Linear functions
53 flashcards covering Linear functions for the SAT Math section.
Linear functions represent straight-line relationships between two variables, such as time and distance or cost and quantity. At its core, a linear function follows the form y = mx + b, where m is the slope (indicating the rate of change) and b is the y-intercept (the starting point on the graph). This concept is fundamental for modeling real-world scenarios with constant rates, like how a car's speed affects its position over time. Understanding linear functions helps build a strong foundation in algebra, as they simplify complex problems into predictable patterns.
On the SAT Math section, linear functions appear in various question types, including graphing lines, solving equations, and interpreting word problems that involve rates or proportions. Common traps include confusing linear functions with nonlinear ones, miscalculating slopes from points, or overlooking key details in graphs. Focus on mastering equation manipulation, identifying intercepts, and translating real-world contexts into linear equations, as these skills are tested frequently. Practice these to avoid errors and improve speed on the exam. A quick tip: Always check if a graph is truly linear by verifying the slope consistency.
Terms (53)
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Linear function
A linear function is a function whose graph is a straight line, typically expressed as f(x) = mx + b, where m is the slope and b is the y-intercept.
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Slope
Slope measures the steepness and direction of a line, calculated as the change in y-coordinates divided by the change in x-coordinates between two points on the line.
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Y-intercept
The y-intercept is the point where a line crosses the y-axis, represented by the value of b in the equation y = mx + b.
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X-intercept
The x-intercept is the point where a line crosses the x-axis, found by setting y to zero in the equation and solving for x.
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Slope-intercept form
Slope-intercept form is the equation of a line written as y = mx + b, where m is the slope and b is the y-intercept, making it easy to graph.
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Point-slope form
Point-slope form is the equation of a line written as y - y1 = m(x - x1), using a known point (x1, y1) and the slope m to define the line.
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Standard form
Standard form of a line is written as Ax + By = C, where A, B, and C are constants, and it can be used to easily find intercepts.
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Equation of a line through two points
To find the equation of a line through two points, first calculate the slope using the two points, then use point-slope form and convert if needed.
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Parallel lines
Parallel lines have the same slope but never intersect, meaning if one line has slope m, another parallel line will also have slope m.
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Perpendicular lines
Perpendicular lines have slopes that are negative reciprocals of each other, so if one slope is m, the other is -1/m.
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Horizontal line
A horizontal line has a slope of zero and is written as y = k, where k is a constant, representing a flat line parallel to the x-axis.
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Vertical line
A vertical line has an undefined slope and is written as x = h, where h is a constant, representing a line parallel to the y-axis.
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Graphing a linear equation
Graphing a linear equation involves plotting the y-intercept and using the slope to find another point, then drawing a straight line through those points.
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Rate of change
Rate of change in a linear function is the slope, indicating how much the output changes for each unit increase in the input.
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Systems of linear equations
Systems of linear equations involve two or more equations representing lines, and solving them means finding the point or points where the lines intersect.
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Substitution method
The substitution method solves a system of equations by solving one equation for one variable and substituting that expression into the other equation.
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Elimination method
The elimination method solves a system by adding or subtracting the equations to eliminate one variable, allowing you to solve for the remaining variable.
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Graphing method for systems
The graphing method for systems involves plotting both equations on the same graph and identifying the intersection point as the solution.
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Consistent system
A consistent system of equations has at least one solution, meaning the lines intersect at one point or are the same line.
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Inconsistent system
An inconsistent system of equations has no solution, occurring when the lines are parallel and distinct.
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Dependent system
A dependent system has infinitely many solutions, happening when the equations represent the same line.
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Linear inequality
A linear inequality is an inequality involving a linear expression, such as y > mx + b, and its graph is a region on one side of the line.
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Graphing linear inequalities
Graphing linear inequalities involves graphing the corresponding line and shading the region that satisfies the inequality, using a dashed or solid line based on the symbol.
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Positive slope
A positive slope indicates that the line rises from left to right, meaning as x increases, y increases.
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Negative slope
A negative slope indicates that the line falls from left to right, meaning as x increases, y decreases.
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Zero slope
Zero slope means the line is horizontal, with no change in y for any change in x, as in y = constant.
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Undefined slope
Undefined slope occurs in vertical lines, where x is constant and there is no change in x, making the slope calculation impossible.
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Increasing linear function
An increasing linear function has a positive slope, so its graph rises and the function value grows as x increases.
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Decreasing linear function
A decreasing linear function has a negative slope, so its graph falls and the function value shrinks as x increases.
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Intercept form
Intercept form of a line is written as x/a + y/b = 1, where a is the x-intercept and b is the y-intercept, useful for quick graphing.
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Midpoint formula
The midpoint formula finds the midpoint between two points (x1, y1) and (x2, y2) as ((x1 + x2)/2, (y1 + y2)/2), often used with lines.
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Distance formula
The distance formula calculates the distance between two points (x1, y1) and (x2, y2) as sqrt((x2 - x1)^2 + (y2 - y1)^2), related to line segments.
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Break-even point
The break-even point is the value where cost equals revenue in a linear model, found by solving the system of equations for cost and revenue.
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Linear cost function
A linear cost function models total cost as a straight-line relationship, typically C(x) = mx + b, where m is the variable cost per unit and b is the fixed cost.
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Linear revenue function
A linear revenue function models total revenue as R(x) = px, where p is the price per unit and x is the number of units, assuming constant price.
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Profit function
The profit function is profit equals revenue minus cost, often P(x) = R(x) - C(x), and for linear functions, it's also linear.
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Mixture problems
Mixture problems involve combining solutions or items at different concentrations, often modeled with linear equations to find the required amounts.
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Work-rate problems
Work-rate problems model tasks completed at constant rates, using linear equations to determine time or rates, such as one job taking a certain number of hours.
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Distance-speed-time problems
Distance-speed-time problems use the linear relationship distance = speed × time to set up equations, often for objects moving at constant speeds.
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Converting between line forms
Converting between line forms involves algebraic manipulation, such as rearranging y = mx + b into Ax + By = C or vice versa.
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Finding intersection of two lines
Finding the intersection means solving the system of equations for the two lines to get the point (x, y) where they cross.
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Common mistake: Confusing slope and intercept
A common mistake is mixing up slope as the y-intercept or vice versa, so always identify m as the rate of change and b as the starting point.
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Identifying linear functions
To identify a linear function, check if the equation is of the form f(x) = mx + b or if the graph is a straight line without curves.
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Using linear functions for predictions
Linear functions can predict future values by extending the line, such as using the equation to find y for a given x beyond the data points.
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Slope from a table
To find slope from a table, calculate the change in y values divided by the change in x values between any two points in the table.
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Slope from a graph
To find slope from a graph, select two points on the line, count the rise over run, or use the formula with the coordinates.
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Linear models in context
Linear models apply to real-world situations with constant rates, like population growth at a steady rate or depreciation of an asset.
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Absolute value equations
Absolute value equations can represent V-shaped graphs, but on the SAT, they are often treated as piecewise linear functions for solving.
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Piecewise linear functions
Piecewise linear functions consist of linear segments, and on the SAT, you may need to evaluate or graph them based on given conditions.
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Domain of a linear function
The domain of a linear function is all real numbers unless restricted, meaning x can be any value without limitations.
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Range of a linear function
The range of a linear function is all real numbers, as the line extends infinitely in both directions.
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Linear function transformations
Transformations like shifting or stretching a linear function change its graph, such as adding a constant to shift it vertically.
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Interpreting slope in context
Interpreting slope means understanding it as a rate, like dollars per hour in a wage problem, to make sense of the function's meaning.