Answered

Set up a table of $x , y$ pairs that fits the following linear function. For your convenience, x values have been selected. Just calculate the corresponding y-values, plot the points on a Cartesian coordinate system, and draw the line.



$y=2 x-1$




x		y

-2		A

-1		B

0		C

1		D

2		E






Choose the correct sequence of $y$-values ($A, B, C, D, E$) for the table shown below. Graph the line on your own paper.


A-2, -1, 0, 1, 2
B0, 1, 2, 3, 4
C-4, -3, -2, -1, 0
D$-5,-3,-1,1,3$

Verified Answer :

  • Substitute each \(x\)-value (\(-2, -1, 0, 1, 2\)) into the linear function \(y = 2x - 1\).
  • Calculate the resulting \(y\)-values: \(-5, -3, -1, 1, 3\).
  • Match the calculated sequence to choice (d).
  • The final answer is \(\boxed{\text{d}}\).

Explanation

  1. Problem Analysis

We are given the linear function \(y = 2x - 1\) and a set of input \(x\)-values: \(-2, -1, 0, 1, 2\). We need to determine the corresponding \(y\)-values \(A, B, C, D, E\) to complete the table and identify the correct sequence from the provided options. This involves systematic substitution of each \(x\)-value into the given linear equation.

  1. Calculations and Selection

We calculate the corresponding \(y\)-values for each \(x\) in the table:

  • For \(x = -2\), \(y = 2(-2) - 1 = -4 - 1 = -5\). So, \(A = -5\).
  • For \(x = -1\), \(y = 2(-1) - 1 = -2 - 1 = -3\). So, \(B = -3\).
  • For \(x = 0\), \(y = 2(0) - 1 = 0 - 1 = -1\). So, \(C = -1\).
  • For \(x = 1\), \(y = 2(1) - 1 = 2 - 1 = 1\). So, \(D = 1\).
  • For \(x = 2\), \(y = 2(2) - 1 = 4 - 1 = 3\). So, \(E = 3\).

The resulting sequence of \(y\)-values \((A, B, C, D, E)\) is \(-5, -3, -1, 1, 3\). Comparing this result with the given options, we find that choice (d) corresponds to our findings.

Examples

Linear functions are used in everyday life to model relationships that change at a constant rate. For example, if you are renting a bike, the cost might be a flat base fee plus a hourly rate. If the base fee is \(-1\) (like a discount) and the hourly rate is \(2\) dollars, the total cost \(y\) for \(x\) hours could be modeled by \(y = 2x - 1\). By calculating values for different hours, you can create a table to help budget your expenses efficiently. This simple linear model helps in planning and resource allocation.

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