To find the exponential model of best fit for the given points, we can use the general form of an exponential function: y = ab^x, where a is the initial value and b is the base.
Step 1: Identify the initial value, a.
Looking at the given points, we can see that when x = -3, y = 5. This means that (x, y) = (-3, 5) is of the form (x, y) = (-3, a * b^-3). Therefore, a * b^-3 = 5. We will use this equation later to solve for a.
Step 2: Identify the base, b.
To find the base, we can compare the ratios of consecutive y-values to see the common ratio. Considering the points (-3, 5), (1, 12), (5, 72), and (7, 137), let's check the ratios:
12/5 = 2.4
72/12 = 6
137/72 ≈ 1.9028
These ratios are not exactly equal, so we can assume some error in the data or rounding. However, they are quite similar, suggesting that the base might be approximately 2. Thus, we try setting b = 2 and see if the exponential model fits the data well.
Step 3: Find the value of a.
We can substitute the known values of x and y from the initial point (-3, 5) into the equation a * b^-3 = 5:
a * (2)^(-3) = 5
a * (1/8) = 5
a = 5 * 8
a = 40
Step 4: Write the exponential model.
Using the values of a = 40 and b = 2, the exponential model is:
y = 40 * 2^x
Rounding the values to the nearest hundredth, the exponential model of best fit for the given points is:
y = 40 * 2^x
Find the exponential model of best fit for the points (-3, 5), (1,12), (5, 72), (7,137) Explain how you got your answer. Round values to the nearest hundredth
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