Answer:
- first difference: -0.5
- linear
Explanation:
You want to use the ideas of common difference and common ratio to determine whether the given table represents a linear, quadratic, or exponential function.
First differences
When the x-values are evenly spaced, it is often useful to look at first differences of y-values.
Here, the x-values increase by 1 from any row of the table to the next. They are evenly spaced.
The first differences are the differences between a y-value and the one on the previous row:
3 -3.5 = -0.5
2.5 -3 = -0.5
2 -2.5 = -0.5
1.5 -2 = -0.5
Here, the first differences have the constant value -0.5. That means the function is linear.
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Additional comment
Sometimes, the first differences are not constant. Sometimes, they have a constant difference. That difference is called the second difference. When the second differences are not constant, you can compute the differences of those, called the third difference.
The degree of the polynomial relation describing the table will be the same as the index of the constant differences. That is, constant 2nd differences will indicate a 2nd-degree (quadratic) polynomial describes the relation.
When the table values and/or the differences have a common ratio, it indicates the function is exponential. The common ratio is found the same way the common difference is: compute the ratio of each term to the one before.
In this table, the ratios are 3/3.5 = 6/7; 2.5/3 = 5/6, ratios that are not constant. If the table were described by 3·2^x, its values would be ...
(x, y) = (0, 3), (1, 6), (2, 12), (3, 24)
The ratios would be 6/3 = 2; 12/6 = 2; 24/12 = 2. That is, the common ratio would be 2, the same as the base of the exponential term. The ratio can be positive or negative, with magnitude greater than 1 or less than 1. It will never be 0 or 1.
Advanced idea
In very rare cases, the ratios will not be constant, but the ratios of first differences will be. This indicates the exponential function has an added constant, such as 1+3·2^x. This gives the sequence for x=0, 1, 2, ... having values y = 4, 7, 13, 25. First differences are 3, 6, 12, which have a common ratio of 2.