In Exercises 23-28, write a rule for $$ g $$ and then graph each function. Describe the graph of $$ g $$ as a transformation of th graph of $$ f $$. (See Example 4.) $$ \begin{array}{ll} ext { 23. } f(x)=x^{4}+1, g(x)=f(x+2) & ext { 24. } f(x)=2 x^{3}-2 x^{2}+6, g(x)=-\frac{1}{2} f(x) \ ext { 25. } f(x)=x^{6}-3 x^{3}+2, g(x)=f(x-1)-3 & ext { 26. } f(x)=x^{5}-2 x+3, g(x)=3 f(x)+1 \ ext { rule: }(x-1)^{6}-3(x-1)^{3}+2-3 & ext { 28. } f(x)=\frac{1}{2} x^{5}+x^{3}-4 x-4, g(x)=-f(2 x)+1 \ ext { 27. } f(x)=x^{4}+x^{3}-1, g(x)=f(-x)-5 & \end{array} $$

Question

In Exercises 23-28, write a rule for $$ g $$ and then graph each function. Describe the graph of $$ g $$ as a transformation of th graph of $$ f $$. (See Example 4.) $$ \begin{array}{ll}	ext { 23. } f(x)=x^{4}+1, g(x)=f(x+2) & 	ext { 24. } f(x)=2 x^{3}-2 x^{2}+6, g(x)=-\frac{1}{2} f(x) \ 	ext { 25. } f(x)=x^{6}-3 x^{3}+2, g(x)=f(x-1)-3 & 	ext { 26. } f(x)=x^{5}-2 x+3, g(x)=3 f(x)+1 \ 	ext { rule: }(x-1)^{6}-3(x-1)^{3}+2-3 & 	ext { 28. } f(x)=\frac{1}{2} x^{5}+x^{3}-4 x-4, g(x)=-f(2 x)+1 \ 	ext { 27. } f(x)=x^{4}+x^{3}-1, g(x)=f(-x)-5 & \end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's tackle Exercises **23-28** one by one. For each exercise, we'll:

1. **Write the rule for $$ g(x) $$.**
2. **Describe the transformation of the graph of $$ g $$ relative to the graph of $$ f $$.**

---

### **Exercise 23**

**Given:**
$$ f(x) = x^4 + 1 $$
$$ g(x) = f(x + 2) $$

**1. Rule for $$ g(x) $$:**
$$ g(x) = (x + 2)^4 + 1 $$

**2. Transformation Description:**
- **Horizontal Shift:** The term $$ (x + 2) $$ inside the function causes the graph of $$ f $$ to shift **2 units to the left**.
- **No Vertical Shift or Scaling:** There's no change in the vertical position or scaling since there are no additional constants or coefficients outside the function besides the existing $$ +1 $$.

**Graphical Interpretation:**
- If the original graph of $$ f(x) = x^4 + 1 $$ has its vertex at $$ (0,1) $$, the graph of $$ g(x) $$ will have its vertex moved to $$ (-2,1) $$.

---

### **Exercise 24**

**Given:**
$$ f(x) = 2x^3 - 2x^2 + 6 $$
$$ g(x) = -\frac{1}{2} f(x) $$

**1. Rule for $$ g(x) $$:**
$$ g(x) = -\frac{1}{2}(2x^3 - 2x^2 + 6) = -x^3 + x^2 - 3 $$

**2. Transformation Description:**
- **Vertical Scaling:** The factor $$ \frac{1}{2} $$ scales the graph of $$ f $$ **vertically by a factor of $$ \frac{1}{2} $$** (compresses it vertically).
- **Reflection:** The negative sign $$ (-) $$ reflects the graph of $$ f $$ **across the x-axis**.

**Graphical Interpretation:**
- Peaks and troughs of $$ f $$ will be half as high/lower in $$ g $$, and all points will be mirrored over the x-axis.

---

### **Exercise 25**

**Given:**
$$ f(x) = x^6 - 3x^3 + 2 $$
$$ g(x) = f(x - 1) - 3 $$

**1. Rule for $$ g(x) $$:**
$$ g(x) = (x - 1)^6 - 3(x - 1)^3 + 2 - 3 = (x - 1)^6 - 3(x - 1)^3 - 1 $$

**2. Transformation Description:**
- **Horizontal Shift:** The term $$ (x - 1) $$ shifts the graph of $$ f $$ **1 unit to the right**.
- **Vertical Shift:** The subtraction of 3 shifts the graph **3 units downward**.

**Graphical Interpretation:**
- If the original graph of $$ f $$ has a particular feature at $$ x = a $$, the graph of $$ g $$ will have the same feature at $$ x = a + 1 $$ and **3 units lower** vertically.

---

### **Exercise 26**

**Given:**
$$ f(x) = x^5 - 2x + 3 $$
$$ g(x) = 3f(x) + 1 $$

**1. Rule for $$ g(x) $$:**
$$ g(x) = 3(x^5 - 2x + 3) + 1 = 3x^5 - 6x + 10 $$

**2. Transformation Description:**
- **Vertical Scaling:** The factor 3 **stretches the graph vertically by a factor of 3**.
- **Vertical Shift:** Adding 1 shifts the graph **1 unit upward**.

