A graging calcilator is recimmended. Graph the given functions on a cominos wereen. Eoplain how these grathes are relited. y=2
2
,y=e
*
,y=1
*
,y=20
*
M it these ereits appruecti men the hecion increases for x>0
The Correct Answer and Explanation is :
It looks like there are some typos in your question. Based on what I can understand, you are asking about graphing the given functions:
- ( y = 2^x )
- ( y = e^x )
- ( y = 1^x )
- ( y = 20^x )
These functions are exponential functions of the form ( y = a^x ), where the base ( a ) determines the rate of growth for ( x > 0 ). Let’s analyze and compare them.
Step 1: Graphing the Functions
I’ll plot these functions on a common graph and display them for better visualization.
Step 2: Explanation of the Graphs and Their Relationships
- Growth Rate Differences:
- The function ( y = 2^x ) grows at a moderate rate.
- The function ( y = e^x ) (where ( e \approx 2.718 )) grows slightly faster than ( 2^x ) because ( e ) is a larger base.
- The function ( y = 20^x ) grows extremely fast since the base is significantly larger.
- The function ( y = 1^x ) remains constant at 1 for all values of ( x ), meaning it has no growth.
- Behavior for ( x > 0 ):
- As ( x ) increases, all exponential functions (except ( 1^x )) increase, but at different rates.
- The larger the base, the faster the function grows. ( 20^x ) skyrockets, while ( 2^x ) grows much slower.
- Behavior for ( x < 0 ):
- ( y = 2^x ), ( y = e^x ), and ( y = 20^x ) approach zero but never reach it, forming asymptotes along the x-axis.
- The function ( y = 1^x ) remains at 1 always, as ( 1^x = 1 ) for all ( x ).
- Common Properties:
- All exponential functions pass through ( (0,1) ), since any nonzero base raised to the power of 0 equals 1.
- They exhibit exponential growth for ( x > 0 ) and approach zero for ( x < 0 ).
Conclusion
Exponential functions show different rates of growth depending on their base. A higher base results in faster growth. The function ( y = 1^x ) is an exception because it is always equal to 1. As ( x ) increases, the function with the largest base (in this case, ( 20^x )) grows the fastest, making exponential functions crucial in modeling real-world scenarios like population growth and compound interest.