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An exponential function is graphed in the xy-plane. If the graph of the function is always increasing and its y-intercept is (0,6), which of the following could be the equation of the function? Choose 1 answer: (A) y=6(0.5)^(x) (B) y=12(0.5)^(x) (C) y=6(2)^(x)+1 (D) y=4(2)^(x)+2

An exponential function is graphed in the xy x y -plane. If the graph of the function is always increasing and its y y -intercept is (0,6) (0,6) , which of the following could be the equation of the function?\newlineChoose 11 answer:\newline(A) y=6(0.5)x y=6(0.5)^{x} \newline(B) y=12(0.5)x y=12(0.5)^{x} \newline(C) y=6(2)x+1 y=6(2)^{x}+1 \newline(D) y=4(2)x+2 y=4(2)^{x}+2

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Q. An exponential function is graphed in the xy x y -plane. If the graph of the function is always increasing and its y y -intercept is (0,6) (0,6) , which of the following could be the equation of the function?\newlineChoose 11 answer:\newline(A) y=6(0.5)x y=6(0.5)^{x} \newline(B) y=12(0.5)x y=12(0.5)^{x} \newline(C) y=6(2)x+1 y=6(2)^{x}+1 \newline(D) y=4(2)x+2 y=4(2)^{x}+2
  1. Exponential Function Behavior: An exponential function of the form y=abxy = ab^x is always increasing if the base bb is greater than 11. This is because as xx increases, the value of bxb^x increases when b > 1.
  2. Finding Y-Intercept: The yy-intercept of a function is the value of yy when x=0x = 0. For an exponential function y=abxy = ab^x, the yy-intercept is found by setting xx to 00, which gives y=ab0=ay = ab^0 = a. Therefore, the yy-intercept is (0,a)(0, a).
  3. Determining Correct Function: Given that the y-intercept is (0,6)(0,6), we can determine that aa must be 66 for the correct function. We can now check the options to see which one has a y-intercept of 66 and a base greater than 11.
  4. Option (A) Analysis: Option (A) y=6(0.5)xy=6(0.5)^x has a base of 0.50.5, which is less than 11. This means the function is decreasing, not increasing. Therefore, option (A) is not the correct function.
  5. Option (B) Analysis: Option (B) y=12(0.5)xy=12(0.5)^x has a base of 0.50.5, which is less than 11. This means the function is decreasing, not increasing. Additionally, the yy-intercept is 1212, not 66. Therefore, option (B) is not the correct function.
  6. Option (C) Analysis: Option (C) y=6(2)x+1y=6(2)^x+1 has a base of 22, which is greater than 11, indicating an increasing function. However, when x=0x = 0, y=6(2)0+1=6(1)+1=7y = 6(2)^0 + 1 = 6(1) + 1 = 7, not 66. Therefore, option (C) does not have the correct yy-intercept.
  7. Option (D) Analysis: Option (D) y=4(2)x+2y=4(2)^x+2 has a base of 22, which is greater than 11, indicating an increasing function. However, when x=0x = 0, y=4(2)0+2=4(1)+2=6y = 4(2)^0 + 2 = 4(1) + 2 = 6. This function has the correct y-intercept of 66. Therefore, option (D) is the correct function.

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