Consider this linear function:
y = -2z - 5.
Plot all of the ordered pairs for the values in the domain.
D: (-6, -5, -2, 0, 1)
Answer (2)
Calculate the y-values for each z in the domain using the function y = − 2 z − 5 .
Determine the ordered pairs: ( − 6 , 7 ) , ( − 5 , 5 ) , ( − 2 , − 1 ) , ( 0 , − 5 ) , ( 1 , − 7 ) .
Plot the ordered pairs on the coordinate plane.
The ordered pairs are ( − 6 , 7 ) , ( − 5 , 5 ) , ( − 2 , − 1 ) , ( 0 , − 5 ) , ( 1 , − 7 ) .
Explanation
Understanding the Problem We are given the linear function y = − 2 z − 5 and the domain D = { − 6 , − 5 , − 2 , 0 , 1 } . Our goal is to find the corresponding y values for each z in the domain and plot the ordered pairs ( z , y ) .
Calculating y values For each z in the domain, we will calculate the corresponding y value using the function y = − 2 z − 5 .
Finding the Ordered Pairs
When z = − 6 , we have y = − 2 ( − 6 ) − 5 = 12 − 5 = 7 . So the ordered pair is ( − 6 , 7 ) .
When z = − 5 , we have y = − 2 ( − 5 ) − 5 = 10 − 5 = 5 . So the ordered pair is ( − 5 , 5 ) .
When z = − 2 , we have y = − 2 ( − 2 ) − 5 = 4 − 5 = − 1 . So the ordered pair is ( − 2 , − 1 ) .
When z = 0 , we have y = − 2 ( 0 ) − 5 = 0 − 5 = − 5 . So the ordered pair is ( 0 , − 5 ) .
When z = 1 , we have y = − 2 ( 1 ) − 5 = − 2 − 5 = − 7 . So the ordered pair is ( 1 , − 7 ) .
Plotting the Points Therefore, the ordered pairs are ( − 6 , 7 ) , ( − 5 , 5 ) , ( − 2 , − 1 ) , ( 0 , − 5 ) , and ( 1 , − 7 ) . We plot these points on the coordinate plane.
Final Answer The ordered pairs for the given domain are: ( − 6 , 7 ) , ( − 5 , 5 ) , ( − 2 , − 1 ) , ( 0 , − 5 ) , ( 1 , − 7 ) These points can now be plotted on the coordinate plane.
Examples
Understanding linear functions and plotting points is crucial in many real-world applications. For instance, if you're tracking the depreciation of an asset over time, you might model the asset's value as a linear function of time. By plotting points from this function, you can visualize how the asset's value decreases and make informed decisions about when to replace it. Similarly, in physics, understanding linear relationships helps in analyzing motion, such as the constant speed of an object, and predicting its position at different times. This skill is also valuable in economics for modeling supply and demand curves, where the intersection of these lines helps determine market equilibrium.
To find the ordered pairs for the function y = − 2 z − 5 based on the domain D = { − 6 , − 5 , − 2 , 0 , 1 } , we calculated corresponding y-values resulting in pairs: ( − 6 , 7 ) , ( − 5 , 5 ) , ( − 2 , − 1 ) , ( 0 , − 5 ) , ( 1 , − 7 ) . These points can be plotted on a coordinate plane to visually analyze the function's behavior. Each ordered pair illustrates the linear relationship defined by the equation.
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