UNIT 3: LINEAR FUNCTIONS

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!!

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)

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

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

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

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.

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.

Joint Variation

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

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.

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??

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

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

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

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?

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

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.

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:

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

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

3-7 Transformation of Linear Functions

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)

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)

3-7 Transformation of Linear Functions

3-7 Transformation of Linear Functions

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

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

Scatter Plots and Trend Lines

Scatter Plots and Trend Lines df

Problems/Need to Know