The graph of $y=12(0.25)^{x}+1$ is shown 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 $x$ becomes very large

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Algebra 2

Conjugate root theorems

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Q. The graph of

y=12(0.25)^{x}+1

is shown 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

becomes very large

Identify Characteristics: We need to identify which characteristic of the graph is directly represented in the equation $y=12(0.25)^{x}+1$ . Let's start by examining the equation and its components.
Find Y-Intercept: The y-intercept of a graph is the value of y when x is $0$ . To find the y-intercept from the equation, we substitute x with $0$ and solve for y. $\newline$ $y = 12(0.25)^{0} + 1$ $\newline$ $y = 12(1) + 1$ $\newline$ $y = 12 + 1$ $\newline$ $y = 13$ $\newline$ The y-intercept is $13$ , which is displayed in the equation as the sum of the constant term $1$ and the coefficient $12$ multiplied by $(0.25)^0$ .
No X-Intercept: The $x$ -intercept is the value of $x$ when $y$ is $0$ . However, in the equation $y=12(0.25)^{x}+1$ , there is no $x$ -intercept represented because the term $+1$ ensures that $y$ is never $0$ for any real value of $x$ .
No Slope Represented: The slope of a graph is represented by the coefficient of $x$ in a linear equation, which is of the form $y = mx + b$ . However, the given equation is not linear; it is an exponential decay function. Therefore, there is no slope represented as a constant or coefficient in this equation.
Horizontal Asymptote: The value $y$ approaches as $x$ becomes very large is known as the horizontal asymptote. In the equation $y=12(0.25)^{x}+1$ , as $x$ becomes very large, the term $(0.25)^{x}$ approaches $0$ , and the value of $y$ approaches the constant term $+1$ . This constant term represents the horizontal asymptote of the graph.