Difference between revisions of "Slope"
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===Slope Between Two Points=== | ===Slope Between Two Points=== | ||
| − | Given two points <math> (x_1, y_1) </math> and <math> (x_2, y_2) </math>, the slope between these two points is <math> m = \frac{y_2 - y_1}{x_2 - x_1}</math>. That is, the slope between two points is the difference between the y-coordinates of the points, divided by the difference between the x-coordinates of the points. For example, the slope between the two points (1, 3) and (5, 6) is <math> \frac{6 - 3}{5 - 1} = \frac{3}{4}</math>. The slope between (-1, -1) and (15, -21) is <math> \frac{-21 - (-1)}{15 - (-1)} = \frac{-21 + 1}{15 + 1} = \frac{ | + | Given two points <math> (x_1, y_1) </math> and <math> (x_2, y_2) </math>, the slope between these two points is <math> m = \frac{y_2 - y_1}{x_2 - x_1}</math>. That is, the slope between two points is the difference between the y-coordinates of the points, divided by the difference between the x-coordinates of the points. For example, the slope between the two points (1, 3) and (5, 6) is <math> \frac{6 - 3}{5 - 1} = \frac{3}{4}</math>. The slope between (-1, -1) and (15, -21) is <math> \frac{-21 - (-1)}{15 - (-1)} = \frac{-21 + 1}{15 + 1} = -\frac{20}{16} = -\frac{5}{4}</math>. |
| − | ===Point-Slope Form=== | + | ===Point-Slope Form of a Line=== |
The equation for a line with a slope of <math> m </math> that goes through some point <math> (x_1, y_1) </math>, in point-slope form, is <math> y - y_1 = m(x - x_1) </math>. For example, the equation of a line with a slope of 3 that goes through the point (1, 4) is <math> y - 4 = 3(x - 1) </math>. The equation of a line with a slope of <math> -\frac{1}{2} </math> that goes through point (-7, -7) is <math> y + 7 = -\frac{1}{2}(x + 7) </math>. | The equation for a line with a slope of <math> m </math> that goes through some point <math> (x_1, y_1) </math>, in point-slope form, is <math> y - y_1 = m(x - x_1) </math>. For example, the equation of a line with a slope of 3 that goes through the point (1, 4) is <math> y - 4 = 3(x - 1) </math>. The equation of a line with a slope of <math> -\frac{1}{2} </math> that goes through point (-7, -7) is <math> y + 7 = -\frac{1}{2}(x + 7) </math>. | ||
| − | ===Slope-Intercept Form=== | + | ===Slope-Intercept Form of a Line=== |
Another form of an equation of a line is slope-intercept form. The equation of a line with a y-intercept of b (that is, a line that intersects the y-axis at the point (0, b)) and a slope of m is <math> y = mx + b </math>. For example, the equation of a line with a y-intercept of 5 and slope of 6 is <math> y = 5x + 6 </math>. Note that this equation is equivalent to point-slope form. A line with a y-intercept of 5 goes through the point (0, 5), so the point-slope form of this same line is <math> y - 5 = 6(x - 0) = 6x </math>. By adding 5 to each side of the equation, we get the slope-intercept form of the line. | Another form of an equation of a line is slope-intercept form. The equation of a line with a y-intercept of b (that is, a line that intersects the y-axis at the point (0, b)) and a slope of m is <math> y = mx + b </math>. For example, the equation of a line with a y-intercept of 5 and slope of 6 is <math> y = 5x + 6 </math>. Note that this equation is equivalent to point-slope form. A line with a y-intercept of 5 goes through the point (0, 5), so the point-slope form of this same line is <math> y - 5 = 6(x - 0) = 6x </math>. By adding 5 to each side of the equation, we get the slope-intercept form of the line. | ||
==Resources== | ==Resources== | ||
* [https://tasks.illustrativemathematics.org/content-standards/tasks/1537 Slope Between Points], Illustrative Mathematics | * [https://tasks.illustrativemathematics.org/content-standards/tasks/1537 Slope Between Points], Illustrative Mathematics | ||
| + | * [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:point-slope-form/v/idea-behind-point-slope-form Point-Slope Form], Khan Academy | ||
| + | * [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:intro-to-slope-intercept-form/v/slope-intercept-form Slope-Intercept Form], Khan Academy | ||
* [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:summary-forms-of-two-variable-linear-equations/a/forms-of-linear-equations-review Forms of Linear Equations], Khan Academy | * [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:summary-forms-of-two-variable-linear-equations/a/forms-of-linear-equations-review Forms of Linear Equations], Khan Academy | ||
Latest revision as of 12:50, 20 September 2021
Contents
Slope Between Two Points
Given two points and , the slope between these two points is . That is, the slope between two points is the difference between the y-coordinates of the points, divided by the difference between the x-coordinates of the points. For example, the slope between the two points (1, 3) and (5, 6) is . The slope between (-1, -1) and (15, -21) is .
Point-Slope Form of a Line
The equation for a line with a slope of that goes through some point , in point-slope form, is . For example, the equation of a line with a slope of 3 that goes through the point (1, 4) is . The equation of a line with a slope of that goes through point (-7, -7) is .
Slope-Intercept Form of a Line
Another form of an equation of a line is slope-intercept form. The equation of a line with a y-intercept of b (that is, a line that intersects the y-axis at the point (0, b)) and a slope of m is . For example, the equation of a line with a y-intercept of 5 and slope of 6 is . Note that this equation is equivalent to point-slope form. A line with a y-intercept of 5 goes through the point (0, 5), so the point-slope form of this same line is . By adding 5 to each side of the equation, we get the slope-intercept form of the line.
Resources
- Slope Between Points, Illustrative Mathematics
- Point-Slope Form, Khan Academy
- Slope-Intercept Form, Khan Academy
- Forms of Linear Equations, Khan Academy
