$ with respect to $x=h$: The values
$f$ at $x=h+a$ and $x=h-a$ will be equal since $((h+a)-h)^2 +
k= a^2+k =(-a)^2 +k =((h-a)-h)^2$ +k.
Thus $f$ takes on equal values for numbers symmetrically
located by either adding/subtracting the same number to/from
$h$ in the domain.
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 Quadratic
Functions: Graphs and Mapping Diagrams