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