Complete the table for each function. Then answer the questions that follow. [tex]a=[/tex] $\square$ [tex]b =[/tex] $\square$ [tex]c =[/tex] $\square$ [tex]d =[/tex] $\square$ [tex]e =[/tex] $\square$; [tex]f =[/tex] $\square$; [tex]g =[/tex] $\square$ | x | [tex]y _{ 1 }= 4 x[/tex] | [tex]y _{ 2 }= 4 x ^{ 2 }[/tex] | [tex]y _{ 3 }= 4 ^{ x }[/tex] | |---|---|---|---| | 0 | 0 | 0 | 1 | | 1 | 4 | a | b | | 2 | c | 16 | d | | 3 | e | f | g |
Answer (2)
Calculate a by substituting x = 1 into y 2 = 4 x 2 , which gives a = 4 ( 1 ) 2 = 4 .
Calculate b by substituting x = 1 into y 3 = 4 x , which gives b = 4 1 = 4 .
Calculate c by substituting x = 2 into y 1 = 4 x , which gives c = 4 ( 2 ) = 8 .
Calculate d , e , f , g similarly, and the final answers are: a = 4 , b = 4 , c = 8 , d = 16 , e = 12 , f = 36 , g = 64 .
Explanation
Understanding the Problem We are given three functions: y 1 = 4 x , y 2 = 4 x 2 , and y 3 = 4 x . We need to complete the table by finding the values of a , b , c , d , e , f , g by evaluating the functions at the given x values.
Calculating a To find the value of a , we substitute x = 1 into the function y 2 = 4 x 2 . Thus, we have a = 4 ( 1 ) 2 = 4 ( 1 ) = 4. So, a = 4 .
Calculating b To find the value of b , we substitute x = 1 into the function y 3 = 4 x . Thus, we have b = 4 1 = 4. So, b = 4 .
Calculating c To find the value of c , we substitute x = 2 into the function y 1 = 4 x . Thus, we have c = 4 ( 2 ) = 8. So, c = 8 .
Calculating d To find the value of d , we substitute x = 2 into the function y 3 = 4 x . Thus, we have d = 4 2 = 16. So, d = 16 .
Calculating e To find the value of e , we substitute x = 3 into the function y 1 = 4 x . Thus, we have e = 4 ( 3 ) = 12. So, e = 12 .
Calculating f To find the value of f , we substitute x = 3 into the function y 2 = 4 x 2 . Thus, we have f = 4 ( 3 ) 2 = 4 ( 9 ) = 36. So, f = 36 .
Calculating g To find the value of g , we substitute x = 3 into the function y 3 = 4 x . Thus, we have g = 4 3 = 4 × 4 × 4 = 64. So, g = 64 .
Final Answer Therefore, the completed table has the following values: a = 4 , b = 4 , c = 8 , d = 16 , e = 12 , f = 36 , and g = 64 .
Examples
Understanding different types of functions (linear, quadratic, and exponential) and how they change as the input variable changes is crucial in many real-world applications. For example, in physics, the distance traveled by an object moving at a constant speed is a linear function of time. The area of a circle is a quadratic function of its radius. The growth of a population can be modeled by an exponential function. By understanding these functions, we can make predictions and solve problems in various fields.
We calculated the values of the functions at different x values to complete the table. The final values obtained are: a=4, b=4, c=8, d=16, e=12, f=36, and g=64. Each value corresponds to the results of evaluating each function for the specified x values.
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