options: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
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 $P_2$.


correct_option_index: 2