[Music] welcome to linear function a linear function is of the form y equals ax plus b where A and B are constants intercepts these are the points on the graph that intersect the X axis are the x-intercepts to find them you set y equal to 0 and solve the equation for x the points where the graph intersect the y-axis or the y-intercept are the y-intercept and there's only one if it's a function to solve to find this one you set x equal to zero and solve for y note the a function can have more than one X but has it most one y and the number of X depends on the highest power of X so for a linear function you're only going to have one for quadratic you're going to have two for cubic you're going to have three and so on this section is linear so we're only going to have one slope is the rate of change this is m equals Y2 minus y1 over X2 minus X1 also Delta X over Delta y well Delta means the change in parallel parallel lines have the same slope I typically note it with two bars that look like parallel lines m equals m perpendicular lines have negative reciprocal slope the symbol for perpendicular looks like an upside down t m equals negative 1 over m the equation of a line you have two forms slope intercept form where m is the slope and the point 0 B is the y-intercept and this is y equals MX plus b this is the form I like to use you have point-slope form m is the slope X1 Y2 is a point on the line and the equation is y minus y1 equals m minus X1 the problem that I have with this formula is that whenever you use it students think okay that's my answer but it's not you always need to put the equation of your line in slope intercept form then you have standard form which uses capital letters and this is a B and C are integers not whole numbers so they don't have denominators and they're not decimals they're not fractions and they're not decimals they're integers and that is a X plus b y plus C equals zero a vertical line is a line that goes straight up and down a is a real number and it's x equals a a horizontal line where B is a real number and it's y equals B and it's a totally flat line so it looks like the Horizon hence horizontal all right a phone company determines the monthly bill for customer by adding an energy charge of point three three cents per minute to the base charge of 18.36 per month write an equation for the monthly charge Y in terms of X the number of minutes used let y be measured in dollars that y be measured in dollars is important you have a rate of dollars .0833 you have to change that cents per minute to dollars that's your slope because you're charged that rate your slope is a rate it's the rate of change you're charged that rate every minute you use your phone your base rate is your y-intercept and that's 18.36 this gives you the equation of the line or the equation of your cost function to be Y equals .0833 X Plus 18.36 A supplier will buy 10 packages of paper goods from a wholesaler if the price is 25 dollars and five packages of paper goods from a wholesaler for fifty dollars the wholesaler is willing to supply three packages for sixteen dollars and 11 packages for sixty four dollars find the demand equation find the supply equation find the equilibrium point we need both the demand and Supply functions these are both linear functions and we're going to have to find them separately for the demand we have the points 10 25 and 550. for the supply we have the points 316 and 1164. we have to find the equations of these lines demand 10 25 and 550. find the slope n is Y2 minus y1 over X2 minus X1 so that's 50 minus 25 over 5 minus 10. equals 25 over negative 5. which is negative 5. then to find the line we use y equals MX plus b so we have 25 equals negative 5 time which is our slope times 10 plus b we need to find B we simplify that to get 25 equals negative 50 plus b so b equals 75. our demand and I'm denoting it y sub D equals negative 5x plus 75. I put that D there so that I know that this is the demand function to find the supply we have 316 and 1164. to find the slope we have m equals Y2 minus y1 over X2 minus X1 that 64 minus 16 over 11 minus 3 equals 48 over 8 which equals 6. our line we have y equals MX plus b so we get 16 equals 6 times 3 plus b ixteen equals 18 plus b so negative 2 equals B so s y sub s equals 6X minus 2. to find the equilibrium point we have to set the demand equal to the supply so that is setting negative 5x plus 75 which is our demand equal to 6X minus 2 which is our supply we're going to add 5x to both sides and add 2 to both sides which gives us 77 equals 11x divide by 11 we get x equals 7. so we still need to take that 7 and put it back into the supply to find the number to find the Y value so that gets us 42 minus 2 which is 40. so we have seven units and forty dollars forty dollars is our equilibrium point thank you let me know if you have any questions [Music] foreign