Algebra 1 Module 3 Test Review

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Questions and Answers

What is a piecewise function?

A function defined by two or more different equations applied to different parts of the function's domain.

How do you find the equation for lines on a piecewise function?

Use a calculator, put in 2 coordinates, then find the linear regression.

Which values are most critical to include in order to sketch the graph of a piecewise function?

The starting and the end values.

What are x-values that are graphed?

Domain. Signup and view all the answers

What are y-values that are graphed?

Range. Signup and view all the answers

What is the difference between brackets and parentheses in interval notation?

Brackets include the numbers on the end, parentheses do not. Signup and view all the answers

What is a transformation in the context of graphing functions?

A change in the graph, such as a reflection, compared to the original graph. Signup and view all the answers

What shape do absolute value graphs make?

V-shaped. Signup and view all the answers

Why is infinity always in parentheses instead of brackets?

Infinity is not a number; it is a concept. Signup and view all the answers

What are the 4 transformations of functions?

<ol> <li>y=|x|+k, moves it up or down; 2) y=k|x|, changes width; 3) y=-k|x|, reflects over the x-axis; 4) y=|x|+k, moves left or right.</li> </ol> Signup and view all the answers

What is a sequence?

An ordered list of elements where the first term is often found by substituting 1 into a formula. Signup and view all the answers

What is a recursive sequence?

A sequence defined by initial terms and a recursive formula that relies on previous terms. Signup and view all the answers

What are the two types of sequences?

Arithmetic sequence and geometric sequence. Signup and view all the answers

What is simple interest?

Interest calculated once per year on the original amount borrowed or invested. Signup and view all the answers

What is compound interest?

Interest calculated on the current amount borrowed or invested, where interest becomes part of the principal. Signup and view all the answers

How can you tell if a table is linear, exponential, or quadratic?

Linear if the 1st difference is constant; quadratic if the 2nd difference is constant; exponential if there's a multiplication pattern. Signup and view all the answers

What are the characteristics of linear equations?

Linear equations have an x with no exponent and make a straight line. Signup and view all the answers

What are the characteristics of quadratic equations?

Quadratic equations have an x^2 term and create a U-shaped graph. Signup and view all the answers

What are the characteristics of exponential equations?

Exponential equations have an x in the exponent and change more quickly than linear functions. Signup and view all the answers

What is the difference between linear function growth and exponential function growth?

Exponential function will eventually surpass linear function since it is being multiplied. Signup and view all the answers

What does the explicit formula f(t) = ab^t represent?

It models exponential decay. Signup and view all the answers

What is the general form of a linear model?

f(x) = ax + b. Signup and view all the answers

What is the general form of an exponential model?

f(x) = a(b)^x. Signup and view all the answers

What is a linear function?

A function where the difference in outputs is constant for given differences in inputs. Signup and view all the answers

What is an exponential function?

A function where the quotient of outputs is constant for given differences in inputs. Signup and view all the answers

How does a linear function compare to an exponential function?

An increasing exponential function will eventually exceed any linear function. Signup and view all the answers

Study Notes

Piecewise Functions

A piecewise function is defined by multiple equations, each applicable to specific parts of its domain.
To find the equation for lines in a piecewise function, use a calculator for linear regression with two coordinates.
Critical values for graphing include the starting and ending points.

Domain and Range

The x-values that are graphed represent the domain.
The y-values that are graphed represent the range.

Interval Notation

Brackets [ ] indicate that the endpoints are included; for example, [8, 7) includes 8 but not 7.
Parentheses ( ) indicate that the endpoints are not included.

Transformations

Graph transformations denote changes in the graph's position or shape, such as reflections.
Absolute value graphs produce a V shape.

Infinity in Interval Notation

Infinity is always symbolized with parentheses ( ) because it represents a concept, not a specific number.

Function Transformations

Vertical shifts occur when adding or subtracting outside the function.
Multiplying by a value affects width; greater than 1 narrows, less than 1 widens.
Reflection over the x-axis is achieved by multiplying by a negative value.
Horizontal shifts occur inside the function; negative shifts move the graph left, positive shifts move it right.

Sequences

A sequence is an ordered list defined by a pattern, often expressed through a formula.
Recursive sequences specify initial terms and use a formula to define subsequent terms.
Arithmetic sequences involve a constant difference between terms, while geometric sequences involve a constant ratio.

Interest Calculations

Simple interest is calculated annually on the principal. Formula: I(t) = P * r * t.
Compound interest adds calculated interest to the principal, where future value is calculated using FV = PV(1 + r)^n.

Recognizing Function Types

Linear tables have a constant first difference.
Quadratic tables show a consistent second difference.
Exponential tables reveal a common multiplication pattern.

Characteristics of Equations

Linear equations lack exponents and produce straight lines; examples include y = mx + b.
Quadratic equations include x^2 and form U-shaped graphs.
Exponential equations have x as an exponent, resulting in rapidly changing y values.

Growth Comparisons

Exponential functions will eventually outpace linear functions due to their multiplicative nature.

Models and Parameter Meanings

Linear Model: f(x) = ax + b; 'a' denotes slope and 'b' denotes y-intercept. Growth is constant.
Exponential Model: f(x) = a(b)^x; 'a' is the y-intercept, and 'b' is the constant growth/decay factor.

Differentiating Functions

Linear functions maintain a constant difference in output for equal input differences.
Exponential functions have a constant output ratio for the same input differences, leading to rapid growth.
The rates of increase diverge significantly over time, with exponential functions exhibiting sharp rises compared to linear functions.

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Description

This quiz focuses on key concepts from Algebra 1 Module 3, specifically dealing with piecewise functions. Review essential definitions and methods for finding equations of lines in piecewise contexts. Perfect for preparing for your upcoming test!