Find the rule for these tables of values:（x,y) (-2,-10)(2,6)(6,22)(10,38)
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Let's examine this table of coordinate pairs and find the mathematical rule that connects the x and y values. We have four data points: negative 2 comma negative 10, 2 comma 6, 6 comma 22, and 10 comma 38. Our goal is to discover the function that can predict any y value given an x value.
Now let's analyze the differences between consecutive values. Looking at the x-values, we see that each increase by 4: from negative 2 to 2 is 4, from 2 to 6 is 4, and from 6 to 10 is 4. Similarly, the y-values increase by 16 each time: from negative 10 to 6 is 16, from 6 to 22 is 16, and from 22 to 38 is 16. These constant differences tell us we have a linear relationship.
Now let's calculate the slope using the formula m equals y2 minus y1 over x2 minus x1. Using the first two points, we get m equals 6 minus negative 10 over 2 minus negative 2, which equals 16 over 4, giving us a slope of 4. Let's verify this with other point pairs. Using points 2 and 3, we get 22 minus 6 over 6 minus 2, which is 16 over 4, again equals 4. Using points 3 and 4 gives us the same result. The slope is consistently 4, meaning y increases by 4 units for every 1-unit increase in x.
Now we need to find the y-intercept using the point-slope form y equals mx plus b. We know the slope m is 4, so our equation becomes y equals 4x plus b. Let's use the point (2, 6) to find b. Substituting: 6 equals 4 times 2 plus b, which gives us 6 equals 8 plus b, so b equals negative 2. Let's verify this with another point. Using (-2, -10): negative 10 equals 4 times negative 2 plus negative 2, which equals negative 8 plus negative 2, giving us negative 10. This confirms our y-intercept is negative 2.
Our complete rule is y equals 4x minus 2. Let's verify this works for all our data points. For x equals negative 2, we get y equals 4 times negative 2 minus 2, which equals negative 10. For x equals 2, we get 6. For x equals 6, we get 22. And for x equals 10, we get 38. All values match perfectly! This linear function can now predict the y-value for any given x-value, not just those in our original table.