A linear function<\/strong> is a function with a constant rate of change, represented by a polynomial of degree 1. Its graph is always a straight line.<\/p>\r\n Key Features of Linear Graphs<\/strong><\/p>\r\n\r\n Slope measures how one quantity changes in relation to another:<\/p>\r\n [latex]m = \\dfrac{\\text{change in output}}{\\text{change in input}} = \\dfrac{\\text{rise}}{\\text{run}} = \\dfrac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/p>\r\n The units of slope are \"output units per input unit\" (like miles per hour or dollars per year).<\/p>\r\n Graphing by Plotting Points<\/strong><\/p>\r\n\r\n Graphing Using Slope and y-intercept<\/strong><\/p>\r\n\r\n Three Forms of Linear Equations<\/strong><\/p>\r\n Slope-Intercept Form:<\/strong> [latex]y = mx + b[\/latex]<\/p>\r\n\r\n Point-Slope Form:<\/strong> [latex]y - y_1 = m(x - x_1)[\/latex]<\/p>\r\n\r\n Standard Form:<\/strong> [latex]Ax + By = C[\/latex]<\/p>\r\n\r\n Writing Equations of Lines<\/strong><\/p>\r\n Given two points:<\/strong><\/p>\r\n\r\n Given a graph:<\/strong><\/p>\r\n\r\n Horizontal and Vertical Lines<\/strong><\/p>\r\n\r\n Parallel and Perpendicular Lines<\/strong><\/p>\r\n\r\n Parallel and Perpendicular to Horizontal or Vertical Lines:<\/strong><\/p>\r\n\r\n Building Linear Models from Word Problems<\/strong><\/p>\r\n\r\n Scatter Plots and Lines of Best Fit<\/strong><\/p>\r\n\r\n Interpolation vs. Extrapolation<\/strong><\/p>\r\n\r\n Absolute value<\/strong> measures distance from zero on a number line, always non-negative: [latex]|x| = \\begin{cases} x & \\text{if } x \\geq 0 \\ -x & \\text{if } x < 0 \\end{cases}[\/latex]<\/p>\r\n Graphing Absolute Value Functions<\/strong><\/p>\r\n The basic absolute value function [latex]f(x) = |x|[\/latex] has a V-shape with a corner point at the origin. It can be transformed using:<\/p>\r\n\r\n Solving Absolute Value Equations<\/strong><\/p>\r\n For [latex]|A| = B[\/latex] where [latex]B \\geq 0[\/latex]:<\/p>\r\n\r\n Special cases:<\/p>\r\n\r\n Solving Absolute Value Inequalities<\/strong><\/p>\r\n For [latex]|X| < k[\/latex] where [latex]k > 0[\/latex]:<\/p>\r\n\r\n For [latex]|X| > k[\/latex] where [latex]k > 0[\/latex]:<\/p>\r\n\r\n Same logic applies for [latex]\\leq[\/latex] and [latex]\\geq[\/latex]:<\/p>\r\n\r\n absolute value<\/strong><\/p>\r\n the distance of a number from zero on the number line, always non-negative; denoted [latex]|x|[\/latex]<\/p>\r\n absolute value equation<\/strong><\/p>\r\n an equation of the form [latex]|A| = B[\/latex] with [latex]B \\geq 0[\/latex]; has solutions when [latex]A = B[\/latex] or [latex]A = -B[\/latex]<\/p>\r\n absolute value inequality<\/strong><\/p>\r\n a relationship in the form [latex]|A| < B[\/latex], [latex]|A| \\leq B[\/latex], [latex]|A| > B[\/latex], or [latex]|A| \\geq B[\/latex]<\/p>\r\n decreasing linear function<\/strong><\/p>\r\n a linear function with a negative slope; as input increases, output decreases<\/p>\r\n extrapolation<\/strong><\/p>\r\n predicting a value outside the domain and range of observed data<\/p>\r\n horizontal line<\/strong><\/p>\r\n a line defined by [latex]y = b[\/latex] where [latex]b[\/latex] is a constant; has slope of 0<\/p>\r\n increasing linear function<\/strong><\/p>\r\n a linear function with a positive slope; as input increases, output also increases<\/p>\r\n interpolation<\/strong><\/p>\r\n predicting a value inside the domain and range of observed data<\/p>\r\n linear function<\/strong><\/p>\r\n a function with a constant rate of change, represented as a polynomial of degree 1; graph is a straight line<\/p>\r\n line of best fit<\/strong><\/p>\r\n a line that best represents the trend in a set of data points; can be found using linear regression<\/p>\r\n parallel lines<\/strong><\/p>\r\n two or more lines with the same slope<\/p>\r\n perpendicular lines<\/strong><\/p>\r\n two lines that