To analyze the given function [tex]\( f(x) = -2 \cdot (0.5)^x \)[/tex], let's break it down step by step.
First, we identify the coefficients:
1. Coefficient [tex]\( a \)[/tex]:
In the standard exponential form [tex]\( f(x) = a \cdot b^x \)[/tex], [tex]\( a \)[/tex] is the coefficient in front of the base [tex]\( b \)[/tex]. Here, [tex]\( a = -2 \)[/tex].
2. Base [tex]\( b \)[/tex]:
The base [tex]\( b \)[/tex] is the value that is raised to the power of [tex]\( x \)[/tex]. Here, [tex]\( b = 0.5 \)[/tex].
Next, let's determine the y-intercept of the function:
3. Y-intercept:
The y-intercept is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex]. In other words, it's [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = -2 \cdot (0.5)^0 = -2 \cdot 1 = -2
\][/tex]
So the y-intercept is [tex]\( -2 \)[/tex].
Now, let's describe the end behavior of the function:
4. End Behavior as [tex]\( x \to +\infty \)[/tex]:
As [tex]\( x \)[/tex] increases without bound (i.e., [tex]\( x \to +\infty \)[/tex]), the term [tex]\( (0.5)^x \)[/tex] gets closer and closer to 0 because [tex]\( 0.5 \)[/tex] is a fraction (less than 1). Therefore:
[tex]\[
\lim_{{x \to +\infty}} f(x) = \lim_{{x \to +\infty}} -2 \cdot (0.5)^x = 0
\][/tex]
So, as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
5. End Behavior as [tex]\( x \to -\infty \)[/tex]:
As [tex]\( x \)[/tex] decreases without bound (i.e., [tex]\( x \to -\infty \)[/tex]), the term [tex]\( (0.5)^x \)[/tex] grows indefinitely large because the exponentiation of a fraction with a negative exponent leads to a larger result. Since [tex]\( a \)[/tex] is negative:
[tex]\[
\lim_{{x \to -\infty}} -2 \cdot (0.5)^x = -\infty
\][/tex]
So, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
In summary, the key details are:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 0.5 \)[/tex]
- The y-intercept is [tex]\( -2 \)[/tex]
- As [tex]\( x \to +\infty \)[/tex], [tex]\( y \to 0 \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex]
