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.

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.

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).
problem_activity: Task
Complete the table. In the third column, show your work as demonstrated. What do you notice about the 3rd column?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$y=2x+5$"}, {"_value": "$\\Delta y$"}]}, {"td": [{"_value": "1"}, {"_value": "7"}, {"_value": "----"}]}, {"td": [{"_value": "2"}, {"_value": "9"}, {"_value": "9 - 7 = 2"}]}, {"td": [{"_value": "3"}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}]}, {"td": [{"_value": "5"}, {}, {}]}]}]}]}
Complete the table, showing your work as above. What do you notice about the 3rd column? What is the graphical interpretation of this?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$y=ax+b$"}, {"_value": "$\\Delta y$"}]}, {"td": [{"_value": "1"}, {"_value": "$a\\cdot\u00a01 +b$"}, {"_value": "---"}]}, {"td": [{"_value": "2"}, {"_value": "$a \\cdot\u00a02 +b$"}, {"_value": "$a\\cdot\u00a02+b - (a\\cdot\u00a01 + b) = a$"}]}, {"td": [{"_value": "3"}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}]}, {"td": [{"_value": "5"}, {}, {}]}]}]}]}
Let $y=ax+b$. Let $x_0$ be any particular $x$-value. Show that if $x_0$ is increased by 1, the corresponding $\Delta y$ is a constant that does not depend on $x_0$. What is this constant?
Does (a) serve as an example of the result in (c) ? Explain.


correct_option_index: 1