Example QF.FORM.1: Visualize $q(x)=2x^2-4$ with a mapping
diagram that illustrates the effect of $= A=2$ and the "Y-intercept"
is $-4$ .
Find the X-intercepts for $q$ and visualize this on the mapping
diagram as well.
Draw a mapping diagram yourself or use the GeoGebra figure.
Given a point / number, $x$, on the source line, there is a unique
arrow meeting the target line at the point / number, $2x^2 -
4$, which corresponds to the quadratic function's value for $x$
When the point in the domain corresponds to the number $0$, the arrow points to $q(0) = -4$
visualizes the "Y-intercept" of $q$, the "initial value" of $q$.
The value at $x=0$ is the minimum value for the function.
If we consider $q(1)= -2$ then the difference in the value of $q$
for a unit change in $x$ from the extreme point is $q(1)-q(0) = -2-
(-4) = 2.$
Since a unit step from the extreme point is used in this mapping
diagram, we see the leading coefficient visualized in the gap
between the heads of consecutive arrows on the mapping
The "X- intercepts" for the graph or "zeroes' for the function $q$ are the values for $a$ on the domain axis from
which the arrow hits the number $q(a) =0$ on the target. For this
function, those values is $a=\pm \sqrt 2$. You can check this by
entering "-sqrt(2)" $=- \sqrt 2 $ and "sqrt(2) " $= \sqrt 2$
for $a = x_0$ on the GeoGebra figure.