Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.
The points , and transform to the
points

Ask by Marshall Zimmerman. in the United States

Mar 25,2025

Question

Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.
The points , and transform to the
points

Ask by Marshall Zimmerman. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

**1. Express the Function Using Transformations**

The function is  
$$
f(x) = 4^x + 2.
$$  
This can be seen as a vertical translation of the parent function  
$$
g(x) = 4^x
$$  
upward by 2 units.

---

**2. Domain**

Since exponential functions are defined for all real numbers, the domain of  
$$
g(x) = 4^x
$$  
is  
$$
(-\infty,\infty),
$$  
and the vertical translation does not affect the domain. Therefore, the domain of $$ f(x) $$ is  
$$
(-\infty, \infty).
$$

---

**3. Horizontal Asymptote**

The parent function $$ g(x)=4^x $$ approaches 0 as $$ x \to -\infty $$. After shifting upward by 2, the horizontal asymptote shifts from  
$$
y=0
$$  
to  
$$
y=2.
$$  
Thus, the horizontal asymptote is  
$$
y=2.
$$

---

**4. Range**

For the parent function $$ g(x)=4^x $$, the range is  
$$
(0, \infty).
$$  
After the vertical translation by 2, every output increases by 2, so the range of $$ f(x) $$ becomes  
$$
(0+2, \infty+2) \quad \text{or} \quad (2, \infty).
$$

---

**5. Monotonicity**

Since the base 4 is greater than 1, the function $$ 4^x $$ is strictly increasing. A vertical translation does not change the monotonicity; therefore, the function  
$$
f(x) = 4^x + 2
$$  
remains **increasing**.

---

**6. Transformation of Key Points**

The key points on the graph of the parent function $$ g(x) = 4^x $$ are:
- $$\left(-1, \frac{1}{4}\right)$$
- $$(0, 1)$$
- $$(1, 4)$$

Since the transformation is a vertical shift upward by 2 units, add 2 to each $$ y $$-coordinate:

- For $$\left(-1, \frac{1}{4}\right)$$:
  $$
  y = \frac{1}{4} + 2 = \frac{1}{4} + \frac{8}{4} = \frac{9}{4}.
  $$
  Transformed point: $$\left(-1, \frac{9}{4}\right)$$.

- For $$(0, 1)$$:
  $$
  y = 1 + 2 = 3.
  $$
  Transformed point: $$(0, 3)$$.

- For $$(1, 4)$$:
  $$
  y = 4 + 2 = 6.
  $$
  Transformed point: $$(1, 6)$$.

---

**Summary**

- **Domain:** $$(-\infty, \infty)$$
- **Range:** $$(2, \infty)$$
- **Horizontal Asymptote:** $$y = 2$$
- **Monotonicity:** Increasing
- **Transformed Points:**  
  $$\left(-1, \frac{9}{4}\right)$$, $$(0, 3)$$, and $$(1, 6)$$.
**Transformed Function and Graph Analysis** - **Function:** $$ f(x) = 4^x + 2 $$ - **Domain:** All real numbers $$(-∞, ∞)$$ - **Range:** $$ (2, ∞) $$ - **Horizontal Asymptote:** $$ y = 2 $$ - **Monotonicity:** Increasing - **Transformed Points:** $$\left(-1, \frac{9}{4}\right)$$, $$(0, 3)$$, and $$(1, 6)$$

Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.
The points , and transform to the
points

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Use transformations to help graph the function . Identify the domain, range, and horizontal asymptote. Determine whether the function is increasing or decreasing. The points , and transform to the points

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Use transformations to help graph the function
. Identify the domain, range, and horizontal
asymptote. Determine whether the function is increasing
or decreasing.
The points , and transform to the
points