If $y=15(1.08)^{12 x}$ is graphed in the $x y$ -plane, which of the following characteristics of the graph is displayed as a constant or coefficient in the equation? $\newline$ Choose $1$ answer: $\newline$ (A) $x$ -intercept $\newline$ (B) $y$ -intercept $\newline$ (C) Slope $\newline$ (D) The value $y$ approaches as $x$ increases

View step-by-step help

Home

Math Problems

Algebra 2

Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q. If

y=15(1.08)^{12 x}

is graphed in the

x y

-plane, which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?

\newline

Choose

1

answer:

\newline

(A)

x

-intercept

\newline

(B)

y

-intercept

\newline

\newline

(D) The value

y

approaches as

x

increases

Identifying Constants and Coefficients: We need to identify which characteristic of the graph is represented by a constant or coefficient in the given equation $y=15(1.08)^{12x}$ . Let's analyze the equation.
Meaning of Constant 'a': The equation is in the form $y=a(b)^{cx}$ , where $a$ , $b$ , and $c$ are constants. In this case, $a=15$ , $b=1.08$ , and $c=12$ . We need to determine what each of these constants represents in the graph.
Meaning of Constant 'b': The constant ' $a$ ' (which is $15$ in this equation) represents the initial value of $y$ when $x$ is $0$ . This is because anything raised to the power of $0$ is $1$ , so $y=a(1)$ when $x=0$ , which simplifies to $y=a$ . Therefore, the constant ' $a$ ' represents the $y$ -intercept of the graph.
Meaning of Constant 'c': The constant ' $b$ ' (which is $1.08$ in this equation) represents the base of the exponential function. It affects the rate of growth or decay of the graph but does not directly represent a characteristic like the $y$ -intercept, $x$ -intercept, or slope.
X-Intercept and Exponential Functions: The constant $c$ (which is $12$ in this equation) is the coefficient of $x$ in the exponent. It affects the rate at which the function grows or decays, but like $b$ , it does not directly represent a characteristic like the y-intercept, x-intercept, or slope.
Slope of an Exponential Function: The $x$ -intercept is the value of $x$ when $y=0$ . Since an exponential function like this one never touches the $x$ -axis (assuming $b>1$ , which it is in this case), the $x$ -intercept is not represented by any constant or coefficient in the equation.
Horizontal Asymptote and Exponential Growth: The slope of an exponential function is not constant; it changes at every point on the graph. Therefore, it is not represented by a single constant or coefficient in the equation.
Horizontal Asymptote and Exponential Growth: The slope of an exponential function is not constant; it changes at every point on the graph. Therefore, it is not represented by a single constant or coefficient in the equation.The value $y$ approaches as $x$ increases is known as the horizontal asymptote. For exponential growth functions where $b>1$ , $y$ approaches infinity as $x$ increases, and this behavior is not represented by a specific constant or coefficient in the equation.