Find the rule for these tables of values:（x,y) (-2,6)(-1,4)(0,2)(1,0)(2,-2)

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Let's examine this table of coordinate pairs and find the mathematical rule that connects each x-value to its corresponding y-value. When we say 'find the rule,' we're looking for a function or equation that describes the relationship between these variables. This systematic approach will help us understand the pattern hidden in the data. Now let's analyze the pattern systematically. Looking at our table, we can calculate the differences between consecutive y-values. From 6 to 4, that's negative 2. From 4 to 2, again negative 2. From 2 to 0, negative 2. And from 0 to negative 2, that's also negative 2. Notice that the y-values decrease by exactly 2 each time x increases by 1. This constant difference is a key indicator of a linear relationship. Now that we've identified the constant difference, we can recognize this as a linear relationship. Linear functions follow the slope-intercept form y equals mx plus b, where m is the slope and b is the y-intercept. Our slope m equals negative 2, which is the constant rate of change we found. To find the y-intercept, we look at the point where x equals 0. From our table, when x is 0, y is 2, so b equals 2. Therefore, our rule is y equals negative 2x plus 2. Now let's verify our rule by testing it with each given point. For the point negative 2, 6: y equals negative 2 times negative 2 plus 2, which gives us 4 plus 2 equals 6. Check! For negative 1, 4: y equals negative 2 times negative 1 plus 2, which is 2 plus 2 equals 4. Check! For 0, 2: y equals negative 2 times 0 plus 2, which is 0 plus 2 equals 2. Check! For 1, 0: y equals negative 2 times 1 plus 2, which is negative 2 plus 2 equals 0. Check! And finally for 2, negative 2: y equals negative 2 times 2 plus 2, which is negative 4 plus 2 equals negative 2. Check! All points satisfy our rule, confirming that y equals negative 2x plus 2 is indeed the correct solution. Now let's visualize our solution on a coordinate plane. Here we can see all five given points plotted, and the line y equals negative 2x plus 2 passes perfectly through each point. The graph clearly shows the key features of our linear function: it has a negative slope of negative 2, meaning it decreases from left to right. The line crosses the y-axis at the point 0, 2, which is our y-intercept. It also crosses the x-axis at the point 1, 0, which is the x-intercept. This visual representation confirms our algebraic solution and helps us understand the relationship between the variables.