 Hello and welcome to the session. In this session, we will use a table and a graph to compare the changes in linear and exponential expressions as the value of x increases. I also will recognize that as x increases, a linear expression increases at a constant rate, additively, while an exponential function increases multiplicatively. Now let us consider three functions. That is y is equal to 3x, y is equal to 3x square and y is equal to 3 raised to power x. The first function is linear, second function is quadratic and third function is exponential. Now consider this table. Now we will complete the table using three functions and we will observe which column is growing faster. Now in this column, we have linear function y is equal to 3x. Now when x is equal to 0, y is equal to 3 into x, that is 3 into 0, which is equal to 0. Now when x is equal to 1, then y is equal to 3 into 1, that is 3. When x is equal to 2, y is equal to 3 into 2, that is 6. When x is equal to 3, y is equal to 3 into 3, that is 9. When x is equal to 4, y is equal to 3 into 4, that is 12. when x is equal to 5, y is equal to 3 into 5 that is 15. Now in this column we have quadratic function y is equal to 3 x square. Now here when x is equal to 0, y is equal to 3 into 0 square that is 3 into 0 which is 0. When x is equal to 1, y is equal to 1 square that is equal to 3. When x is equal to 2, y is equal to 3 into 2 square that is 12. When x is equal to 3, y is equal to 3 into 3 square which is equal to 27. When x is equal to 4, y is equal to 3 into 4 square which is equal to 48. Now when x is equal to 5, y is equal to 3 into 5 square which is equal to 75. Now in this column we have exponential function y is equal to 3 raised to power x. Now when x is equal to 0, y is equal to 3 raised to power 0 that is 1. When x is equal to 1, y is equal to 3 raised to power 1 that is 3. When x is equal to 2, y is equal to 3 raised to power 2 that is 9. When x is equal to 3, y is equal to 3 raised to power 3 that is 27. When x is equal to 4, y is equal to 3 raised to power 4 that is 81 and 1, x is equal to 5, y is equal to 3 raised to power 5 that is 233. So, for different values of x we have completed this table. Now here you can see in this column that is the column of linear function the values are increasing by a difference of 3 as x increases by 1 in a pattern 3 3 plus 3 that is 6, then 3 plus 3 plus 3 that is 9, then 3 plus 3 plus 3 plus 3 that is 12 and then 3 plus 3 plus 3 plus 3 plus 3 that is 15. So, the values are increasing in this pattern. Now in the column of quadratic function you can see the values are increasing at a higher rate than the values in the column of linear function. Now in this column of linear function you can see when x is 3, y is 9, but in this column of quadratic function you can see when x is 3, y is 27. So, the values are increasing at a higher rate. Now in the column of exponential function the values are increasing at a very high rate, then the values in these two columns they are following the pattern 3, then 3 into 3 that is 9, then 3 into 3 into 3 that is 27, then 3 into 3 into 3 into 3 that is 81, then 3 into 3 into 3 into 3 into 3 that is 243. So, when x is 3 the value of given linear function is 9 that the value given by exponential function is 3 that is 27. So, from these patterns we can see that as x increases a linear expression increases at a constant rate additively while an exponential function increases multiplicatively. Thus from table we can see that the values for y is equal to 3 h to power x are increasing faster than the values for y is equal to 3x and for y is equal to 3x power. Also we see that at x is equal to 1 the value of all three functions is same that is 3 and as x is equal to 5 linear function gives value 15, quadratic function gives value 75 that exponential function give a very high value 233. Thus we see that the values increasing exponentially eventually exceed the values increasing linearly or quadratically. Now, let us see the difference in the graphs of the three functions. So, we have the following graph showing the graphs of the three functions. Now, you can see this pink straight line and this straight line is the graph of linear function y is equal to 3x. Now, this curve is the graph of quadratic function that is this red curve is the graph of quadratic function y is equal to 3x square. This blue curve is the graph of exponential function y is equal to 3 raised to power x. Now, this linear function increase in values at a constant rate and the values of y are not higher. Now, in quadratic function there is an increase in values as compared to linear function. Now, see the exponential curve is lower than or equal to the quadratic curve is equal to 3 but after x is equal to 3 it shows a great increase in the values of y as compared to both the curves. Thus we see that the exponential function increases multiplicatively as compared to linear and quadratic functions. So, in this fashion we have learned that the values increasing exponentially eventually exceed the values increasing linearly or quadratically. So, this completes our fashion. Hope you all have enjoyed this fashion.