Example LF.FORM.2: Suppose $f$ is a linear function with slope
$=m=2$ and $f(−1)=3$. Find the point-slope form of the line.
Visualize $f$ with a mapping diagram that illustrates the slope$ =
m=2$ and $f(−1)=3$. Find the X- and Y-intercepts for $f$ and
visualize them on the mapping diagram as well.
Solution: 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,
$3+2(x-(-1))= 2x+5$, which corresponds to the linear function's
value for $x$
When the point in the domain is $-1$, the arrow points to $f(-1) =
3$ visualizing the point $(-1,3)$ on the graph of $f$.
If we
consider $f(0)= 2 +2(0+1) = 5$ then the difference in the value of
$f$ for a unit change in $x$ is $\Delta y=f(0)-f(-1) = 5-3=2.$
Since a unit step is used, we see the slope
(magnification, rate) visualized in the gap between the heads of
consecutive arrows on the mapping diagram.
As $m > 0$, the arrows never cross illustrating that the
function is increasing at a rate of 2 units on the target
value for every unit increase in the domain value.
The X- intercept is the value $a$ on the domain axis from which the
arrow hits the number $0$ on the target. For this function,
that value is $a=-\frac 5 2$. You can check this by checking the box
"Show a where f(a) = 0 " and entering $-\frac 5 2$ for $a$ on the
GeoGebra figure.
