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>. | ||
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==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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, y_1) } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_2, y_2) } , the slope between these two points is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}} . 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{6 - 3}{5 - 1} = \frac{3}{4}} . The slope between (-1, -1) and (15, -21) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-21 - (-1)}{15 - (-1)} = \frac{-21 + 1}{15 + 1} = -\frac{20}{16} = -\frac{5}{4}} .
Point-Slope Form of a Line
The equation for a line with a slope of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m } that goes through some point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, y_1) } , in point-slope form, is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - y_1 = m(x - x_1) } . For example, the equation of a line with a slope of 3 that goes through the point (1, 4) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - 4 = 3(x - 1) } . The equation of a line with a slope of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{2} } that goes through point (-7, -7) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y + 7 = -\frac{1}{2}(x + 7) } .
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = mx + b } . For example, the equation of a line with a y-intercept of 5 and slope of 6 is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 5x + 6 } . 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - 5 = 6(x - 0) = 6x } . 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
