Complete the table column by column:
$x$ | $-2 x$ | $y=-2 x+3$ |
-3 | ||
-2 | ||
-1 | ||
0 | ||
1 | ||
2 | ||
3 | ||
The goal is to complete a table of values for two linear functions based on a given set of input values for \(x\). We are given the input set \(x \in \{-3, -2, -1, 0, 1, 2, 3\}\) and the two functions:
We will proceed by calculating the output for each function for every value of \(x\). Notice that the second function is simply the first function plus a constant: \(g(x) = f(x) + 3\). This relationship will make our calculations straightforward.
We perform the calculations for each \(x\):
We now compile our results into the requested table:
| \(x\) | \(-2x\) | \(y = -2x + 3\) \) -3 | 6 | 9 \) -2 | 4 | 7 \) -1 | 2 | 5 \) 0 | 0 | 3 \) 1 | -2 | 1 \) 2 | -4 | -1 \) 3 | -6 | -3 \) |
This completes the table as required.
Linear functions are fundamental in modeling real-world changes. For instance, imagine a company that charges a flat fee of \(\$3\) to start a rental, plus a rate that depends on the time used, represented by a function like \(y = -2x + 3\). If \(x\) represents hours, this function might model a scenario where the price decreases over time due to a specific rebate program, allowing you to easily calculate the total cost for any given duration. Mastery of these tables helps in predicting outcomes and understanding how variables relate to each other in everyday financial planning.