Slope
50 flashcards covering Slope for the SAT Math section.
Slope measures the steepness of a straight line on a graph, indicating how much the line rises or falls as you move from left to right. It's calculated as the change in the y-coordinates divided by the change in the x-coordinates between any two points on the line—essentially, rise over run. For instance, if you have points (1, 2) and (3, 6), the slope shows that for every 2 units you move right, the line goes up 4 units, helping you understand relationships in linear equations and real-world scenarios like speed or costs.
On the SAT Math section, slope appears in multiple-choice questions involving linear functions, graphs, or word problems, where you might calculate it from points, equations, or tables. Common traps include confusing slope with the y-intercept, mishandling negative values, or overlooking undefined slopes for vertical lines. Focus on practicing the formula, recognizing parallel and perpendicular lines, and interpreting slope in context to boost accuracy on these problems.
Remember to always plot points if needed to visualize the line.
Terms (50)
- 01
Slope
Slope measures the steepness and direction of a straight line on a graph, calculated as the change in y-coordinates divided by the change in x-coordinates between any two points on the line.
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Slope formula
The formula for slope between two points (x1, y1) and (x2, y2) is m = (y2 - y1) / (x2 - x1), where m represents the slope, provided the line is not vertical.
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Rise over run
Rise over run is another way to describe slope, where rise is the vertical change (difference in y-values) and run is the horizontal change (difference in x-values) between two points.
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Positive slope
A positive slope indicates that the line rises from left to right on a graph, meaning as x increases, y also increases.
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Negative slope
A negative slope indicates that the line falls from left to right on a graph, meaning as x increases, y decreases.
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Zero slope
Zero slope means the line is horizontal, with no change in y-values as x changes, so the slope equals 0.
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Undefined slope
Undefined slope occurs when a line is vertical, with no change in x-values, making the denominator in the slope formula zero and thus undefined.
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Slope from two points
To find slope from two points, subtract the y-coordinate of the first point from the second, subtract the x-coordinate of the first from the second, and divide the results.
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Slope from a graph
Slope from a graph is determined by picking two points on the line, calculating the rise (vertical distance) and run (horizontal distance) between them, and dividing rise by run.
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Slope-intercept form
Slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept, allowing quick identification of the slope from the equation.
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Point-slope form
Point-slope form of a line is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, useful for writing equations when slope and a point are known.
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Standard form of a line
Standard form of a line is Ax + By = C, and to find the slope, rearrange it to y = mx + b form, where m = -A/B, provided B is not zero.
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Parallel lines
Parallel lines have the same slope and never intersect, so if one line has a slope of 2, any line parallel to it also has a slope of 2.
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Perpendicular lines
Perpendicular lines have slopes that are negative reciprocals of each other, meaning if one slope is 3, the perpendicular slope is -1/3.
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Rate of change
Rate of change is the slope of a line in a real-world context, representing how one quantity changes relative to another, such as speed as distance over time.
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Average rate of change
Average rate of change between two points on a function is the slope of the line connecting them, calculated as the change in output values divided by the change in input values.
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Slope from a table
Slope from a table is found by selecting two rows, treating the x-values as inputs and y-values as outputs, and computing the difference in y divided by the difference in x.
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Horizontal line slope
The slope of a horizontal line is always zero because there is no vertical change, regardless of the horizontal distance.
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Vertical line slope
The slope of a vertical line is undefined because there is no horizontal change, making division by zero in the slope formula impossible.
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Calculating slope with fractions
When calculating slope with fractional coordinates, subtract the fractions carefully, simplify the resulting fraction, and ensure the final slope is in simplest terms.
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Negative reciprocal
The negative reciprocal of a number is what you multiply it by to get -1, used for perpendicular lines; for example, the negative reciprocal of 4/5 is -5/4.
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Graphing using slope
To graph a line using slope, start at a point on the line, then move according to the slope's rise and run values to plot additional points and draw the line.
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Finding equation from slope and point
To find the equation from slope and a point, use point-slope form y - y1 = m(x - x1), then rearrange to slope-intercept or standard form if needed.
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Finding equation from two points
To find the equation from two points, first calculate the slope, then use one point and the slope in point-slope form to write the equation.
