standard_description: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
problem_activity: Task
Suppose $P_1 = (0,5)$ and $P_2 = (3,−3)$. Sketch $P_1$ and
$P_2$. For which real numbers $m$ and $b$ does the graph of a linear function described by the equation $f(x) = mx+b$ contain $P_1$? Explain. Do any of these graphs also contain $P_2$? Explain.
Suppose $P_1 = (0,5)$ and $P_2 = (0,7)$.  Sketch $P_1$ and $P_2$.  Are there real numbers $m$ and $b$ for which the graph of a linear function described by the equation $f(x)=mx+b$ contains $P_1$ and $P_2$? Explain.
Now suppose $P_1= (c,d)$ and $P_2 = (g,h)$ and $c$ is not equal to $g$. Show that there is only one real number $m$ and only one real number $b$ for which the graph of $f(x)=mx+b$ contains the points $P_1$ and $P_2$.


aligns: no