 Hello and welcome to the session. In this session we will discuss how to construct exponential functions from a graph, table or description of a relationship. Now we know that exponential function is a function that can be described by an equation of the form y is equal to a into b raised to power x, where a is not equal to 0, b is greater than 0 and b is not equal to 1. Also a and b are constants and a is the initial value of starting value, b is the factor i.e. the amount by which the value of expression gets multiplied for each unit increase in x. We also call b as the base of exponential expression. Now exponential functions are nonlinear and non-paralleled functions. Now let us learn how to construct exponential function from a graph. Now suppose we are given a graph of an exponential function that passes through the points 0, 3, 1, 6. Now we have to write the exponential function of the form y is equal to a into b raised to power x. Now the given graph passes through the points 0, 3 and 1, 6. Now the exponential expression is of the form y is equal to a into b raised to power x. So we have to find values of a and b and put them back in the expression. So we find the values of a and b. Now let this be equation number 1. Now this point with coordinates 0, 3 lies on the curve. So we will put x is equal to 0 and y is equal to 3 in equation 1. So we have 3 is equal to a into b raised to power 0 which implies 3 is equal to a into 1. Now we know that when exponent is 0 the value is 1. So b raised to power 0 will be 1. So this implies 3 is equal to now a into 1 is a and we can write it as a is equal to 3. Now the point with coordinates 1, 6 also lies on the given curve. So we put x is equal to 1 and y is equal to 6 in equation 1 and we have 6 is equal to a into b raised to power 1 which implies 6 is equal to a b. Now we know that a is equal to 3. So this implies 6 is equal to 3 b which implies 6 upon 3 is equal to b and 3 into 2 is 6. So this implies 2 is equal to b or we can write it as b is equal to 2. Now putting a is equal to 3 and b is equal to 2 in 1. We have y is equal to 3 into 2 raised to power x. So this is the required exponential function. Now let us see how to construct the exponential function from an input output table. Now see the following input output table. Now we have to construct an exponential function from this table. Now general exponential function is of the form y is equal to a into b raised to power x where b is the constant factor that is the amount by which the value of expression gets multiplied for each unit in 3's in x. So let us find b from the table. For this we will find ratio of convective values of y. Now here 12 upon 24 is equal to 1 upon 2. Then see the next ratio. That is 6 upon 12 which is again 1 upon 2. Then next is 3 upon 6 which is again 1 upon 2 which is again 1 upon 2 and 1.5 upon 3 is also 1 upon 2. So here we have got the constant factor which is equal to 1 upon 2. So b is equal to 1 upon 2. So we put b is equal to 1 upon 2 in this equation and we have y is equal to a into 1 upon 2 raised to power x. Now we have to find value of a. For this let us put any other pair from the table. The most convenient will be where x is 0. So let us take this point that is the point where x is equal to 0 and y is equal to 3. So put x is equal to 0 and y is equal to 3 in this equation. And here we have x is equal to 0 and 3 is equal to a into 1 by 2 whole raised to power 0. For this implies 3 is equal to a into now 1 by 2 whole raised to power 0 will be 1. So this implies 3 is equal to a or we can write it as a is equal to 3. Now let us put a is equal to 3 in this equation. So we have y is equal to 3 into 1 by 2 whole raised to power x. So this is the required exponential function. Now we can also check that b is equal to 1 by 2 is factor by writing down each value of y by 1 by 2. We will get the subsequent values of y for each unit included in the equation. So we have x is equal to 3 in x. Now here we can see 24 into 1 by 2 is 12, 12 into 1 by 2 is 6, 6 into 1 by 2 is 3 and 3 into 1 by 2 is 1.5. Now let us see how to construct the exponential function from a description. Now we know that exponential function is of the form y is equal to a into b raised to power x where a is the initial amount, b is growth of decay factor, x is time and y is ending amount. Now this equation represents exponential growth when a is greater than 0 and b is greater than 1 and here b is called growth factor and this equation represents exponential decay when a is greater than 0 and b lies between 0 and 1 and here b is called decay factor. Now let us see one example. Now in this example we have to write a function that models the population of Smithville a town that in 2003 was estimated to have 35000 people that improves by 2.4 percent every year describe a reasonable way to use a function to predict future population in Smithville. Now we are given initial population has 35000 numbers where a is equal to 35000 which is said that population increases by 2.4 percent every year. It means growth factor is 1 plus the whole where r is the total population 2.4 percent. So growth factor is equal to 1 plus 2.4 upon 100 which is equal to 1 plus 0.024 which is equal to 1.024. So growth factor is equal to 1.024. So b is equal to 1.024. Now we know that exponential function is what the form y is equal to a into b is to balance. So we will put a is equal to 35000 and b is equal to 1.024 and b is equal to a into b is to balance. So we have y is equal to 35000 into 1.024 whole is to balance. Now suppose we want to estimate the population in the year 2007. Now here the initial year is the year 2003. So time x from the year 2003 to the year 2007 is 4 years. Now let this be equation 1. Now we put x is equal to 4 in equation 1 and we have x is equal to 4 in equation 1. So y is equal to 35000 into 1.024 whole is to power 4 which is approximately equal to 38482.91 which is approximately equal to 38500. Thus in the year 2007 we have the population will be 38500. So in this session we have discussed how to construct exponential functions from a graph table or a description of a relationship and this completes our session. Hope you all have enjoyed the session.