Geometry and Exponential Functions Practice Flashcards

Comprehensive practice flashcards covering geometric constructions, linear equations of parallel lines, geometric sequences, recursive/explicit formulas, and exponential growth/decay models.

What are the two primary characteristics of a perpendicular bisector in a construction?

It cuts a line segment into two equal halves and forms a $90^\text{o}$ angle with that segment.

What is the function of an angle bisector in geometric constructions?

It cuts an angle exactly in half.

If line $w$ is parallel to line $j$ which has the equation $y = -\frac{1}{5}x + 2$, and line $w$ passes through points $(0,1)$ and $(10,t)$, what is the value of $t$?

$$t = -1$$

If line $q$ is parallel to line $z$ which has the equation $y = \frac{1}{2}x - 1$, and line $q$ passes through points $(0,3)$ and $(4,r)$, what is the value of $r$?

$$r = 5$$

What is the general equation to represent an exponential function based on a table?

$$f(x) = \text{zero term} \times \text{common ratio}^x$$

When writing an explicit equation from a geometric sequence starting with the first term, what is the standard formula used?

$$f(n) = f(1) \times r^{n-1}$$

In a geometric sequence, what does it indicate about the common ratio if the sequence alternates between positive and negative values (e.g., $+, -, +, -$)?

The common ratio is negative.

What is the explicit formula for the geometric sequence $5, -10, 20, -40, \text{...}$?

$$f(n) = 5 \times (-2)^{n-1}$$

Given the sequence $10, 20, 40, 80, 160, \text{...}$, identify the Recursive Function.

$f(1) = 10$, $f(n) = 2f(n - 1)$

Given the recursive function $f(1) = 3$ and $f(n) = 5f(n - 1)$ for $n \text{\textgreater 2}$, what is the equivalent explicit equation for $n \text{\textgreater 1}$?

$$f(n) = 3(5)^{n-1}$$

If $f(x) = 2^x$, what is the equation for $g(x)$ if it is translated $5$ units up?

$$g(x) = 2^x + 5$$

If $f(x) = 3^x$, what is the equation for $w(x)$ if it is translated $7$ units to the right?

$$w(x) = 3^{x-7}$$

A car is purchased for $\text{\$}35,200$ and depreciates at a rate of $5\%$ per year. What is the equation for the value of the car $V(t)$ after $t$ years?

$$V(t) = 35,200 \times (0.95)^t$$

A town has a population of $55,500$ and shrinks at a rate of $6\%$ every year. What is the remaining decimal factor used in the exponential equation?

$$0.94$$

An investment of $\text{\$}15,000$ is placed in a savings account with an annual interest rate of $2.3\%$. What is the exponential function that models this situation?

$$f(x) = 15,000 \times (1.023)^x$$

In the population equation $f(x) = 95,000(0.92)^x$, is the population increasing or decreasing, and by what percentage?

Decreasing by $8\%$

In the equation $f(x) = 100 \times 3^x$ representing a bacteria population where $x$ is days, what does the base $3$ represent?

The growth factor.

A boat purchased for $\text{\$}45,000$ loses $7.5\%$ of its value each year. What is the Decay Factor used in the formula?

$$0.925$$