Modelling with Graphs and Analyzing Linear Relations
Connecting Variation, Slope, and first differences
A mathematical relation can be described in four different ways:
- Using words - Using a diagram or graph - Using numbers - Using an equation
Equation: E = 9/100F + 25, where E is Abby's earnings in dollars, and F is he numbers of flyers delivered. The slope is Abby's rate of pay in relation to the number of flyers delivered.
Numbers:
Words: Abby earns $25 per day plus $9 per 100 flyers foe delivering newspapers.
Algebraically, The equation of a line has the form y = mx + b, where m represents the slope and b represents the vertical intercept, or the value of the dependent variable where the line intersects the vertical or y-axis.
Slope can be symbolised as m= delta-y/delta-x, where delta represents change in.
The slope, m, of a line can be calculated by dividing the change in y by the change in x.
The slope of a linear relation remains constant. The first differences also remain constant when the changes in the x-values are constant.
Since the slope of a linear relation is constant, you can use any pair of points and the slope will be the same.
First Differences
If the differences of a relation are not constant, the relation is non-linear
If the differences of a relation are constant, the relation is linear
To find first differences, subtract consecutive values of y (dependent variale)
Differences between consecutive y-values with evenly spaced x-values
To work with first differences, the values of x (independent variable) must be change by a constant amount.
Slope as a Rate of Change
To find the slope of a line segment joining 2 points, subtract the y-values to get the rise in the same order to get the run.
m = rise/run
= 5 - 65 /500 - 0
=-60/500
=-0.12
The rate of change is -0.12/km. The car uses 0.12 litre of gasoline per km driven. The rate of change is negative since the gasoline is always decreasing.
When a relation is graphed, the slope describes the rate of change.
Rate of change requires units, such as kilometres per hour.
0.25 km/min
Rate of change is the changes in one quantity relative to the change in another.
Alexa runs every morning before school. One day she ran 5 km in 20 min. To calculate the rate of change you need to divide the change in distance by change in time. So it would be 5/20 = 0.25. Therefore the rate of change is 0.25 km/min.
Slope
For safety, the slope of a staircase must be greater than 0.58 and less than 0.70. A staircase has a vertical rise of 2.5m over a horizontal run of 3.7m. Is the staircase safe?
m=rise/run
=2.5/3.7
= 0.68
Vertical Slope
rise=-4 run=0
The slope is unidentified
Horizontal Slope
rise=0 run=4
m=0/4. Slope is zero.
Use the slope to find a point
If a line segment has one endpoint, A(4,7), and slope of -5/3. To find the coordinates of another possible endpoint, B you need to 3 right and 5 down. so the answer could be B(7,2).
Negative graph
Rise=-2 Run=3 Slope=-2/3
positive graph
Rise=3 Run= 2 slope= 3/2
Rise shows if you are going up or down and the run tells you if the direction is left or right. A rise of 3 means go up 3, and run of 4 means go right 4.
The steepness of a line is measured by its slope. The slope is the ratio of the rise to the run and is often represented by
the letter m. We use the formula slope = rise/run to calculate slope. The rise is the vertical distance between two points, while the run is the horizontal distance between two points.
Partial Varition
Kristina works as a sales representative at the local toys store. She earns a weekly salary of $100 plus 15%
commission on her sales.
Equation: let E represent earnings and let s represent number of sales. E=0.15s+100
Initial value: 100. The constant of variation: 0.15
Table of values
The graph's line is straight line that doesn't pass through the origin.
A partial variation has an equation of the form y=mx+b, where b represents the fixed, or initial value of y and m represents the constant of variation.
Partial variation is a relationship between 2 variables in which the dependent variable is the sum of a constant number and a constant multiple of the independent variable
Direct Variation
Example: The cost of oranges varies directly with total mass bought. 2 kg of oranges costs $4.50. To write the relationship in words you would say,"to get the cost of C, of oranges, multiply the mass r, in kg, of oranges, by $2.25. In this case the constant of variation represents the constant average cost, $2.25/ kg. C = 2.25 r.
The graph's line is a straight line that passes through the origin
if d varies directly as t, then the constant of variation, k, is given by k = d/t or d = kt
Constant of variation: in a direct the ratio of corresponding values of the variables, often represented by k, or the constant multiple by which one variable is multiplied. In the phrase, "The total cost varies directly with the number of books bought. 5 books cost $35" is a direct variation. So the constant of variation would be 7 because 35 divided by 5 is 7.
A relationship between 2 variables in which 1 variable is a constant multiple of the other.
Find the Equation of a Line given 2 Points
You can find an equation for a line if you know two points on the line.