intersect at right angles and have slopes that are negative reciprocals of each other<\/p>\r\n point-slope form<\/strong><\/p>\r\n the equation of a line in the form [latex]y - y_1 = m(x - x_1)[\/latex] where [latex]m[\/latex] is the slope and [latex](x_1, y_1)[\/latex] is a point on the line<\/p>\r\n scatter plot<\/strong><\/p>\r\n a graph of plotted points that may show a relationship between two sets of data<\/p>\r\n slope<\/strong><\/p>\r\n the ratio of the change in output to the change in input; measures the steepness and direction of a line; calculated as [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/p>\r\n slope-intercept form<\/strong><\/p>\r\n the equation of a line in the form [latex]y = mx + b[\/latex] where [latex]m[\/latex] is the slope and [latex]b[\/latex] is the y-intercept<\/p>\r\n standard form<\/strong><\/p>\r\n the equation of a line in the form [latex]Ax + By = C[\/latex] where [latex]A[\/latex], [latex]B[\/latex], and [latex]C[\/latex] are integers<\/p>\r\n vertical line<\/strong><\/p>\r\n a line defined by [latex]x = a[\/latex] where [latex]a[\/latex] is a constant; has undefined slope<\/p>\r\n x-intercept<\/strong><\/p>\r\n the point where a graph crosses the x-axis; has coordinates [latex](a, 0)[\/latex]<\/p>\r\n y-intercept<\/strong><\/p>\r\n the point where a graph crosses the y-axis; has coordinates [latex](0, b)[\/latex]; also called the initial value<\/p>","rendered":" A linear function<\/strong> is a function with a constant rate of change, represented by a polynomial of degree 1. Its graph is always a straight line.<\/p>\n Key Features of Linear Graphs<\/strong><\/p>\n Slope measures how one quantity changes in relation to another:<\/p>\n [latex]m = \\dfrac{\\text{change in output}}{\\text{change in input}} = \\dfrac{\\text{rise}}{\\text{run}} = \\dfrac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/p>\n The units of slope are “output units per input unit” (like miles per hour or dollars per year).<\/p>\n Graphing by Plotting Points<\/strong><\/p>\n Graphing Using Slope and y-intercept<\/strong><\/p>\n Three Forms of Linear Equations<\/strong><\/p>\n Slope-Intercept Form:<\/strong> [latex]y = mx + b[\/latex]<\/p>\n Point-Slope Form:<\/strong> [latex]y - y_1 = m(x - x_1)[\/latex]<\/p>\n Standard Form:<\/strong> [latex]Ax + By = C[\/latex]<\/p>\n Writing Equations of Lines<\/strong><\/p>\n Given two points:<\/strong><\/p>\n Given a graph:<\/strong><\/p>\n Horizontal and Vertical Lines<\/strong><\/p>\n Parallel and Perpendicular Lines<\/strong><\/p>\n Parallel and Perpendicular to Horizontal or Vertical Lines:<\/strong><\/p>\n Building Linear Models from Word Problems<\/strong><\/p>\n Scatter Plots and Lines of Best Fit<\/strong><\/p>\n\r\n \t
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Linear Functions<\/h3>\r\n
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Linear Models<\/h3>\r\n
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Absolute Value Functions<\/h3>\r\n
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Key Equations<\/h2>\r\n
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\r\n Slope formula<\/strong><\/td>\r\n [latex]m = \\frac{y_2 - y_1}{x_2 - x_1}[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Slope-intercept form of a line<\/strong><\/td>\r\n [latex]y = mx + b[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Point-slope form of a line<\/strong><\/td>\r\n [latex]y - y_1 = m(x - x_1)[\/latex]<\/td>\r\n<\/tr>\r\n \r\n Standard form of a line<\/strong><\/td>\r\n [latex]Ax + By = C[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n Glossary<\/h2>\r\n
Essential Concepts<\/h2>\n
Graphs of Linear Functions<\/h3>\n
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Linear Functions<\/h3>\n
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Linear Models<\/h3>\n
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