Week 4 Handout MAT 1033C Professor Niraj Wagh J Sections 3.2 & 3.3 Introduction to Functions & Graphing Function A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B. Basically for every input there is ONE output. The notation is y = f(x) which is read f of x. Simply put, it means y is a function of x, or that y depends on x. Because of this y is called the dependent variable and x is called the independent variable. For example, We could have f(x) = x 2 f is the function name, x is the input, and x 2 is what is outputted. If we wanted to find f(4), we are taking the input 4 and it becomes an output of 16 because 4 is being squared as our function f says! Hello! I m your function! f of x! If you ever say f times x I won t be happy L N. Wagh 1
Q: How can you tell if a graph is a function or not? A: Vertical Line Test!! If you can draw a vertical line anywhere on a graph so that it hits the graph in more than one spot, then the graph is not a function. If it does not hit the graph in more than one spot, then the graph is a function. N. Wagh 2
Domain A domain is the set of x-values where a function is DEFINED. It is the set of all possible input values (often the x variable), which produce a valid output from a particular function. -> Remember that if you have a fraction the denominator cannot equal zero, and if you have a square root it cannot be negative (in the real values). Range Conversely, the range is the set of all possible output values (usually the variable y ), which result from using a particular function. N. Wagh 3
3.2 & 3.3 Examples Find the domain and range of the relation. Also determine whether the relation is a function. 1. {( 1, 7), (0, 6), ( 2, 2), (5, 6) } Determine whether the following equation is a function. 2. y = x 2 Given the following functions, find the indicated values. 3. f (x) = 2x 2 + 4 (a) f ( 11) 1 (b) f 2 N. Wagh 4
4. Kerbela Cruz is planning on moving from Kansas City to Los Angeles! She has dreams of becoming the next Jessica Alba. She plans on renting a rental truck to do this move. The cost of renting a truck for a day is 24 dollars and it costs 20 cents per mile. (a) Create a linear function that models the above scenario. (b) Find the cost of driving the truck 200 miles. N. Wagh 5
Sections 3.4 & 3.5 Slope & Equations of Lines Slope Let P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ) be two distinct points. The slope m containing P 1 and P 2 is: m = rise run m = y y 2 1 OR m = y y 1 2 x 2 x 1 x 1 x 2 N. Wagh 6
Cases: If x 1 = x 2, the slope of the line will be undefined since we cannot divide by zero. è The resulting line is vertical. (UNDEFINED SLOPE, Form: X = some #) Mr. Penguin: AHH!!! HELP ME!! L If y 1 = y 2, the slope of the line will be 0 and zero divided by anything will be zero. è The resulting line is horizontal. (NO SLOPE, Form: Y = some #) Mr. Penguin: Umm.. I m not going anywhere, this is boring! N. Wagh 7
Positive Slope Mr. Penguin: I m tired. Negative Slope Mr. Penguin: Weee!!!! Slope-Intercept Form An equation with slope m and y-intercept b is y = mx + b Point-Slope Form of an Equation of a Line An equation of a nonvertical line with slope m that containts point (x 1, y 1 ) is y y 1 = m(x x 1 ) General Form of an Equation of a Line Ax + By = C Where A, B, and C are real numbers and A and B are not both 0. N. Wagh 8
Parallel Lines Two nonvertical lines are said to parallel with one another if and only if they have the same slope and different y-intercepts. è Have the same slope. è Have different y-intercepts. NOTE: If the lines had the same y-intercept they would be the same line. Perpendicular Lines Two nonvertical lines are perpendicular (when two lines intersect at a right angle [90 0 ]) if and only if the product of their slopes is -1. NOTE: You can also say that two nonvertical lines are perpendicular if the slopes are negative reciprocals of one another. è For example, if one line had a slope of 5 and the other line had a slope of -1/5 then they are perpendicular lines. o This is because 5*-1/5 = -1. J 3.4 & 3.5 Examples First, find the slope of the line that goes through the given points. Then find the equation of the line in slope-intercept form. 1. (1, 6), (7, 11) N. Wagh 9
Determine whether the lines are parallel, perpendicular, or neither. 2. 2y = 6x +12 3y + x =15 3. Perpendicular to the line -2x + y = 3; containing the point (1, -2) N. Wagh 10
Let s Practice! J Find an equation for the line with the given properties. Express your answers using slope-intercept form. 1. Given, f (x) = 5x 2 + 3x 1 Find : f (3), f 3 1, and f 2 2. Charlotte rents a car for 20 dollars plus 2 dollars per mile. Create a function based on this and determine how much it will cost Charlotte to drive 30 miles. 3. Slope = 2 ; containing the point (-9, 4) 3 4. Parallel to the line x-2y = -5; containing the point (1, 3) N. Wagh 11
Section 3.7 Graphing Linear Inequalities in Two-Variables RECALL: The graph of a linear equation in two variables is the graph of all ordered pairs that satisfy the equation, and the graph is of a line. For linear inequalities, the idea is very similar except now we graph all the ordered pairs that satisfy the inequality, rather than the equation. Steps to Graphing Linear Inequalities in Two Variables 1. Graph the boundary line found by replacing the inequality sign with an equals sign. -> If the inequality sign is < or >, graph a DASHED line --- indicating that the points on the line are not solutions of the inequality. -> If the inequality sign is or, graph a SOLID line indicating that the points on the line are solutions of the inequality. 2. Choose a test point NOT on the boundary line and substitute the coordinate of this test point into the original inequality. 3. If a true statement is obtained in step 2, then shade the half-plane that contains that test point. If the statement is not true, shade the other half-plane that does not contain the test point. N. Wagh 12
3.7 Examples Solve each linear inequality by graphing. 1. y 3x + 4 2. y + 3 4 x <1 Graph the union or intersection. 3. x y < 3 AND x > 4 4. 3x + 5y 15 OR x y 5 N. Wagh 13
Let s Practice! J Solve each linear inequality by graphing. 1. 3x + y 1 2. y 4 x 4 3. y > x 5 3 Graph the union or intersection. 4. x + y 3 OR x + y > -1 5. 2x + 3y 12 AND 8x 4y >1 AND x < 4 That s all for 3.7! Work on MML 3.7. If you have any questions, let me know. J N. Wagh 14