Chapter 2: Functions, Equations, and Graphs Review

Linear Representations and Interconvertibility:
- Linear functions can be represented using slope-intercept form, point-slope form, or standard form.
- Algebraic manipulation allows for transformation from one form to another as required by the problem context.
Slope-Intercept Form (Lesson 2-3):
- General Equation: $y = mx + b$
- Components:
- $m$ represents the slope of the line.
- $b$ represents the $y$ -intercept, which corresponds to the point $(0, b)$ .
- Example Application:
- Equation: $y = 2x - 1$
- Slope ( $m$ ): $2$
- $y$ -intercept ( $b$ ): $-1$ , located at $(0, -1)$
Point-Slope Form (Lesson 2-4):
- General Equation: $y - y_1 = m(x - x_1)$
- Components:
- $m$ represents the slope of the line.
- $(x_1, y_1)$ represents a specific given point on the line.
- Example Application:
- Equation: $y - 5 = 2(x - 3)$
- Slope ( $m$ ): $2$
- Designated Point $(x_1, y_1)$ : $(3, 5)$
Standard Form of a Linear Equation (Lesson 2-4):
- General Equation: $Ax + By = C$
- Components: $A$ , $B$ , and $C$ are constants (typically integers where $A$ and $B$ are not both zero).
- Example Application:
- Equation: $2x - y = 1$

Transformation Rules for Function Families (Lesson 2-6):
- Given a parent function $f(x)$ , transformations modify the graph based on parameter placement:
- $f(x) + k$ : Vertical translation shifted by $k$ units.
- $f(x - h)$ : Horizontal translation shifted by $h$ units.
- $a f(x)$ : Vertical stretch when $|a| > 1$ ; vertical compression when $0 < |a| < 1$ .
- $-f(x)$ : Reflection across the $x$ -axis.
- $f(-x)$ : Reflection across the $y$ -axis.
Absolute Value Functions and Graphs (Lesson 2-7):
- Parent Function: $y = |x|$
- Features a V-shaped graph with a vertex at the origin $(0, 0)$ .
- General Transformation Form:
- Equation: $y = a|x - h| + k$
- Vertex Coordinates: $(h, k)$
- Parameter Functions:
  - The value of $a$ dictates vertical stretching, vertical compression, or reflection across the horizontal axis.
  - The value of $h$ shifts the parent graph horizontally.
  - The value of $k$ shifts the parent graph vertically.

Using Linear Models (Lesson 2-5):
- When bivariate data clusters along a straight path, it exhibits a linear relationship.
- A trend line (or line of best fit) is fitted to the data to model relationships and make predictions.
- Direction of Correlation:
- Positive correlation occurs when data values increase together (represented by a trend line with a positive slope).

Fundamental Definitions:
- Relation: A set of ordered pairs.
- Domain: The set of all first coordinates ( $x$ -coordinates) of the ordered pairs in a relation.
- Range: The set of all second coordinates ( $y$ -coordinates) of the ordered pairs in a relation.
- Function: A specific type of relation where each element of the domain is paired with exactly one element of the range.
Worked Example: Evaluating a Relation:
- Problem Set: Determine whether the relation is a function, and state its domain and range:
- $\{(5, 0), (8, 1), (1, 3), (5, 2), (3, 8)\}$
- Function Test Analysis:
- The domain element $5$ is paired with two distinct range elements: $0$ and $2$ .
- Because a single input maps to multiple outputs, this relation is not a function.
- Domain Determination:
- The set of all $x$ -coordinates: $\{1, 3, 5, 8\}$
- Range Determination:
- The set of all $y$ -coordinates: $\{0, 1, 2, 3, 8\}$

Fundamental Concepts:
- Direct Variation: A linear relationship between two variables, $x$ and $y$ , represented by an equation of the form:
- $y = kx$ , where $k \neq 0$
- Constant of Variation:
- The nonzero constant $k$ , given by the ratio $k = \frac{y}{x}$ .
- Proportional Reasoning:
- Because the ratio $\frac{y}{x}$ is constant for every data pair, proportions can be used to calculate missing values in direct variation systems.
Worked Example: Identifying Direct Variation and Formulating Rules:
- Problem: Given a data table of paired values for $x$ and $y$ , determine if $y$ varies directly with $x$ . If so, calculate the constant of variation and write the function rule.
- Verification Process:
- Evaluate the ratio of $\frac{y}{x}$ across each data point in the table.
- Ratios evaluate to: $\frac{y}{x} = 3$
- Conclusion:
- Because the ratio is constant across all points, $y$ varies directly with $x$ .
- Constant of variation ( $k$ ): $3$
- Function rule: $y = 3x$

Sentence Completion 1:
- Prompt: The graph of a function is (always/sometimes) a line.
- Correct Term: sometimes
- Explanation: Linear functions produce straight lines, but non-linear functions (such as absolute value functions, quadratic functions, or polynomial functions) produce curved or V-shaped graphs.
Sentence Completion 2:
- Prompt: The equation $y - 5 = 3(x + 2)$ is in (point-slope/slope-intercept) form.
- Correct Term: point-slope
- Explanation: The equation follows the template $y - y_1 = m(x - x_1)$ , where $m = 3$ and the point $(x_1, y_1)$ is $(-2, 5)$ .