equations of lines - §3 - university of utahlam/fa131010/3.4.pdf · 2013-09-23 · point-slope...
TRANSCRIPT
Equations of Lines - §3.4
Fall 2013 - Math 1010
y = mx + b(y − y1) = m(x − x1)Ax + By = C
(Math 1010) M 1010 §3.4 1 / 11
Roadmap
I Discussion/Activity: Graphs and linear equations.
I Form: The Point-Slope Equation
I Form: Vertical, Horizontal, Parallel, and Perpendicular Lines
I Applications
I 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:(y2 − y1) = m(x2 − x1)
This formula becomes the point-slope equation of a line when a slope, m,is known along with only one point, (x1, y1).
Point-slope form:(y − y1) = m(x − x1)
(Math 1010) M 1010 §3.4 3 / 11
Example - Point-Slope
Write an equation of the line passing through the point (2,−7) with aslope of m = 4.
y − (−7) = 4(x − 2)
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 aslope of m = 4.
y − (−7) = 4(x − 2)
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 aslope 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 thepoint-slope form looks like:
y − b = m(x − 0)
.
Write the point-slope form of the line through (−2, 1) and (4, 2), thenwrite its slope-intercept form.
m =2− 1
4− (−2)=
1
6
y − 1 =1
6(x + 2)
y =1
6x +
4
3
(Math 1010) M 1010 §3.4 5 / 11
Example - Point-Slope
Slope-intercept forms y = mx + b pass through the point (0, b). Then thepoint-slope form looks like:
y − b = m(x − 0)
.
Write the point-slope form of the line through (−2, 1) and (4, 2), thenwrite its slope-intercept form.
m =2− 1
4− (−2)=
1
6
y − 1 =1
6(x + 2)
y =1
6x +
4
3
(Math 1010) M 1010 §3.4 5 / 11
Example - Point-Slope
Slope-intercept forms y = mx + b pass through the point (0, b). Then thepoint-slope form looks like:
y − b = m(x − 0)
.
Write the point-slope form of the line through (−2, 1) and (4, 2), thenwrite its slope-intercept form.
m =2− 1
4− (−2)=
1
6
y − 1 =1
6(x + 2)
y =1
6x +
4
3
(Math 1010) M 1010 §3.4 5 / 11
Special Forms
Horizontal lines have a slope of . Each point hasy -coordinate b, from its (0, b).
Vertical lines have an slope. Each point has x-coordinatea, from its (a, 0).
Euclid formulated geometric axioms, one of which is that there is only oneline through a given point that is parallel to another line. Recall thatparallel lines have equal slopes. Perpendicular lines haveopposite-and-reciprocal slopes.
Blanks: zero, y -intercept, undefined, x-intercept
(Math 1010) M 1010 §3.4 6 / 11
Special Forms
Horizontal lines have a slope of . Each point hasy -coordinate b, from its (0, b).
Vertical lines have an slope. Each point has x-coordinatea, from its (a, 0).
Euclid formulated geometric axioms, one of which is that there is only oneline through a given point that is parallel to another line. Recall thatparallel lines have equal slopes. Perpendicular lines haveopposite-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 Name of the Form
y = mx + b Slope-Intercept
(y − y1) = m(x − x1) Point-Slope
Ax + By = C Standard Form
x = a Vertical line
y = b Horizontal line
m1 = m2 Parallel lines
m1 = − 1m2
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 belinear 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 - Depreciation
The value of a car decreases in terms of time t. Let’s assume this to belinear 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 - Depreciation
The value of a car decreases in terms of time t. Let’s assume this to belinear 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 toproduce one unit.
Set-up: To make hats, the total cost is the sum of the overhead of $20and 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 itsgraph to estimate how many hats can be produced under this constraint.
(Math 1010) M 1010 §3.4 9 / 11
Application - Cost
The total cost to produce x items combines the overhead cost and cost toproduce one unit.
Set-up: To make hats, the total cost is the sum of the overhead of $20and 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 itsgraph to estimate how many hats can be produced under this constraint.
(Math 1010) M 1010 §3.4 9 / 11
Application - Cost
The total cost to produce x items combines the overhead cost and cost toproduce one unit.
Set-up: To make hats, the total cost is the sum of the overhead of $20and 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 itsgraph 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. From2011, 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
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. From2011, 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
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. From2011, 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