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.

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).

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.
problem_activity: Task
Complete the table. In the third column, show your work as demonstrated. What do you notice about the 3rd column?
| $x$ | $y=2x+5$ | $\Delta y$ |
|-----|----------|------------|
|  1  |    7     |    ----    |
|  2  |    9     | 9 - 7 = 2  |
|  3  |          |            |
|  4  |          |            |
|  5  |          |            |
Complete the table, showing your work as above. What do you notice about the 3rd column? What is the graphical interpretation of this?
| $x$ |    $y=ax+b$    |            $\Delta y$             |
|-----|----------------|-----------------------------------|
|  1  | $a\cdot 1 +b$  |                ---                |
|  2  | $a \cdot 2 +b$ | $a\cdot 2+b - (a\cdot 1 + b) = a$ |
|  3  |                |                                   |
|  4  |                |                                   |
|  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: 2