Morris-Linear Functions - Direct Variation
y-intercept
"b" in y = mx + b The place on the y-axis where the line touches or crosses (the x-value is always 0.) Example: (0,3)
Let's pretend that over the last 50 years, the average temperature has increased by 2.5 degrees worldwide. What is the rate of change in worldwide temperatures per year?
+.05 degrees each year
A teacher weighed 165 lbs in 1996 and weighs 205 lbs in 2006. What was the rate of change in weight?
+4 pounds per year
A scuba diver is 30 feet below the surface of the water 10 seconds after he entered the water and 100 feet below the surface after 40 seconds. What is the scuba divers rate of change?
-2.33 ft/sec
If k=y/x and the rate is 36:3. Then the constant of proportionality is =
12
If k=y/x and the rate is 6:3. Then the constant of proportionality (k) is =
2
A rocket is 1 mile above the earth in 30 seconds and 5 miles above the earth in 2.5 minutes. What is the rockets rate of change in miles per second? What about miles per minute.
2 miles/minute or .033 miles per second
System of Linear Equations
2 or more equations graphed or in a group
A plane left Chicago at 8:00 A.M. At 1: P.M., the plane landed in Los Angeles, which is 1500 miles away. What was the average speed of the plane for the trip?
300 miles per hour OR 5 miles per minute
A climber is on a hike. After 2 hours he is at an altitude of 400 feet. After 6 hours, he is at an altitude of 700 feet. What is the average rate of change?
75 ft
Average Rate of Change (Slope formula) in Function Notation
Change in "y"/Change in "x" f(b) - f(a)/ b - a
Slope/Rate of Change
Change in y/ Change in x
Ratio
Comparison of two quantities
Ax + By = C Standard Form
Form of a linear equation, where A and B are coefficients of the variables x and y. C is a constant number.
slope formula
Name of the formula shown. You must use two solutions or ordered pairs to determine the value.
y=20x-50
No
y=2x-55
No
y=4x+9
No
y=4x-5
No
y=564x+876
No
y=5x-65
No
y=5x-7
No
y=8x+45
No
Function Notation
Notation where f(x) is the output of the function and x is the input; for example if f(x) = x + 2, then f(3) = 3 + 2 = 5.: f(3) = 5: or Point Form: (5,f(3)) When "x" is 3, f(x) is 5
Intercepts Graphing Method
One way to graph a function; find the "x" and "y" intercepts, graph, draw a line through them.
Table of Values Graphing Method
One way to graph a function; first find some x-values in the domain, and then calculate the y-values using the function rule; points can then be plotted and connected.
Parallel Lines
Same slope, different y-intercepts Example: y = 1/3x -4 y = 1/3x +5
"m" in the y=mx+b
Slope
y = mx + b
Slope-Intercept form of a linear equation, where "m" represents slope and "b" represents y-intercept.
Perpendicular Lines
Slopes are opposite reciprocal (a/b ---- > -b/a) Example: y = 2/3x -4 y = 3/2x +5
Constant of Proportionality
The constant ratio between two quantities: y=kx, where k is the constant of proportionality.
x-intercept
The place on the x-axis where the line touches or crosses (the y-value is always 0) Example: (4,0)
slope
The ratio of change between two points on a line. Rise/Run or change in y over change in x.
zero slope
The slope of a horizontal line (the change in y-values is 0).Example: 0/12
negative slope
The slope of a line that decreases as x-values increase (points go down left to right). Example: m = -1/2
positive slope
The slope of a line that increases as x-values increase (points go up left to right). Example: m = 5
undefined slope
The slope of a vertical line (cannot be found because it would force a division by 0, undefined in mathematics)-Example: 12/0
Evaluate
To find the value of a numerical or algebraic expression
Slope-Intercept Graphing Method
Use the form y=mx+b, first graph the y-intercept, then use the slope to find another point, draw a line through the two points.
Point-Slope form of the linear Equation Version 1
Used to find the linear equation when given a point and a slope.
"b" in the y=mx+b
Y-intercept
y=37x
Yes
y=45x
Yes
y=567x
Yes
y=5x
Yes
y=6x
Yes
y=8x
Yes
Unit Rate
a ratio that compares a quantity to one unit of a different quantity
Solution of a system of equations
point of intersection on the graph
direct variation
proportional relationship between dependent and independent variables y = kx
What is the value of x on the y-axis?
x = 0
What is the value of "y" on the x-axis?
y = 0
Point-Slope form of a linear equation Version 2 Helps to understand Function Transformations
y = m(x - h) + k m = slope (h,k) is any point on the line
Slope-Intercept Form of a linear equation
y = mx + b
