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3.4 Finding an Equation of a Line

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Point-Slope Form y – y 1 = m(x – x 1 ) (x 1, y 1 ) represent a point on the line m represents the slope To graph a line, all you need is the slope and a point with this formula. Example: Find an equation for the line with slope -1/2 that goes through the point (2, 1) Since we have a point and we have slope, we can use the point-slope formula of a line! Substitute the slope in for m Substitute the point into the (x 1, y 1 ) values Solve for y to get the equation into slope-intercept form (y = mx + b)

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Point-Slope Example 2 Write the equation of the line with slope 1/3 and passes through (4, 2) Since we have a point and we have slope, we can use the point-slope formula of a line! Substitute the slope in for m Substitute the point into the (x 1, y 1 ) values Solve for y to get the equation into slope-intercept form (y = mx + b)

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Finding an Equation through two points Since we have two points, we can determine slope with the slope formula. Now, we have a point and slope and can use point slope formula! Example. Find an equation of the line p through (4, 7) and (6, -2) Solve for slope. Since we have a point and we have slope, we can use the point- slope formula of a line! Substitute the slope in for m Substitute the point into the (x 1, y 1 ) values Solve for y to get the equation into slope-intercept form (y = mx + b)

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Finding an Equation of a Line Determine the key pieces you have x-intercept? means we have a point (x, 0) y-intercept? means we have a point (0, y) slope? means we have m one point? means we have a point (x 1, y 1 ) two points? means we can solve for slope and we have two points is the line vertical? means the equation is x = is the line horizontal? means the equation is y = is it parallel to another line? means the slopes are equal is it perpendicular to another line? means the slope is the negative reciprocal of the first line To use this formula, you need… slope-intercept form (y = mx + b)…slope, y intercept standard form (Ax + By = C)… slope, y-intercept (convert from slope-intercept form) point-slope form (y – y 1 ) = m(x –x 1 )…one point and slope slope formula …two points

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Finding an Equation - Example 3 Find an equation for the line containing the points (0,6) and (4, 3) What formula would be good to use in this example? What pieces do we have?

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Finding an equation - Example 4 Find an equation of the line parallel to y = -1/2x + 6 that goes through the point (-12,3) What formula would be good to use in this example? What pieces do we have?

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Writing an equation from a word problem Figure out what is given. Is it a point (i.e. this # goes with this #)? Is it slope (how two pieces of information change)? Based on the pieces, create the equation using the best method. Example. In a physics experiment, a spring is 11 cm long with a 13-gram weight attached. Its length increases 0.5 cm with each additional gram of weight. Write an equation relating spring length L and weight W. What are we given? Which formula(s) should we use to create the equation?

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Writing an equation from a word problem Example 2. A spring is 23 inches long and has a 3-pound weight attached. Its length increases 3 inches with each additional pound of weight added. Write a formula relating the spring length L and the weight w. What are we given? Which formula(s) should we use to create the equation?

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Piecewise Linear Functions Graphs made up of several line segments are considered piecewise linear. Writing a function for a piecewise linear graph requires writing the equation of each line segment. Since the domain of each segment is specific for that segment, it must be mentioned with its equation. We can combine the equations of several segments into one formula using a brace.

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Piecewise Linear Functions - Example 1 Write a piecewise linear function for the first two segments of Frank’s bicycle trip as described in the graph. Figure out what is given. Create an equation using the best method. Determine the domain that fits the equation. Combine the segments into one formula using a brace.

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Piecewise Linear Functions – Example 2 Write a piecewise linear function for the third and fourth segments of Frank’s bicycle trip as described in the graph. Figure out what is given. Create an equation using the best method. Determine the domain that fits the equation. Combine the segments into one formula using a brace.

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You try! Write an equation for the line 1. through (-2, 4) and perpendicular to 2x + 3y = 6 2. that is horizontal, through the point (4.5, 8.2) 3. containing (-11, 4) and (7, 1) 1. A club will be charged $247.50 for printing 150 t-shirts and $501.25 for printing 325 t-shirts. Let c be the cost of printing s shirts. a. Write an equation in slope-intercept form representing the cost of printing s shirts. b. How much will it cost to print 0 t-shirts (this is the set-up cost).

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You try! Answers Write an equation for the line 1. y = 3/2x + 7 2. y = 8.2 3. y = -1/6x + 13/6 1. A club will be charged $247.50 for printing 150 t-shirts and $501.25 for printing 325 t-shirts. Let c be the cost of printing s shirts. a. c = 1.45s + 30 b. $30

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