3) [tex]y=-3(4)^x ; \quad x=2[/tex]
Answer (1)
Substitute x = 2 into the equation y = − 3 ( 4 ) x .
Calculate 4 2 = 16 .
Multiply − 3 by 16 to get y = − 3 ( 16 ) = − 48 .
The final answer is − 48 .
Explanation
Understanding the Problem We are given the equation y = − 3 ( 4 ) x and asked to find the value of y when x = 2 .
Substituting the Value of x To find the value of y , we substitute x = 2 into the equation: y = − 3 ( 4 ) 2
Calculating the Power Now, we calculate 4 2 , which is 4 × 4 = 16 . So the equation becomes: y = − 3 ( 16 )
Finding the Value of y Finally, we multiply − 3 by 16 :
y = − 48
Final Answer Therefore, when x = 2 , the value of y is − 48 .
Examples
Understanding exponential functions is crucial in many real-world scenarios, such as modeling population growth or radioactive decay. For instance, if a bacterial population doubles every hour, we can use an exponential function to predict its size at any given time. Similarly, in finance, compound interest calculations rely on exponential growth to determine the future value of an investment.
