Functions, Linear Equations, and Scatter Plot Analysis

Vocabulary flashcards covering function notation, evaluating linear equations, interpreting contextual word problems, and analyzing scatter plots and residual plots.

Function Table Evaluation

The process of identifying corresponding inputs and outputs using a tabular representation of a function $f(x)$, such as determining that $f(5) = 23$, $f(-5) = 17$, and $f(-2) = -5$.

Fractional Time Input in Function Notation

The representation of elapsed time as a decimal within a function's input, such as writing 1:30 am (1 hour and 30 minutes after midnight) as $t = 1.5$ in the equation $f(1.5) = 20$.

Function Inequality Comparison

Using inequality symbols to compare output values at different inputs, such as $f(9) > f(18)$ to express that the temperature at 9 am ($t = 9$) was warmer than at 6 pm ($t = 18$).

Function Value at Zero Input ($g(0)$)

The base value or initial cost modeled by a function before any additional units are added, such as $g(0) = 10$ representing a Garden salad with 0 additional ingredients costing 10 dollars.

Linear Equation from Table Values

The algebraic rule of the form $y = mx + b$ determined by calculating the slope and $y$-intercept from a data table; for coordinates $(0, 9)$, $(1, 6)$, $(2, 3)$, and $(3, 0)$, the equation is $y = -3x + 9$.

Goodness of Fit (Residual Plot Criterion)

An assessment showing that a linear model appropriately fits data when the line passes through the center of the scatter plot points and the corresponding residuals are randomly dispersed around zero without a distinct pattern.

Optimal Residual Plot

A residual plot corresponding to the best line of fit, characterized by data points scattered closely and randomly both above and below the horizontal zero line rather than showing a systematic curved or slanted trend.

Correlation Coefficient ($r$-value)

A statistical metric that evaluates the strength and direction of a linear relationship between two quantitative variables, such as the number of traffic tickets and car insurance cost.

Negative Slope Line of Fit

A linear model whose output decreases as the input increases, such as $y = \frac{-7}{13}x + 9$, which has a $y$-intercept of $(0, 9)$ and passes through the point $(13, 2)$.

Strong Positive Relationship

A bivariate data pattern where points cluster tightly along an upward-sloping trend, demonstrating that increases in one variable (such as office space size in square feet) directly correspond to increases in the other variable (such as annual rent in dollars).

No Correlation

The type of relationship between two variables that occurs when the $x$ value has no effect on or association with the $y$ value.