1. UNIT 3: LINEAR FUNCTIONS

2. What is a linear function?
It is a graph of a LINE, with 1 dependent variable or output or y and 1 independent variable or input or x.
The rate of change or slope is the SAME for ANY 2 points on the line.
There are infinitely many points on a line!!

3. Rate of Change and Slope
Rate of Change = change in dependent variable
change in independent variable
Slope = vertical change = run
horizontal change rise
Slope = =
or CANNOT be 0 or slope is undefined (vertical line)

4. What is a linear function?
4 types of line, but ONLY 3 qualify as a function (1 unique output for 1 input)
Positive Slope
Negative Slope
Zero Slope
Horizontal Line
Undefined Slope
NOT a function
Does not pass vertical line test!!
X=3 is a linear equation NOT a linear function

5. Problems/Need to Know
Identifying positive, negative, zero, and undefined slope graphs.
Find slope given 2 points.
Find x or y coordinate given slope and points:
Slope=3, points (1,2) and (2,y) lie on the line, find y
Slope=2, points (3,2) and (x, 10) lie on the line, find x
Find the slope or rate of change given a graph/plot

6. x and y intercepts
x intercept is the POINT where the graph crosses the x-axis or when the value of y is 0
Examples (0,0), (3, 0), (-6,0), (-4,0), (12,0)
y intercept is the POINT where the graph crosses the y-axis or when the value of x is 0.
Examples (0,0), (0, 2), (0,10), (0,-3), (0,-10)
Is there an x and y intercept for:
Positive slope line?
Yes, will have both x and y intercepts
Negative slope line?
Zero slope line?
Will have ONLY y-intercept. Remember 0 slope = horizontal line

7. Direct Variation
Graph of a line with a y-intercept of ZERO, meaning the graph passes through the ORIGIN.
Direct variation is defined by an equation of the form y=kx, where k is the constant of variation or the slope of the line.

8. Joint Variation Joint variation:
Ex. y varies jointly with x and z
Ex. Area of Triangle, The area A varies jointly with length of the base b and height h where the constant of variation k is equal to ½.
The graph of a joint variation also has a y-intercept of ZERO, meaning the graph passes through the ORIGIN.
k is the again the constant of variation or the slope of the line.

9. Joint Variation

10. Problems/Need to Know
Identifying the x and y intercepts on a graph/plot
Find the x and y intercept given a linear equation
Ex. Find the x and y intercept for y=2x-10
Ex. Find the x and y intercept for 3x-4y=-20
Identifying whether a linear equation is a direct variation, if it is, what is the constant of variation k
Ex. Is this a direct variation y=3x, yes, k=3
Ex. Is this a direct variation 2x-3y=10, NO
Ex. Is this a direct variation -3x+2y=0, yes, k=3/2
Find the value of k (constant of variation) for both direct and jointly variation problems and use k to find missing value

11. Slope Intercept Form m is your slope
b is your y-intercept or the y-coordinate of your y-intercept
To write an equation in slope intercept form, you need the slope m and b, your y-intercept.
Ex. Given a line with a slope of 2 passes through the point (0,5). The equation for the line in slope intercept form is
There is ONLY 1 slope intercept form for a linear function. WHY??
Any point (x, y) on the line MUST satisfy the equality with slope and y-intercept.

12. Point Slope Form
To write a linear equation in Point Slope Form, you need
Slope m
Any point on the line,
Ex. Given a line with a slope of -3 passes through the point (2,-3), the equation in point slope form is
Infinite number of point slope form equations for a line: WHY??

13. Standard Form A, B, and C are real numbers.
Real number is any number on a number line; pretty much any number you can think of.
A and B CANNOT both be zero. Why??
Standard form allows finding the x and y intercepts of a linear function easier.
Ex. 2x+3y=6

14. Problems/Need to Know
Given two points on a line: write the equation in slope intercept form, point slope form, standard form, and graph.
Ex. Given (2,5) and (-2,-7)
Given slope and 1 point on a line: write the equation in slope intercept form, point slope form, standard form, and graph.
EX. Given m=-2 and a point (1,7)
Given stand form for a linear function, solve for y and write it in slope intercept form.
Ex. Given 3x-5y=10, write it in slope intercept form

15. Problems/Need to Know
Given a table of x and y values: determine if it is a linear function and if yes, write the equation in slope intercept form, point slope form, standard form, and graph.
Determine whether a point is a solution to a linear equation.
Ex. Is (3, 5) a solution to the linear equation 2x-3y=-9?
Rewrite a slope intercept form equation with fractions in standard form using integers
Ex. Rewrite linear equation in standard form using integers

