standard_description: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
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