Question

fill out the table for each equation provided.

1. $y = 3^{(x+4)}-1$
2. $y = - 8(0.5)^{(x+2)}+7$

	growth or decay?	horizontal asymptote	transformations	y-int.	domain	range	as $x \to - \infty$, $f(x) \to$___	as $x \to + \infty$, $f(x) \to$___
6. $y = - 8(0.5)^{(x+2)}+7$

Explanation:

For Equation 5: $y = 3^{(x+4)} - 1$

Step1: Identify Growth/Decay

Base $3>1$, so it is growth.

Step2: Find Horizontal Asymptote

For $y = a \cdot b^{x-h} + k$, asymptote is $y=k$. Here $k=-1$, so $y=-1$.

Step3: List Transformations

Compare to $y=3^x$: shift left 4, down 1.

Step4: Calculate y-intercept

Set $x=0$: $y=3^{(0+4)}-1=3^4 -1=81-1=80$.

Step5: Determine Domain

Exponential functions have all real $x$: $(-\infty, \infty)$.

Step6: Determine Range

$3^{(x+4)}>0$, so $3^{(x+4)}-1 > -1$: $(-1, \infty)$.

Step7: End Behavior as $x\to-\infty$

$3^{(x+4)}\to0$, so $y\to0-1=-1$.

Step8: End Behavior as $x\to+\infty$

$3^{(x+4)}\to+\infty$, so $y\to+\infty$.

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For Equation 6: $y = -8(0.5)^{(x+2)} + 7$

Step1: Identify Growth/Decay

Base $0<0.5<1$, so it is decay.

Step2: Find Horizontal Asymptote

For $y = a \cdot b^{x-h} + k$, asymptote is $y=k$. Here $k=7$, so $y=7$.

Step3: List Transformations

Compare to $y=0.5^x$: reflect over x-axis, vertical stretch by 8, shift left 2, up7.

Step4: Calculate y-intercept

Set $x=0$: $y=-8(0.5)^{(0+2)}+7=-8(0.25)+7=-2+7=5$.

Step5: Determine Domain

Exponential functions have all real $x$: $(-\infty, \infty)$.

Step6: Determine Range

$(0.5)^{(x+2)}>0$, so $-8(0.5)^{(x+2)}<0$, so $-8(0.5)^{(x+2)}+7 <7$: $(-\infty, 7)$.

Step7: End Behavior as $x\to-\infty$

$(0.5)^{(x+2)}\to+\infty$, so $y\to-8(+\infty)+7\to-\infty$.

Step8: End Behavior as $x\to+\infty$

$(0.5)^{(x+2)}\to0$, so $y\to-8(0)+7=7$.

Answer:

	$y = 3^{(x+4)} - 1$	$y = -8(0.5)^{(x+2)} + 7$
Horizontal Asymptote	$y=-1$	$y=7$
Transformations	Left 4, down 1 (from $y=3^x$)	Reflect over x-axis, stretch by 8, left 2, up7 (from $y=0.5^x$)
y-int.	$80$	$5$
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$(-1, \infty)$	$(-\infty, 7)$
As $x \to -\infty$, $f(x)\to$	$-1$	$-\infty$
As $x \to +\infty$, $f(x)\to$	$+\infty$	$7$