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Students Mentoring Students Presents: Ruben Sanchez [email protected] BEFORE WE START: • We are here to help you! Do NOT be afraid to ask questions. • There are no dumb questions! • The only dumb thing to do is not ask for help when you are stuck. Looking at Linear Equations and Graphs • There are many different types of graphs but the most common ones we see, look something like parent function graphs. • To the right, there are various examples of some of the parent functions that exist. So what do these functions mean? • All these functions can be used to express information gathered or it represents a relationship between 2 variables and then plotted on a graph, in this case the Cartesian Coordinate System. • Information that can be read off of graphs include: • Distances • Velocity • Acceleration • Money/Interest Earned Lets look at examples on how to graph on the coordinate system. There are 2 main ways to graph on a coordinate plane. 1. Point- Slope form 2. Slope intercept form We will look at both and see how they are related. We will also see how to convert from point- slope to slope intercept Lets say they give us a point of ( 8, 6) and that we are given a slope of 5. Once we have our information, all we do is plug our variables into the equation to the left. We are going to plug in the coordinates Into the x and y with sub-script 1. We just plug in our variables where it asks us to in the formula and that is all . Once you do that we have our answer, a point-slope formula! Point Slope Form In Point slope Form we are given a point on the graph. And the slope of a line. When we have these 2 main components we are able to plug into an equation to find an equation for a line. 1) Point (7,8) Slope 6 3) Point (2, 8) Slope 5 2) Point (3,9) Slope 2 4) Point (5,2) Slope 8 5) Point (4,4) Slope 9 Practice with Point-Slope Form Practice putting these points/ slope examples into the point slope formula. Only set up. Point-Slope What do we have to do to make it look like this? Isolate Y! Y wants to be on its own! So first we distribute the right side to make things easier and we get Then we add 6 on the left to leave Y by itself. Remember the rule, whatever we do to one side, we do to the other. Afterwards Y is left by itself and we reach our answer. Changing Point-Slope to Slope Intercept Form Once we have our equation in point slope form, simple algebra is used to convert this equation to slope intercept form, making it possible to graph on graph calculators. Change from point-slope to slope-intercept Using and understanding the Pythagorean Theorem Pythagorean theorem deals with measurement of triangle sides and simple algebra used to find lengths of sides. NOTE- this method only works when you have a right triangle, a triangle with at least one angle measuring 90 degrees. Take the Square root of both sides FINAL ANSWER IS Example: Let us find the length of the hypotenuse for the triangle MLO above. We are given the side lengths but missing the hypotenuse. All we have to do now is substitute into the equation a2+b2=c2 Main Street Commercial Street 4 Miles Park Avenue 3 Miles Real Life Example with 90 Degree Triangles If Park Ave. is 3 miles long and Commercial Street is 4 miles long, how long is Main Street? (hint- set up Park Ave. as side A, and Commercial St. as side B) Trowbridge Drive 6 Miles Paisano Drive Montana Street 4 Miles Real Life Example with 90 Degree Triangles If Trowbridge is 6 miles long and Montana Street is 4 miles long, how long is Paisano Street? (hint- set up Montana as side A, and Trowbridge Drive as side C ) Didn’t see this coming! Let us say that the top bar (olive) of the bicycle is 3 feet, and the bar that goes down from the seat to the pedals (violet) is 2 feet. How long is the bar that that connects the handlebars to the pedals (blue)? More real life applications with pre-calculus We found out that right triangles could be used with distances or lengths but what about other useful math terms or calculations? We can use perimeters and area and relate it to real life applications. Fore example the length of track, perimeter of a farm, etc… Radius of the track circle is 25 Meters Find the total distance around the track What we need to do is find the “perimeter” of the track to find out the length of one lap around the field. How do we do this problem? Put a fence around this barn The owner of this barn decides to put up a fence around his bar. Looking at it from a top view it makes a rectangle measuring 35 feet long and 20 feet wide. If the farmer wants to have “extra” space and put the fence 5 feet away from the barn on every side, what will be the amount of fence the farmer will have to buy? What will be the new total area of the enclosed land including the barn? What is the Perimeter of the cardboard box? If each can has a diameter of 2.5 inches, and the box is 4 cans wide and 6 cans long, what is the perimeter of the cardboard? What is the area of the base of the cardboard box? If each side of the parking aisle can Park 9 cars, how many cars can 8 Aisles hold? Multiplication practice How many cars can be parked if each parking aisle can park 9 cars on each side? What if there are a total of 8 aisles? How many desks will fit? How many desks will fit if the desks are 3 feet wide and 4 feet long and the room measures 16 feet by 21. The teacher wants to leave a gap of 3 feet in between each row of desks. How many test tubes do you need? The doctors office you work for needs to order test tubes but the Doctor doesn’t know how many test tubes to order for the 15 racks he ordered. Each rack’s dimensions are 5 by 12 slots. How many test tubes does he need to order to fill each slot on the racks? How much does it cost to fill up? Lets use the example with the man on the bicycle, if you are to pull up in a big V8 truck that requires you to fuel up with premium grade fuel, how much would it cost to fill up the 24 Gallons the truck has? What if you are in a Prius and only have to fill up 10 Gallons of Regular grade fuel? Rates including speed Nascar engines are built to last and endure very high speeds and accelerations. These fine tuned engines can reach top speeds of 200+ miles per hour (MPH). How long (time) would it take to complete 550 miles at a race? (set up as an equation) Rates with distances The Tour de France typically comprises 20 professional teams of nine riders each and covers some 2,200 miles of flat and mountainous country, mainly in France, with occasional and brief visits to Belgium, Italy, Germany, and Spain. How long would it take to complete the race at a constant pace of 15 miles per hour? Proportions If 250 songs take up 1 gigabyte of space, how many songs can fit in modern day electronics such as iPod’s/ iPad’s/ iPhone’s/ etc… that can hold 16 gigabytes? How about 32 gigabytes? And finally how about 160 gigabytes? Rates The average rate that water comes out from a residential outlet, or hose, is about 200 gallons per hour. At this rate, how long would it take to fill up a pool of 22,000 gallons? Basic Calculus Calculus can be used in many different applications. One key application that this field of math is used in is Physics and Engineering. Others may use calculus to calculate distances or positions. Whatever you wish to apply it to, we will cover the basics to understand how to carry out processes in calculus. Lets practice an example. Derivatives In our real life applications, derivatives can be understood as a rate of change. One way we can think of a rate of change that we may not even notice is when we drive, miles per hour. This is an example of a rate of change. Derivatives don’t always have a simple chart that we can refer to but for integrals, we have charts that can help us set up the problem. Let’s take the derivative of each function Integrals Here for integrals, there are some set integral guides that can help us set up the problem and In most cases and in textbooks, the constant shown here is is written as a “K” or a “C” Integrals Lets practice an example solving integrals. Let us solve the definite integral