**Graphical Interpretation:**
- All y-values of $$ f(x) $$ are multiplied by 3, making the graph steeper, and then the entire graph is lifted up by 1 unit.

---

### **Exercise 27**

**Given:**
$$ f(x) = x^4 + x^3 - 1 $$
$$ g(x) = f(-x) - 5 $$

**1. Rule for $$ g(x) $$:**
$$ g(x) = (-x)^4 + (-x)^3 - 1 - 5 = x^4 - x^3 - 6 $$

**2. Transformation Description:**
- **Reflection:** Replacing $$ x $$ with $$ -x $$ reflects the graph of $$ f $$ **across the y-axis**.
- **Vertical Shift:** Subtracting 5 shifts the graph **5 units downward**.

**Graphical Interpretation:**
- Features that were on the right side of $$ f $$ will appear on the left side for $$ g $$, and the entire graph will be lowered by 5 units.

---

### **Exercise 28**

**Given:**
$$ f(x) = \frac{1}{2}x^5 + x^3 - 4x - 4 $$
$$ g(x) = -f(2x) + 1 $$

**1. Rule for $$ g(x) $$:**
$$ g(x) = -\left(\frac{1}{2}(2x)^5 + (2x)^3 - 4(2x) - 4\right) + 1 $$
Simplify step-by-step:
$$
\begin{align*}
g(x) &= -\left(\frac{1}{2} \cdot 32x^5 + 8x^3 - 8x - 4\right) + 1 \
&= -(16x^5 + 8x^3 - 8x - 4) + 1 \
&= -16x^5 - 8x^3 + 8x + 4 + 1 \
&= -16x^5 - 8x^3 + 8x + 5
\end{align*}
$$

**2. Transformation Description:**
- **Horizontal Scaling:** The argument $$ 2x $$ compresses the graph **horizontally by a factor of $$ \frac{1}{2} $$**.
- **Reflection:** The negative sign $$ (-) $$ reflects the graph **across the x-axis**.
- **Vertical Shift:** Adding 1 shifts the graph **1 unit upward**.

**Graphical Interpretation:**
- The original graph is first compressed horizontally, then flipped over the x-axis, and finally moved up by 1 unit.

---

### **Summary of Transformations**

| Exercise | Transformation Steps                                                                                |
|----------|------------------------------------------------------------------------------------------------------|
| **23**   | Shift **2 units left** horizontally.                                                                  |
| **24**   | **Vertical compress** by $$ \frac{1}{2} $$ and **reflect** over the x-axis.                          |
| **25**   | Shift **1 unit right** horizontally and **3 units downward** vertically.                            |
| **26**   | **Vertical stretch** by a factor of 3 and **shift up** by 1 unit.                                    |
| **27**   | **Reflect** across the y-axis and **shift down** by 5 units.                                        |
| **28**   | **Compress horizontally** by $$ \frac{1}{2} $$, **reflect** over the x-axis, and **shift up** by 1.    |

---

Feel free to ask if you need further explanations or assistance with any of these exercises!
Here are the simplified transformations for each exercise: 1. **Exercise 23:** - **Rule for $$ g(x) $$:** $$ g(x) = (x + 2)^4 + 1 $$ - **Transformation:** Shift the graph of $$ f(x) = x^4 + 1 $$ **2 units to the left**. 2. **Exercise 24:** - **Rule for $$ g(x) $$:** $$ g(x) = -\frac{1}{2}(2x^3 - 2x^2 + 6) = -x^3 + x^2 - 3 $$ - **Transformation:** **Vertically compress** the graph of $$ f(x) = 2x^3 - 2x^2 + 6 $$ by a factor of $$ \frac{1}{2} $$ and **reflect** it across the x-axis. 3. **Exercise 25:** - **Rule for $$ g(x) $$:** $$ g(x) = (x - 1)^6 - 3(x - 1)^3 - 1 $$ - **Transformation:** Shift the graph of $$ f(x) = x^6 - 3x^3 + 2 $$ **1 unit to the right** and **3 units downward**. 4. **Exercise 26:** - **Rule for $$ g(x) $$:** $$ g(x) = 3x^5 - 6x + 10 $$ - **Transformation:** **Vertically stretch** the graph of $$ f(x) = x^5 - 2x + 3 $$ by a factor of 3 and **shift up** by 1 unit. 5. **Exercise 27:** - **Rule for $$ g(x) $$:** $$ g(x) = x^4 - x^3 - 6 $$ - **Transformation:** **Reflect** the graph of $$ f(x) = x^4 + x^3 - 1 $$ across the y-axis and **shift down** by 5 units. 6. **Exercise 28:** - **Rule for $$ g(x) $$:** $$ g(x) = -16x^5 - 8x^3 + 8x + 5 $$ - **Transformation:** **Compress** the graph of $$ f(x) = \frac{1}{2}x^5 + x^3 - 4x - 4 $$ horizontally by a factor of $$ \frac{1}{2} $$, **reflect** it across the x-axis, and **shift up** by 1 unit. --- These transformations describe how the graph of $$ g(x) $$ is derived from the graph of $$ f(x) $$ through various shifts and scaling operations.

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