Writing Linear Equations and Graphing

Writing Linear Equations

• To find the equation of a line given two points, calculate the slope ( m ) and y-intercept ( b ). Example: from points (-3, 5) and (0, 2), ( m = -1 ) and ( b = 2 ), resulting in ( y = -1x + 2 ).
• For a linear function given specific inputs: If ( f(3) = 5 ) and ( f(0) = -7 ), slope ( m = 4 ) and y-intercept ( b = -7 ) lead to ( y = 4x - 7 ).

Slope-Intercept Form

• Use point coordinates to derive an equation in slope-intercept form: From (3, 2) and (4, 9), slope ( m = 7 ) leads to the equation ( y = 7x - 19 ).
• Given points (−4, -8) and (4, 4), with slope ( m = 3/2 ), results in ( y = \frac{3}{2}x - 2 ).

Graphing and Points-Slope Form

• For the equation ( y = -\frac{2}{3}x + 7 ), the slope ( m = -\frac{2}{3} ) and y-intercept is 7, which determines the graph.
• From points (5, -2) and (9, 6), in point-slope form with ( m = 2 ): ( y + 2 = 2(x - 5) ).

Equivalent Equations

• An equation can be rewritten in multiple forms. ( 8x + 4y = 6 ) can be converted to ( 16x + 8y = 12 ) or ( 4x + 2y = 3 ).

Horizontal and Vertical Lines

• The equation for vertical lines through (3, -7) is ( x = 3 ), while the horizontal line is ( y = -7 ).

Standard Form and Finding B

• Standard form can be generated from two points: using (1,5) and (5,-3), derive ( 2x + y = 7 ).
• To ensure a line passes through a specific point, such as (-3, 7) for the equation ( 2x + By = -4 ), solve for ( b = \frac{2}{7} ).

Parallel and Perpendicular Lines

• Lines can be constructed based on characteristics of other lines: For parallel line through (2, -3) to ( y = \frac{2}{3}x - 5 ), use slope ( m = \frac{2}{3} ).
• For perpendicular lines, such as one from (2, -3) to ( y = -\frac{4}{5}x + 8 ), use the negative reciprocal slope ( m = \frac{5}{4} ).

Correlation and Line of Best Fit

• The correlation of a line can vary: Positive when moving left to right, negative when right to left, or no correlation when points are scattered.
• A line of best fit involves calculating slope and intercept from data points. For instance, from points (10, 100) and (18, 400), ( m = 37.5 ) and ( b = -275 ).

Finding Zeros

• Zeros in functions, such as for ( f(x) = 5x - 7 ), can be found where the function equals zero; here, ( x = \frac{7}{5} ).

Linear Regression Output

• Linear regression can determine the best-fit line for given data sets, producing coefficients ( A ) and ( B ). In one case, given arrays produce ( A = 1.47 ) and ( B = 3.79 ).