16. Interpreting Real Life Examples/Situations of Linear Relationship
Ex. I already made 5 baskets to sell at the fair. After working for another 3 hours, I made 6 more baskets for a total of 11 to sell at the fair. Write a linear equation relating the number of baskets made (y) to the amount of time spent (x) making the baskets. How many baskets total will I have for the fair if I spend another 7 hours on making baskets?
Ex. A hot air balloon descends at a linear rate of 2 meters per second from a height of 600 meters above ground. Write the linear equation for the descend? What does the x and y intercept mean?

17. Parallel and Perpendicular Lines
Parallel Lines: lines that never intersect.
Slope: nonvertical parallel lines have the same slope but different y-intercept. Vertical parallel lines are parallel if they have different x-intercept.
Perpendicular Lines: lines that intersect to form a 90 degrees or right angle.
Slope: Two nonvertical lines are perpendicular if the product of their slopes is -1. A vertical line and a horizontal line are also perpendicular.
Opposite reciprocals: two numbers whose product is -1.
Ex. ½ and -2, -3/4 and 4/3, 5 and -1/5

18. Parallel and Perpendicular Lines Problems
Write an equation of a line passing through point (12, 5) and is parallel to the graph of y=2/3x-1.
Know: slope
Need: the y-intercept for y=mx+b
Classifying Lines: are the graphs of 4y=-5x+12 and y=4/5x-8, parallel, perpendicular, or neither
Slope of second line is 4/5
Slope of first line can be found by solving the equation for y which comes out to be -5/4
They are opposite reciprocals: so the lines are perpendicular.

19. Parent Function: See Notes/Handout
We will learn the following 3 families of functions:
Parent Linear Function:
Equation of lines
Example of linear functions:
Parent Quadratic Function:
Equation with a variable x raised to the 2nd power
Examples of quadratic function:
Parent Absolute Value Function:
Equations with an absolute value symbol around x
Examples of absolute value functions:

20. Absolute Value: l l Distance from 0. NO NEGATIVES
Absolute value of ANYTHING is POSITIVE!
Simplify | 2 + 3(–4) |.
| 2 + 3(–4) | = | 2 – 12 | = | –10 | = 10
Simplify –| –4 |.
–| –4| = –(4) = –4
Simplify –| (–2)2 |.
–| (–2)2 | = –| 4 | = –4
Simplify –| –2 |2
–| –2 |2 = –(2)2 = –(4) = –4
Simplify (–| –2 |)2.
(–| –2 |)2 = (–(2))2 = (–2)2 = 4

21. 3-7 Transformation of Linear Functions
Transformation of a function is a change to the equation of the function that results in a change in the graph of the function.
**Graph of ANY linear function is a transformation of the parent function y(x)=x.
Translation: a vertical or horizontal shift in a graph.
Reflection: flip the graph of a function across the x or y axis

22. 3-7 Transformation of Linear Functions

23. 3-7 Transformation: Translation
Slope DOES NOT change.
Adding a value d to the outputs shifts the graph up. y= f(x) + d
Subtracting a positive value of c from the inputs shifts the graph to the right. y= f(x-c)

24. 3-7:Translation More Examples:
Adding a value d to the outputs shifts the graph up. y= f(x) + d
Subtracting a positive value of c from the inputs shifts the graph to the right. y= f(x-c)

25. 3-7 Transformation of Linear Functions

26. 3-7 Transformation of Linear Functions

27. 3-7 Linear Transformation Vocabulary
X-intercept increases or graph shifts to the right
X-intercept decreases or graph shifts to the left
Y-intercept increases or graph shifts up
y-intercept decreases or graph shifts down
Reflected over the x-axis
Reflected over the y-axis
Steeper: slope increases
Flatter: slope decreases
Parallel: slope stays the same

28. Scatter Plots and Trend Lines
Scatter Plot: a graph the relates to different sets of data by displaying them as order pairs.
Trend line: a line on a scatter plot, drawn near the points, that show a correlation.
Correlation: statistical relationship between 2 variables
Positive correlation: y tends to increase as x increases
Negative correction: y tends to decrease as x decreases
Zero correlation: two sets of data are not related

29. Scatter Plots and Trend Lines

30. Scatter Plots and Trend Linesdf

31. Problems/Need to Know