Unit 4 Linear Functions, Word Problems, and Function Notation

Practice vocabulary flashcards covering the comparison of linear functions, real-world interpretations of slope and intercepts, domain constraints, linear equation forms, and function notation.

Comparing Linear Functions

The process of analyzing linear representations across different forms (such as tables, graphs, and equations) by determining and contrasting their slope (rate of change) and $y$-intercept.

Slope in Context

The rate of change representing the number of items or units incurred per unit of the $x$-value, such as cost per item or inches of plant growth per week.

$y$-intercept in Context

The value of $y$ when $x = 0$, representing the starting amount, initial value, or flat fee (such as the original height of a plant or an initial flat charge).

$x$-intercept in Context

The value of $x$ when $y = 0$, representing the point at which the dependent output reaches zero (such as when the total cost is $0$ or the height is $0$).

Interpreting the $y$-value Given an $x$-value

Determining what the dependent variable represents when a specific input is given, such as calculating the total cost when buying $4$ food items or the height of a plant after $3$ weeks.

Interpreting the $x$-value Given a $y$-value

Determining what the input variable represents when an output value is given, such as finding how many items were bought with $\$25$ or how long it took a plant to reach $12$ inches.

Domain Restrictions in Context

Practical limitations placed on possible $x$-values in a real-world scenario where negative or fractional values are impossible (for example, discrete items requiring positive integers versus continuous time requiring positive rational numbers).

Point-Slope Form Procedure

The method of using $y - y_1 = m(x - x_1)$ when given a slope $m$ and a point $(x_1, y_1)$, or two points (after first calculating the slope), to convert the relationship into slope-intercept form.

Function Notation

A mathematical notation where $f(x)$ is used in place of $y$, designating the number inside the parentheses as the input ($x$-value) and the evaluated result as the output ($y$-value).

Evaluating a Function

The process of substituting a given input value into the variable of a function expression to calculate its corresponding output value (for example, finding $f(12)$ by substituting $12$ into $f(x)$).

Solving for an Input Given Function Output

The process of setting the entire function expression equal to a specified output value and solving for the unknown variable $x$ (for example, finding $x$ when $g(x) = 3$ by solving the equation for $x$).