options: Use functions to model relationships between quantities.

Analyze functions using different representations.

Build a function that models a relationship between two quantities.

Model periodic phenomena with trigonometric functions.

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

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

Prove and apply trigonometric identities.

Extend the domain of trigonometric functions using the unit circle.

Define, evaluate, and compare functions.

Understand the concept of a function and use function notation.

Build new functions from existing functions.

Construct and compare linear and exponential models and solve problems.
problem_activity: Task
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$f(x)=128\\cdot \\left({\\frac {1}{2}}\\right)^x$"}, {"_value": "Successive quotients"}]}, {"td": [{"_value": "0"}, {"_value": "128"}, {"_value": "----"}]}, {"td": [{"_value": "1"}, {"_value": "64"}, {"_value": "$\\frac{f(1)}{f(0)}=\\frac {64}{128} = \\frac{1}{2}$"}]}, {"td": [{"_value": "2"}, {}, {"_value": "$\\frac{f(2)}{f(1)}=$"}]}, {"td": [{"_value": "3"}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}]}]}]}]}
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$f(x)=128\\cdot \\left({\\frac {1}{2}}\\right)^x$"}, {"_value": "Successive quotients"}]}, {"td": [{"_value": "0"}, {"_value": "128"}, {"_value": "---"}]}, {"td": [{"_value": "2"}, {"_value": "32"}, {"_value": "$\\frac{f(2)}{f(0)} = \\frac {32}{128} = \\frac{1}{4}$"}]}, {"td": [{"_value": "4"}, {}, {"_value": "$\\frac{f(4)}{f(2)}=$"}]}, {"td": [{"_value": "6"}, {}, {}]}, {"td": [{"_value": "8"}, {}, {}]}]}]}]}
Let $f(x)=a \cdot b^x$. Let $x_0$ be any particular $x$-value. Show that if $x_0$ is increased by a constant, $\Delta x$, the successive quotient  $$\frac{f(x_0+ \Delta x)}{f(x_0)}$$ is always the same no matter what $x_0$ is.
Is b) an example of the result of c)? Explain.


correct_option_index: 11