Lines and Slope Review for the SAT
A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions. It is a completely straight marker with no curvature. It is made up of (and connects) a series of points together.
A slope (also called the gradient) is the measure of the slant (steepness) of a line. It is the change in y for a unit change in x along the line. It is usually denoted by the letter .
The intercept of a line is the point at which it crosses either the x or y-axis. If it is not specified which one, the y-axis is assumed. It is usually denoted by the letter .
Slope Formula and Properties
Slope is calculated by finding the ratio of the “vertical change” to the “horizontal change” between (any) two distinct points on a line. To find the slope of a line, pick any two points on a line, say A(x1, y1) and B(x2, y2). Slope of the line will be equal to
Direction of a line
The direction of a line is either increasing, decreasing, horizontal or vertical.
- A line is increasing if it goes up from left to right. The slope is positive, i.e. m > 0
- A line is decreasing if it goes down from left to right. The slope is negative, i.e. m < 0
- If a line is horizontal the slope is zero, e. m = 0
- If a line is vertical the slope is undefined since all the points on a vertical line have the same x coordinate. Therefore . Since is the denominator in the formula of slope, the slope for a vertical line is undefined.
Another way to think about this is that the angle a vertical line makes with x-axis is 90 and slope = tan 90 = undefined
Equation of a Line
The equation of a line gives the mathematical relationship between the x and y coordinates of any point on the line. For a straight line, the relationship between x and y is linear. Therefore, any equation of a straight line is of the form
A linear equation in the form
where m and b are real numbers, represents a straight line with slope m and y-intercept equal to b.
An easy phonetic way to remember this is that m represents “move” and b represents “beginning”.
e.g. A line with slope = 2 and y-intercept = -3 will have the equation
To help remember this formula, think of solving it for m where (x,y) ia any arbitrary point on the line and point (x1, y1) is another point on the line:
This formula is also easy to remember if you notice that it is just the same as the point-slope form with the slope m replaced by the definition of slope.
e.g. A line passing through two points A(2,1) and B(3,2) will have the equation
Therefore, is the equation of the line passing through A and B.
Here, we know that m = tan where is the angle that the line makes with x-axis. We know that tan 45 = 1, hence = 45.
Therefore, the line passing through A and B makes an angle of 45 with the x-axis.
- Parallel lines have equal slopes i.e. if a line has a slope = 2, then any line parallel to it will also have slope = 2.
- Perpendicular lines have negative reciprocal slopes i.e. product of slopes of perpendicular lines = -1.
So if your original slope was -3, the slope of the perpendicular line would be positive 1/3.