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Chapter 5: Linear Functions (L28 - 34) - Coggle Diagram
Chapter 5: Linear Functions (L28 - 34)
Rate of Change and Slope
Types of Slope
Vertical line: slope undefined, x = a
Horizontal line: 0 slope, y = a
Positive slope: left to right going up
Negative slope: left to right going down
Definitions
Slope: vertical change over horizontal change
Rate of change: shows the relationship between two changing qualities (x & y)
Rise over run
Change in the dependent variable over the change in the independent variable
y2 - y1 over x2 - x1
Scatter Plots and Trend Lines
Definitions
Scatter plot: a graph that relates two different sets of data by displaying them as ordered pairs
Usually in the first quadrant of the coordinate plane
Correlation: a mutual relationship or connection between two variables
Negative correlation
Positive Correlation
No correlation
Strong
Weak
Trend line: a line on a scatter plot that shows correlation
Drawn near the points
Interpolation: estimating a value between two know values
Extrapolation: predicting a value outside the range of know values
Standard Form
Definitions
x-intercept: x-coordinate, where the line crosses the x - axis
Standard form: Ax + By = C
Allows you to find the intercepts quickly
Draw graphs
A,B,C are real numbers
Both A and B cannot be 0
A,B,C do not denote as slope or intercept points
Parallel and Perpendicular Lines
Definitions
Parallel: two lines that have the same distance apart from each other
The two lines never intersect
m1 = m2 (same slope)
Different y/x-intercepts
Perpendicular: two lines intersect at a 90 degree angle
(m1)(m2) = -1
Opposite reciprocals
Example: What is the equation of a line that passes through (-3,-1) and is parallel to y = 2x + 3?
Step 1: substitute m by 2
y = 2x + b
Step 2: substitute x and y by -3 and -1 to find b (y - intercept) -1 = 2(-3) +b
-1 = -6 +b
b = 5
Step 3: put the equation together in slope-intercept form
y = 2x + 5
Slope-Intercept Form
Definitions
Parent function: the simplest function with common characteristics
Linear parent function: y = x
Linear equation: an equation that models a linear function
y - intercept: the y - coordinate of a point where the graph crosses the y - axis
slope-intercept form: y = mx + b
linear equation of a non vertical line
m = slope
b = y - intercept
Example: what is an equation of the line with slope - 4 and y - intercept 7?
y = -4x + 7
Direct Variation
Definitions
Direct variation: a relationship that can be represented by a function in the form y = kx, k cannot be equal to 0
Must pass through (0,0)
Constant of variation: k, which is the coefficient of x
k is the rate of change/slope
Example: determine whether -4 + 7x + 4 = 3y represents a direct variation
Step 1: substitute the x and y values with 0
-4 + 7(0) + 4 = 3(0)
Step 2: if the equation is true after the substitution, it represents a direct variation -4 + 4 = 0 (true)
Point-Slope Form
Definitions
Point-slope form: (y - y1) = m(x = x1)
x1 = x-coordinate
y1 = y-coordinate
an equation of a non vertical line