For each of the graphs below, select which one of the following functions is depicted by the graph, as well as the equation of the graph's horizontal asymptote.
f
(
x
)
=
3
x
f(x)=3 
x
 
,
,
g
(
x
)
=
3
x
+
4
g(x)=3 
x
 +4
,
,
h
(
x
)
=
3
x
+
4
h(x)=3 
x+4
 
,
,
m
(
x
)
=
−
(
3
x
)
+
4
m(x)=−(3 
x
 )+4
,
,
q
(
x
)
=
3
−
x
−
4
q(x)=3 
−x
 −4
,
,
r
(
x
)
=
−
(
3
x
)
r(x)=−(3 
x
 )

To solve this problem, you need to determine which function corresponds to each graph, as well as identify the horizontal asymptote for each.

### Step 1: Analyze the given functions
Let's look at each function and describe its behavior:

1. **\( f(x) = 3^x \):**
   - This is an **exponential growth** function.
   - As \( x \to \infty \), \( f(x) \to \infty \).
   - As \( x \to -\infty \), \( f(x) \to 0 \).
   - **Horizontal asymptote:** \( y = 0 \).

2. **\( g(x) = 3^x + 4 \):**
   - This is an **exponential growth** function shifted **up by 4**.
   - As \( x \to \infty \), \( g(x) \to \infty \).
   - As \( x \to -\infty \), \( g(x) \to 4 \).
   - **Horizontal asymptote:** \( y = 4 \).

3. **\( h(x) = 3^{x+4} \):**
   - This is an **exponential growth** function shifted **horizontally**.
   - The horizontal shift is by \( -4 \), but the overall shape and asymptote remain similar to \( 3^x \).
   - **Horizontal asymptote:** \( y = 0 \).

4. **\( m(x) = -(3^x) + 4 \):**
   - This is a **reflected** version of \( 3^x \), reflected over the x-axis, shifted **up by 4**.
   - As \( x \to \infty \), \( m(x) \to 4 \).
   - As \( x \to -\infty \), \( m(x) \to -\infty \).
   - **Horizontal asymptote:** \( y = 4 \).

5. **\( q(x) = 3^{-x} - 4 \):**
   - This is an **exponential decay** function shifted **down by 4**.
   - As \( x \to \infty \), \( q(x) \to -4 \).
   - As \( x \to -\infty \), \( q(x) \to \infty \).
   - **Horizontal asymptote:** \( y = -4 \).

6. **\( r(x) = -(3^x) \):**
   - This is a **reflected** version of \( 3^x \), reflected over the x-axis.
   - As \( x \to \infty \), \( r(x) \to 0 \).
   - As \( x \to -\infty \), \( r(x) \to -\infty \).
   - **Horizontal asymptote:** \( y = 0 \).

### Step 2: Identify the horizontal asymptote
For each graph, the horizontal asymptote can help you choose the function.

- If the horizontal asymptote is at **\( y = 0 \)**, the function is likely either \( f(x) = 3^x \), \( h(x) = 3^{x+4} \), or \( r(x) = -(3^x) \).
- If the horizontal asymptote is at **\( y = 4 \)**, the function is either \( g(x) = 3^x + 4 \) or \( m(x) = -(3^x) + 4 \).
- If the horizontal asymptote is at **\( y = -4 \)**, the function is \( q(x) = 3^{-x} - 4 \).

### Step 3: Examine the reflection or shift
- If the graph shows **growth**, the function is likely **not reflected**. That eliminates negative leading terms like \( m(x) \) or \( r(x) \).
- If the graph shows **decay**, the function could involve \( 3^{-x} \).

Using this information, you can match each graph with its function.

Do you have the specific graphs to analyze, or should I generate possible examples for you? 

---  
Would you like more clarification on exponential functions or horizontal asymptotes? Here are 5 related questions:

1. What determines the direction of growth or decay in exponential functions?
2. How does horizontal shifting affect the graph of \( 3^x \)?
3. Why does the horizontal asymptote depend on vertical shifts in the equation?
4. Can an exponential function have more than one horizontal asymptote?
5. What are common mistakes when identifying horizontal asymptotes?

**Tip:** Reflected exponential functions flip the graph over the x-axis, which can help differentiate them from regular exponential growth or decay.

For each of the graphs below, select which one of the following functions is depicted by the graph, as well as the equation of the graph's horizontal asymptote. f(x) = 3^x, g(x) = 3^x + 4, h(x) = 3^{x+4}, m(x) = -(3^x) + 4, q(x) = 3^{-x} - 4, r(x) = -(3^x)

Learn how to identify exponential functions and determine their horizontal asymptotes. This problem involves functions such as f(x) = 3^x, g(x) = 3^x + 4, and more. It covers key concepts in exponential growth, decay, reflections, and shifts, suitable for Grades 9-12.

Math Problem Statement

For each of the graphs below, select which one of the following functions is depicted by the graph, as well as the equation of the graph's horizontal asymptote. f ( x )

, , g ( x )

3 x + 4 g(x)=3 x +4 , , h ( x )

, , m ( x )

− ( 3 x ) + 4 m(x)=−(3 x )+4 , , q ( x )

3 − x − 4 q(x)=3 −x −4 , , r ( x )

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