The table below gives the values of the function for selected values of .
| 0 | 1 | 2 | 3 | |
|---|---|---|---|---|
| 5 | 2.5 | 1.25 | 0.625 |
Which of the following equations could define ?
When a function is given in a table, first decide if it looks linear (constant difference) or exponential (constant ratio). Compute ratios like f(1)/f(0), f(2)/f(1), etc.; if they are equal, write the model as f(x)=a·r^x, where a=f(0) and r is the ratio. Then match that model to the answer choices, using exponent rules if needed (for example, rewriting (1/2)^x as 2^{-x}). If you are unsure, quickly plug in one or two x-values from the table into each option and eliminate any equation that does not produce the correct f(x) values.
Hints
Check how f(x) changes as x increases
Look at the sequence of f(x) values: 5, 2.5, 1.25, 0.625. Are you adding or subtracting the same amount each time, or multiplying/dividing by the same amount each time?
Find the constant ratio
Compute f(1)/f(0), f(2)/f(1), and f(3)/f(2). Are these ratios the same number? What is that number?
Write an exponential expression
Once you know the starting value f(0) and the constant ratio r, write f(x) in the form f(x) = f(0)·r^x. Then think about how to express r using powers of 2 so it matches one of the answer choices.
Desmos Guide
Plot the data points from the table
In Desmos, create a table and enter the x-values 0, 1, 2, 3 in the first column and the corresponding f(x) values 5, 2.5, 1.25, 0.625 in the second column. This shows the points the correct function must pass through.
Enter each answer choice as a separate function
Type the four options as functions: y = 52^x, y = 52^(x-1), y = 52^(x+1), and y = 52^(-x). Use different colors if helpful so you can tell them apart.
Compare graphs or use a table for each function
For each function, either visually check whether its graph goes exactly through all four plotted points, or click the gear next to the function and make a table of x-values 0, 1, 2, 3 to see its outputs. The correct choice is the one whose outputs match 5, 2.5, 1.25, and 0.625 for those x-values.
Step-by-step Explanation
Look for the pattern in the table
Compare how f(x) changes as x increases by 1.
From x=0 to x=1, f(x) changes from 5 to 2.5. From x=1 to x=2, it changes from 2.5 to 1.25. From x=2 to x=3, it changes from 1.25 to 0.625.
Notice that each new value is half of the previous one:
- 2.5 = 5 ÷ 2
- 1.25 = 2.5 ÷ 2
- 0.625 = 1.25 ÷ 2
So each time x increases by 1, f(x) is multiplied by 1/2.
Express the exponential model
When a function repeatedly multiplies by the same number as x increases by 1, it is exponential.
Here:
- The starting value at x=0 is f(0)=5.
- The constant ratio is 1/2 (because each step multiplies by 1/2).
An exponential function with initial value 5 and constant ratio r has the form f(x)=5·r^x. In this case, r=1/2, which we will rewrite using base 2 in the next step.
Rewrite using base 2 and a negative exponent
Recall the exponent rule: for any nonzero a, (1/a)^x = a^{-x}.
Since 1/2 = 2^{-1}, it follows that (1/2)^x = 2^{-x}.
Form the function and match to the choices
Combining the starting value with the rewrite gives f(x)=5·2^{-x}.
Comparing with the answer options, this exactly matches choice D, so the correct answer is f(x)=5·2^{-x}.