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Interpreting slope in word problems
In word problems, slope represents the constant rate of change, such as cost per item or speed in miles per hour, helping to predict outcomes based on the relationship.
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Common mistake: slope of vertical line
A common mistake is assigning a number to the slope of a vertical line; remember, it is undefined, not infinite or zero.
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Trap: confusing slope with y-intercept
Students often confuse slope (the rate of change) with y-intercept (the starting value), so identify which part of y = mx + b is which.
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Slope in linear functions
In linear functions, slope indicates the function's steepness and direction, affecting how the output changes with the input in a straight-line relationship.
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Slope and proportionality
Slope represents the constant of proportionality in direct variation equations like y = kx, where k is the slope and shows how y changes with x.
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Unit rate
Unit rate is the slope in a proportional relationship, such as price per item, calculated as the change in the dependent variable per one unit of the independent variable.
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Slope in distance-time graphs
In distance-time graphs, slope represents speed, as it shows the change in distance over change in time for the object moving at a constant rate.
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Speed as slope
Speed is the slope of a distance-time graph, where a steeper positive slope means faster speed and a zero slope means the object is stationary.
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Work problems with slope
In work problems, slope can represent rates like work done per hour, allowing you to model and solve for total work using linear equations.
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Slope of a line segment
The slope of a line segment is the same as the slope of the line it lies on, calculated between its endpoints, even if the segment is part of a larger line.
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Midpoint and slope together
When working with midpoints, you can also calculate slope between endpoints; for points (x1, y1) and (x2, y2), midpoint is ((x1+x2)/2, (y1+y2)/2) and slope is (y2-y1)/(x2-x1).
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Applications in geometry
In geometry, slope helps determine if lines are parallel or perpendicular in coordinate planes, such as in triangles or polygons defined by points.
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Strategy: simplify fractions in slope
Always simplify the fraction obtained for slope by dividing numerator and denominator by their greatest common divisor to avoid errors in comparisons or equations.
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Trap: assuming slope is always positive
A trap is assuming slope must be positive; lines can have negative, zero, or undefined slopes, so check the direction and type carefully.
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Miscalculating with negative coordinates
When points have negative coordinates, subtract carefully, remembering that subtracting a negative is adding a positive, to get the correct slope.
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Deriving slope from equation
To derive slope from an equation, rewrite it in slope-intercept form y = mx + b; the coefficient of x is the slope.
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Converting equations to find slope
Convert equations from standard form to slope-intercept form by solving for y, which reveals the slope as the coefficient of x.
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Slope in systems of equations
In systems of equations, comparing slopes helps determine if lines intersect, are parallel, or are the same; equal slopes mean parallel or identical.
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Slope and angles
Slope relates to the angle a line makes with the positive x-axis, where a slope of 1 corresponds to a 45-degree angle, though exact angles aren't required on the SAT.
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Example: slope of line through (1,2) and (3,4)
For points (1,2) and (3,4), the slope is (4-2)/(3-1) = 2/2 = 1, indicating a line that rises one unit for every one unit it runs right.
This line passes through (1,2) and goes up to (3,4).
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Example: slope of line through (2,3) and (2,5)
For points (2,3) and (2,5), the slope is undefined because the x-coordinates are the same, indicating a vertical line.
Both points share x=2, so the line is straight up.
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Example: slope of line through (-1,4) and (3,-2)
For points (-1,4) and (3,-2), the slope is (-2-4)/(3-(-1)) = (-6)/(4) = -3/2, showing a line that falls as it goes right.
From (-1,4) to (3,-2), it drops 6 units over 4 units.
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Strategy for finding slope in word problems
In word problems, identify the two variables' changes, such as input and output, and compute slope as the ratio of their differences to find rates.
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Trap: order of points in slope formula
A trap is reversing the order of points; the formula works regardless, but consistent subtraction prevents sign errors in the result.
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Slope in piecewise linear functions
In piecewise linear functions, slope varies by section, so calculate it for each linear part separately to understand the function's behavior.
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Comparing slopes
Comparing slopes determines which line is steeper; a larger absolute value means steeper, with positive slopes rising and negative ones falling.