in this video we will practice working with rates of change in linear and quadratic functions guys I really broke it down in the previous video so if you have not seen that yet you should definitely watch that first you can either click the link that appears in the upper right hand corner or find the link in the description this is AP pre-calculus topic 1.3 if you appreciate this content please give it a like select values for several functions are given in the tables below for each table of values determine if the function could be linear quadratic or neither in the previous video we learned that if a function is linear the average rate of change is constant and if a function is quadratic the rate of change of the rate of change is constant in other words the rate of change changes at a constant rate that last statement is only true over consec itive equal length input value intervals so I need you to get really comfortable with that phrasing because every time you do a justification I need you to say over consecutive equal length input value intervals for number one these are the changes in the output values and these are the changes in the input values notice that we have consecutive equal length input value intervals the average rate of change for for each interval is -3 over2 1 / 2 5 over2 and 9 /2 because the rate of change is not constant we know that the function is not linear let's see how the rate of change is changing each time from -3 to 1 the numerator is increasing by four so that means the rate of change is increasing by 4 over 2 which of course is equal to two from 1 to 5 that's another increase of four so again that's 4 over 2 which is equal to 2 from 5 to 9 same story 4 over 2 which is equal to 2 so we are seeing that the average rate of change is increasing at a constant rate that means that f ofx is quadratic when you have consecutive equal length input value intervals you don't have to show all this work to find that the rate of change of the rate of change is constant you can do a shortcut we can get all the information we need from the output values we can see that the rate of change is increasing at a constant rate without actually calculating the rate of change if asked to justify our answer we would say that F ofx could be quadratic because the average rate of change is increasing at a constant rate over consecutive equal length input value intervals notice that saying the average rate of change is increasing at a constant rate is another way of saying the rate of change of the rate of change is constant for number two here are the output value changes and here are the input value changes notice that we do not have consecutive equal length input value intervals for this reason we need to calculate the actual average rate of change by dividing 2 over 1 2 over two 2 over 4 and 2 over 8 since the average rate of change is not constant we know that g ofx is not linear since the rate of change is not changing at a constant rate we know that g of X is not quadratic we can say that g of X is not linear because the average rate of change is not constant and G ofx is not quadratic either because the average rate of change is not changing at a constant rate for number three here are the output value changes and the input value changes since we do not have consecutive equal length input value intervals we need to actually calculate the average rate of change 4 over 2 is 2 2 over 1 is 2 10 over 5 is 2 and 2 over 1 is 2 if asked to justify we would say that h of X is linear because the average rate of change is constant notice that for linear we do not need equal length input value intervals that's just for quadratic for number four here are the output value changes and here are the input value changes since we have consecutive equal length input value intervals we do not need to actually calculate the average rate of change to see that the average rate of change is not constant and we do not need to calculate the rate of change of the rate of change to see that the rate of change is not increasing at a constant rate so it's back to this answer K ofx is not linear because the average rate of change is not constant K ofx is not quadratic either because the average rate of change is not changing at a constant rate for number five here are the output value changes and here are the input value changes since we do have consecutive equal length input value intervals we don't have to actually show the calculation of the average rate of change we can see that the average rate of change is constant we can say that m ofx is linear because the average rate of change is constant for number six here are the changes in the output values and here are the changes in the input values since we once again have consecutive equal length input value intervals we don't actually have to show the calculation of the average rate of change to see that the average rate of change is not constant and we don't have to to calculate the rate of change of the rate of change to see that the average rate of change is decreasing at a constant rate we can say that P of X could be quadratic because the average rate of change is decreasing at a constant rate over consecutive equal length input value intervals for quadratic you have to say that statement looking back at two of my previous answers I should have said could be linear here instead of just saying it is linear technically these functions could do something crazy in between the values that we are given selected values for several functions are given in the tables below for each table of values determine if the function could be concave up concave down or neither here's another chart that I have showed you in previous videos pause the video and memorize this if you have not done so already for these problems focus on this part of the chart F ofx will be concave up wherever the rate of change is increasing and F ofx will be concave down wherever the rate of change is decreasing for number seven these are the output value changes and these are the input value changes because we do have consecutive equal length input value intervals we don't have to actually calculate the average rate of change to see that the average rate of change is decreasing we know that Q ofx could be concave down because the average rate of change is decreasing over consecutive equal length input value intervals for number eight these are the output value changes and these are the input value changes again we have consecutive equal length input value intervals so we don't need to actually calculate the average rate of change to see that the average rate of change is constant s oft is neither concave up nor concave down because the average rate of change is neither increasing nor decreasing over consecutive equal length input value intervals sft is most likely linear for number nine here are the changes in the input values and here are the changes in the output values notice that we once again have concept consecutive equal length input value intervals so we don't need to actually calculate the average rate of change to see that the average rate of change is increasing over some intervals but then decreasing over others vfc is neither concave up nor concave down because the average rate of change is increasing and then decreasing over consecutive equal length input value intervals hey guys don't forget to like And subscribe but also if you found this video helpful there's a lot more where that came from you can click the upper link which will take you to the whole unit playlist or you can click the lower link which will take you to the next video in the playlist see you there