Equations of Lines - 3.4 Fall 2013 - Math 1010 y = mx + b (y y 1 ) = m(x x 1 ) Ax + By = C (Math 1010) M 1010 3.4 1 / 11
Roadmap Discussion/Activity: Graphs and linear equations. Form: The Point-Slope Equation Form: Vertical, Horizontal, Parallel, and Perpendicular Lines Applications Discussion on homework, quizzes, and exams. (Math 1010) M 1010 3.4 2 / 11
Point-Slope Recall the slope formula re-imagined without fractions. Slope: (y 2 y 1 ) = m(x 2 x 1 ) This formula becomes the point-slope equation of a line when a slope, m, is known along with only one point, (x 1, y 1 ). Point-slope form: (y y 1 ) = m(x x 1 ) (Math 1010) M 1010 3.4 3 / 11
Example - Point-Slope Write an equation of the line passing through the point (2, 7) with a slope of m = 4. (Math 1010) M 1010 3.4 4 / 11
Example - Point-Slope Write an equation of the line passing through the point (2, 7) with a slope of m = 4. y ( 7) = 4(x 2) (Math 1010) M 1010 3.4 4 / 11
Example - Point-Slope Write an equation of the line passing through the point (2, 7) with a slope of m = 4. y ( 7) = 4(x 2) y + 7 = 4(x 2) (Math 1010) M 1010 3.4 4 / 11
Example - Point-Slope Slope-intercept forms y = mx + b pass through the point (0, b). Then the point-slope form looks like:. y b = m(x 0) (Math 1010) M 1010 3.4 5 / 11
Example - Point-Slope Slope-intercept forms y = mx + b pass through the point (0, b). Then the point-slope form looks like:. y b = m(x 0) Write the point-slope form of the line through ( 2, 1) and (4, 2), then write its slope-intercept form. (Math 1010) M 1010 3.4 5 / 11
Example - Point-Slope Slope-intercept forms y = mx + b pass through the point (0, b). Then the point-slope form looks like:. y b = m(x 0) Write the point-slope form of the line through ( 2, 1) and (4, 2), then write its slope-intercept form. m = 2 1 4 ( 2) = 1 6 y 1 = 1 (x + 2) 6 y = 1 6 x + 4 3 (Math 1010) M 1010 3.4 5 / 11
Special Forms Horizontal lines have a slope of y-coordinate b, from its Vertical lines have an a, from its (a, 0).. Each point has (0, b). slope. Each point has x-coordinate Euclid formulated geometric axioms, one of which is that there is only one line through a given point that is parallel to another line. Recall that parallel lines have equal slopes. Perpendicular lines have opposite-and-reciprocal slopes. (Math 1010) M 1010 3.4 6 / 11
Special Forms Horizontal lines have a slope of y-coordinate b, from its Vertical lines have an a, from its (a, 0).. Each point has (0, b). slope. Each point has x-coordinate Euclid formulated geometric axioms, one of which is that there is only one line through a given point that is parallel to another line. Recall that parallel lines have equal slopes. Perpendicular lines have opposite-and-reciprocal slopes. Blanks: zero, y-intercept, undefined, x-intercept (Math 1010) M 1010 3.4 6 / 11
Summary of Forms of Equations of Lines Algebraic Form y = mx + b Name of the Form Slope-Intercept (y y 1 ) = m(x x 1 ) Point-Slope Ax + By = C x = a y = b m 1 = m 2 m 1 = 1 m 2 Standard Form Vertical line Horizontal line Parallel lines Perpendicular lines (Math 1010) M 1010 3.4 7 / 11
Application - Depreciation The value of a car decreases in terms of time t. Let s assume this to be linear depeciation. Set-up: The car s initial value is $38,000. After 7 years it will be valued at $7,000. Write an equation for the straight-line depreciation of the value of the car. (Math 1010) M 1010 3.4 8 / 11
Application - Depreciation The value of a car decreases in terms of time t. Let s assume this to be linear depeciation. Set-up: The car s initial value is $38,000. After 7 years it will be valued at $7,000. Write an equation for the straight-line depreciation of the value of the car. Use the equation to find the value of the car 2 years from its initial value. (Math 1010) M 1010 3.4 8 / 11
Application - Depreciation The value of a car decreases in terms of time t. Let s assume this to be linear depeciation. Set-up: The car s initial value is $38,000. After 7 years it will be valued at $7,000. Write an equation for the straight-line depreciation of the value of the car. Use the equation to find the value of the car 2 years from its initial value. Graph the equation. When does the value of the car become $0? (Math 1010) M 1010 3.4 8 / 11
Application - Cost The total cost to produce x items combines the overhead cost and cost to produce one unit. Set-up: To make hats, the total cost is the sum of the overhead of $20 and unit cost of $6 per item. Write an equation for the total cost of producing x hats. (Math 1010) M 1010 3.4 9 / 11
Application - Cost The total cost to produce x items combines the overhead cost and cost to produce one unit. Set-up: To make hats, the total cost is the sum of the overhead of $20 and unit cost of $6 per item. Write an equation for the total cost of producing x hats. Use the equation to find the cost of make 40 products. (Math 1010) M 1010 3.4 9 / 11
Application - Cost The total cost to produce x items combines the overhead cost and cost to produce one unit. Set-up: To make hats, the total cost is the sum of the overhead of $20 and unit cost of $6 per item. Write an equation for the total cost of producing x hats. Use the equation to find the cost of make 40 products. A budget constraint of $300 is introduced. Use either the equation or its graph to estimate how many hats can be produced under this constraint. (Math 1010) M 1010 3.4 9 / 11
Application - Demand Demand relates the price p of a service and the demand d at that price. This relationship may be linear. Set-up: From 2010, raffle tickets priced at $4 sold 2000 tickets. From 2011, raffle tickets priced at $5 sold 1800 tickets. Write a linear equation for the demand of tickets sold priced at p dollars. (Math 1010) M 1010 3.4 10 / 11
Application - Demand Demand relates the price p of a service and the demand d at that price. This relationship may be linear. Set-up: From 2010, raffle tickets priced at $4 sold 2000 tickets. From 2011, raffle tickets priced at $5 sold 1800 tickets. Write a linear equation for the demand of tickets sold priced at p dollars. Use the equation to find the demand of tickets sold at $10 per ticket. (Math 1010) M 1010 3.4 10 / 11
Application - Demand Demand relates the price p of a service and the demand d at that price. This relationship may be linear. Set-up: From 2010, raffle tickets priced at $4 sold 2000 tickets. From 2011, raffle tickets priced at $5 sold 1800 tickets. Write a linear equation for the demand of tickets sold priced at p dollars. Use the equation to find the demand of tickets sold at $10 per ticket. Use the equation to find the demand of tickets sold at $2 per ticket. (Math 1010) M 1010 3.4 10 / 11
Assignment Assignment: For Wednesday: 1. Exercises from 3.4 due Wednesday, September 25. 2. Quiz # 3: Graphs, Linear Equations 3. Read section 3.6. (Skip 3.5) (Math 1010) M 1010 3.4 11 / 11