Question

the data in the table represents the predicted price of a gallon of milk, y, for number of years, x. price of milk

years	price
1	$3.12
2	$3.24
3	$3.37
4	$3.51

which form of an equation was used to model the price of a gallon of milk?
○ ( y = ax + b )
○ ( y = ax^2 + b )
○ ( y = ab^x )
○ ( y = a + bx )

Explanation:

Step 1: Analyze linear model (\(y = ax + b\) or \(y=a + bx\))

For a linear model, the difference in \(y\)-values (price) should be constant for equal differences in \(x\)-values (years). Let's calculate the differences:

• From \(x = 0\) to \(x = 1\): \(3.12 - 3.00 = 0.12\)
• From \(x = 1\) to \(x = 2\): \(3.24 - 3.12 = 0.12\)
• From \(x = 2\) to \(x = 3\): \(3.37 - 3.24 = 0.13\) (not equal to 0.12)
• From \(x = 3\) to \(x = 4\): \(3.51 - 3.37 = 0.14\) (not equal to 0.12)

So, linear model is not appropriate.

Step 2: Analyze quadratic model (\(y = ax^2 + b\))

For a quadratic model, the second - differences should be constant. First, find the first differences (as above): \(0.12, 0.12, 0.13, 0.14\). Then find the second differences:

• \(0.12 - 0.12 = 0\)
• \(0.13 - 0.12 = 0.01\)
• \(0.14 - 0.13 = 0.01\)

Second differences are not constant, so quadratic model is not appropriate.

Step 3: Analyze exponential model (\(y=ab^x\))

Let's check the ratio of consecutive \(y\)-values:

• From \(x = 0\) to \(x = 1\): \(\frac{3.12}{3.00}=1.04\)
• From \(x = 1\) to \(x = 2\): \(\frac{3.24}{3.12}\approx1.038\approx1.04\)
• From \(x = 2\) to \(x = 3\): \(\frac{3.37}{3.24}\approx1.04\)
• From \(x = 3\) to \(x = 4\): \(\frac{3.51}{3.37}\approx1.04\)

The ratio of consecutive \(y\)-values is approximately constant (around 1.04), which is a characteristic of an exponential model of the form \(y = ab^x\) (where \(a = 3.00\) when \(x = 0\) and \(b\approx1.04\)).

Answer:

\( y = ab^x \)