Answer:
Since graph D is experiencing exponential growth (meaning its b value is greater than 1) and is steeper than graphs E and F (which also experience exponential growth and have b values greater than 1), it has the largest b value.
Explanation:
- It's clear from the shapes of the 6 graphs that all 6 are exponential functions, whose general form is given by:
y = a(b)^x, where
- (x, y) is any point on the exponential curve,
- a is the initial value (value of y when x = 0),
- and b is the base.
There are two facts we must remember about the base, which determines the shape of the exponential curve:
- When 0 < b < 1, the graph experiences exponential growth and y decreases as x increases.
- When b > 1, the graph experiences exponential growth and y increases as x increases.
For curves A, B, and C, y increases as x increases, so we know that the b values of these three curves must be between 0 and 1 (i.e. a decimal).
Thus, they can't have the largest b value out of all the graphs.
For curves D, E, and F, y increases as x increases, so we know that the b values of these three curves must be greater than 1.
Thus, one of these three graphs has the largest b value out of all the graphs.
For any exponential graph with exponential growth (meaning b > 1), a larger b value will result in a steeper curve, meaning that:
- bigger b values result in the curve being closer to the y-axis, as the curve becomes more vertical/steep;
- smaller b values result in the curve being closer to the x-axis, as the curve becomes more horizontal/flat.
Since graph D is experiencing exponential growth (meaning its b value is greater than 1) and is steeper than graphs E and F (which also experience exponential growth and have b values greater than 1), it has the largest b value.