options: Analyze functions using different representations.

Define, evaluate, and compare functions.

Model periodic phenomena with trigonometric functions.

Prove and apply trigonometric identities.

Interpret functions that arise in applications in terms of the context.

Understand the concept of a function and use function notation.

Construct and compare linear and exponential models and solve problems.

Build a function that models a relationship between two quantities.

Use functions to model relationships between quantities.

Build new functions from existing functions.

Extend the domain of trigonometric functions using the unit circle.

Interpret expressions for functions in terms of the situation they model.
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: 6