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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 $x y$ -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? $\newline$ Choose $1$ answer: $\newline$ (A) $y=6(0.5)^{x}$ $\newline$ (B) $y=12(0.5)^{x}$ $\newline$ (C) $y=6(2)^{x}+1$ $\newline$ (D) $y=4(2)^{x}+2$

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Compare linear and exponential growth

Full solution

Q. An exponential function is graphed in the

x y

-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?

\newline

Choose

1

answer:

\newline

(A)

y=6(0.5)^{x}

\newline

(B)

y=12(0.5)^{x}

\newline

(C)

y=6(2)^{x}+1

\newline

(D)

y=4(2)^{x}+2

Exponential Function Behavior: An exponential function of the form $y = ab^x$ is always increasing if the base $b$ is greater than $1$ . This is because as $x$ increases, the value of $b^x$ increases when b > 1.
Finding Y-Intercept: The $y$ -intercept of a function is the value of $y$ when $x = 0$ . For an exponential function $y = ab^x$ , the $y$ -intercept is found by setting $x$ to $0$ , which gives $y = ab^0 = a$ . Therefore, the $y$ -intercept is $(0, a)$ .
Determining Correct Function: Given that the y-intercept is $(0,6)$ , we can determine that $a$ must be $6$ for the correct function. We can now check the options to see which one has a y-intercept of $6$ and a base greater than $1$ .
Option (A) Analysis: Option (A) $y=6(0.5)^x$ has a base of $0.5$ , which is less than $1$ . This means the function is decreasing, not increasing. Therefore, option (A) is not the correct function.
Option (B) Analysis: Option (B) $y=12(0.5)^x$ has a base of $0.5$ , which is less than $1$ . This means the function is decreasing, not increasing. Additionally, the $y$ -intercept is $12$ , not $6$ . Therefore, option (B) is not the correct function.
Option (C) Analysis: Option (C) $y=6(2)^x+1$ has a base of $2$ , which is greater than $1$ , indicating an increasing function. However, when $x = 0$ , $y = 6(2)^0 + 1 = 6(1) + 1 = 7$ , not $6$ . Therefore, option (C) does not have the correct $y$ -intercept.
Option (D) Analysis: Option (D) $y=4(2)^x+2$ has a base of $2$ , which is greater than $1$ , indicating an increasing function. However, when $x = 0$ , $y = 4(2)^0 + 2 = 4(1) + 2 = 6$ . This function has the correct y-intercept of $6$ . Therefore, option (D) is the correct function.

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