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 = =  and 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  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?  Negative slope line?  Zero slope 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.

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  Ex. Is this a direct variation 2x-3y=10  Ex. Is this a direct variation -3x+y=0

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 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??

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.  A and B CANNOT both be zero.  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  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.

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 is the x and y intercept mean?

Parallel and Perpendicular Lines