When analyzing how the graph of the function [tex]\( f(x) = 10(2)^x \)[/tex] changes if the base [tex]\( b \)[/tex] is decreased but remains greater than 1, we note the following points:
1. The graph will begin at a lower point on the [tex]\( y \)[/tex]-axis:
- False. The initial point of the graph, [tex]\( f(0) = 10 \cdot (2)^0 = 10 \)[/tex], remains unchanged regardless of the value of [tex]\( b \)[/tex]. This is because any number raised to the power of 0 is 1, and hence multiplying it by 10 gives the same starting point. The initial [tex]\( y \)[/tex]-value only depends on the coefficient, which remains 10.
2. The graph will increase at a faster rate:
- False. If [tex]\( b \)[/tex] is decreased but still greater than 1, the rate at which the function grows becomes slower. This is because the exponential growth factor is less than it was when [tex]\( b \)[/tex] was 2.
3. The graph will increase at a slower rate:
- True. As [tex]\( b \)[/tex] decreases while remaining greater than 1, the rate of exponential growth decreases. This means the graph will rise more gradually compared to when the base was 2.
4. The [tex]\( y \)[/tex]-values will continue to increase as [tex]\( x \)[/tex]-increases:
- True. Since [tex]\( b \)[/tex] is still greater than 1, [tex]\( b^x \)[/tex] will continue to increase as [tex]\( x \)[/tex] increases. Thus, [tex]\( f(x) \)[/tex] will still rise as [tex]\( x \)[/tex] becomes larger, ensuring that the [tex]\( y \)[/tex]-values increase.
5. The [tex]\( y \)[/tex]-values will each be less than their corresponding [tex]\( x \)[/tex]-values:
- False. For an exponential function [tex]\( f(x) = 10 \cdot b^x \)[/tex], the [tex]\( y \)[/tex]-values grow exponentially. Even with a smaller base [tex]\( b \)[/tex] (greater than 1), [tex]\( y \)[/tex]-values are typically much larger than their corresponding [tex]\( x \)[/tex]-values, especially as [tex]\( x \)[/tex] increases.
Therefore, the correct statements are:
- The graph will increase at a slower rate.
- The [tex]\( y \)[/tex]-values will continue to increase as [tex]\( x \)[/tex]-increases.
Thus, we check statements 3 and 4.