A line that passes through the points (1,2) and (5,10). Find the equation for the line
Step 1. find the slope
m = 10 -2/5 -1
= 8/4 or 2
Step 2. Find the y-intercept
2 = 2(1) + b
2 = 2 + b
0 = b
Step 3. write the equation
y = (2)x + (0)
= 2x + 0
= 2x
Write the equation by substituting m and b into y=mx+b
- Find the y-intercept by substituting the slope and one of the two points into y=mx+b , and then solve for b (the choice of which point you choose does not matter).
- Find the slope by substituting the two points into the slope formula
Find an Equation for a Line Given the Slope and a point
Parallel and Perpendicular lines
Perpendicular Lines:
Lines that intersect at right angles – i.e. 90°. A small box is often used to indicate that the angle between the two lines is 90°. The slopes are negative reciprocals. Negative reciprocals are 2 numbers whose product is -1
the equation of a line is y = 3/5x + 2. Give the slope pf the perpendicular line. The line y = 3/5x + 2 has the slope ⅗.
The perpendicular line will have slope -5/3.
(Turn the fraction upside down and use opposite operation)
Parallel Lines:
Lines that run in the same direction and never cross. Matching arrow symbols are used to indicate that the lines are parallel. The slopes are the same
The equation of the line is y = 3x -4. Give the slope of the parallel line. The line y = 3x -4 has the slope 3.
A parallel line will have the same slope, 3.
Graph a Line Using Intercepts
Find slope using the intercepts
If the x-intercept is -4 and the y-intercept is -6, you would substitute these points in the formula. m = y1-y2/x1-x2
= -6 -6/0- (-4)
= -6/4
= -3/2
The slope of the line -3/2
Note: Graphing using the intercept method may not be the best method to choose when the values that we are working with in an equation are not divisible by the coefficient by which we end up working with equations
y-intercept
3x - 2y - 6 = 0
Let y = 0
3x - 2(o) - 6 = 0
3x - 6 +6= 0 +6
3x/3 = 6/3
x = 2
x-intercept
3x - 2y - 6 = 0
Let x = 0
3(0) - 2y - 6 = 0
-2y - 6 +6 = 0 +6
-2y/2 = 6/-2
y = -3
Graph
The x-intercept is the x-coordinate of the point where a line crosses the x-axis. At this point, y=0.
The y-intercept is the y-coordinate of the point where a line crosses the y-axis.
At this point, x=0.
The Equation of a Line in a Slope y-intercept Form: y = mx+ b
To graph a line when given m and b,
y = mx + b
= 3/4x + (-2)
Start off by plotting -2 first. Then move 4 to the right and up 3 to find the second point. Repeat this process until a longer line can be made.
The y-intercept is -2. The slope is 3/4
A vertical line is written in the form
x = a, where a is the x-intercept. The slope of a
vertical line is undefined.
m = 1 - 0/ 0-0
= 1/0
Since division by zero is unidentified and there is no y-intercept, the equation for this line be x = 2
A horizontal line is written in the form
y = b, where b is the y-intercept.
m = 4-4/ 2-0
= 0/2
=0
The slope is 0
The y-intercept is 4
y= 0x +4
y= 4
The equation of a line can be written in slope y-intercept form: y = mx + b , where m is the slope of the line and b is the y-intercept of the line.
Example:
Equation =y = mx + b
y = 2/3x + (-5)
The equation of the line is y = 2/3x + (-5)
The y-intercept is -5
m = y2 - y1/x2 - x1
= -1(--5)/ 6-0
= -1+5/ 6
= 4/6
= 2/3
The slope is 2/3
The Equation of a Line in Standard form: Ax+ By+ C = 0
Identify fixed and variable costs in a partial variation
25n - C + 1250 = 0
-C = 25n - 1250
C/-1 = -25n/-1 = C
C = 25n + 1250
You can convert an equation from standard form to slope y-intercept form and vice versa.
y = y = -3x + 5
3x + y - 5 = 0
Get all the values on the left side by doing opposite operation.
x + 2y - 4 = 0
2y = -x + 4
2y/2 = -x +4/2
y = -1x/2 + 4/2
y = - ½x + 2
1. Subtract x from both sides and add 4 to both sides.
2. Divide both sides by 2
3. Divide each term on the right side by 2
The equation of a line can be expressed in different ways. The second way that we can express the equation of a line is standard form.
It is expressed as Ax+ By+ C = 0 where A, B, and C are not fractions and A is positive. The equations 3x + 2 -5=0 and 2x -5y +7=0 are two examples of linear relations which are in standard form.
