QUESTION 4
Sketch on the same set of axes the graphs of and .
Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

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Jan 03,2025

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QUESTION 4 Sketch on the same set of axes the graphs of $$ f(x)=-2 x^{2}-4 x+6 $$ and $$ g(x)=-2 \cdot 2^{x-1}+1 $$. Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

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To sketch the graphs of $$ f(x) = -2x^{2} - 4x + 6 $$ (a quadratic function) and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ (an exponential function) on the same set of axes, follow these steps:

---

### **1. Graph of $$ f(x) = -2x^{2} - 4x + 6 $$**
 
**a. Nature of the Function:**
- **Type:** Quadratic (Parabola)
- **Orientation:** Opens **downwards** (since the coefficient of $$ x^2 $$ is negative).

**b. Key Features:**

1. **Vertex (Turning Point):**
   - **Formula:** $$ x_v = -\frac{b}{2a} $$
   - **Calculation:** 
     $$
     x_v = -\frac{-4}{2 \cdot (-2)} = -\frac{4}{-4} = -1
     $$
     $$
     f(-1) = -2(-1)^2 - 4(-1) + 6 = -2(1) + 4 + 6 = 8
     $$
   - **Vertex:** $$ (-1, 8) $$

2. **X-Intercepts:**
   - **Set $$ f(x) = 0 $$:**
     $$
     -2x^2 - 4x + 6 = 0 \implies 2x^2 + 4x - 6 = 0
     $$
     $$
     x = \frac{-4 \pm \sqrt{16 + 48}}{4} = \frac{-4 \pm 8}{4}
     $$
     - $$ x = 1 $$ and $$ x = -3 $$
   - **X-Intercepts:** $$ (1, 0) $$ and $$ (-3, 0) $$

3. **Y-Intercept:**
   - **Set $$ x = 0 $$:**
     $$
     f(0) = -2(0)^2 - 4(0) + 6 = 6
     $$
   - **Y-Intercept:** $$ (0, 6) $$

**c. Sketching the Parabola:**
- Plot the vertex at $$ (-1, 8) $$.
- Plot the x-intercepts at $$ (-3, 0) $$ and $$ (1, 0) $$.
- Plot the y-intercept at $$ (0, 6) $$.
- Draw a symmetric downward-opening parabola passing through these points.

---

### **2. Graph of $$ g(x) = -2 \cdot 2^{x-1} + 1 $$**

**a. Nature of the Function:**
- **Type:** Exponential
- **Transformation Breakdown:**
  - **Base Function:** $$ 2^x $$
  - **Horizontal Shift:** Right by 1 unit ($$ 2^{x-1} $$)
  - **Vertical Stretch & Reflection:** Multiplied by -2 ($$ -2 \cdot 2^{x-1} $$)
  - **Vertical Shift:** Up by 1 unit ($$ -2 \cdot 2^{x-1} + 1 $$)

**b. Key Features:**

1. **Horizontal Asymptote:**
   - **Equation:** $$ y = 1 $$
   - **Behavior:** As $$ x \to -\infty $$, $$ g(x) $$ approaches 1.

2. **Intercepts:**
   - **Y-Intercept (Set $$ x = 0 $$):**
     $$
     g(0) = -2 \cdot 2^{-1} + 1 = -1 + 1 = 0
     $$
     - **Point:** $$ (0, 0) $$
     
   - **X-Intercept (Set $$ g(x) = 0 $$):**
     $$
     -2 \cdot 2^{x-1} + 1 = 0 \implies 2^{x} = 1 \implies x = 0
     $$
     - **Point:** $$ (0, 0) $$
     - *Note:* The x-intercept coincides with the y-intercept.

3. **Additional Points for Clarity:**
   - **At $$ x = 1 $$:**
     $$
     g(1) = -2 \cdot 2^{0} + 1 = -2 + 1 = -1
     $$
     - **Point:** $$ (1, -1) $$
     
   - **At $$ x = 2 $$:**
     $$
     g(2) = -2 \cdot 2^{1} + 1 = -4 + 1 = -3
     $$
     - **Point:** $$ (2, -3) $$
     
   - **At $$ x = -1 $$:**
     $$
     g(-1) = -2 \cdot 2^{-2} + 1 = -0.5 + 1 = 0.5
     $$
     - **Point:** $$ (-1, 0.5) $$

**c. Sketching the Exponential Curve:**
- Draw the horizontal asymptote $$ y = 1 $$ as a dashed line.
- Plot the intercept at $$ (0, 0) $$.
- Plot additional points such as $$ (1, -1) $$, $$ (2, -3) $$, and $$ (-1, 0.5) $$.
- Draw a smooth curve decreasing from left to right, approaching the asymptote $$ y = 1 $$ as $$ x \to -\infty $$ and heading towards $$ -\infty $$ as $$ x \to \infty $$.

---

### **3. Combined Sketch Overview**

- **Axes:** Draw both $$ f(x) $$ and $$ g(x) $$ on the same Cartesian plane.
  
- **For $$ f(x) $$:**
  - Plot the vertex at $$ (-1, 8) $$.
  - Plot x-intercepts at $$ (-3, 0) $$ and $$ (1, 0) $$.
  - Plot the y-intercept at $$ (0, 6) $$.
  - Draw a downward-opening parabola through these points.

- **For $$ g(x) $$:**
  - Draw the horizontal asymptote $$ y = 1 $$.
  - Plot the intercept at $$ (0, 0) $$.
  - Plot additional points $$ (1, -1) $$, $$ (2, -3) $$, and $$ (-1, 0.5) $$.
  - Draw the exponential curve decreasing from left to right, approaching $$ y = 1 $$ as $$ x $$ decreases and plummeting towards $$ -\infty $$ as $$ x $$ increases.

### **4. Visual Representation**

While I cannot provide a visual sketch here, here's a conceptual representation:

- **Parabola $$ f(x) $$:** Centered around the vertex $$ (-1, 8) $$, crossing the x-axis at $$ (-3, 0) $$ and $$ (1, 0) $$, and intersecting the y-axis at $$ (0, 6) $$.

- **Exponential Curve $$ g(x) $$:** Starting near $$ y = 1 $$ for large negative $$ x $$, passing through the origin $$ (0, 0) $$, descending through $$ (1, -1) $$ and $$ (2, -3) $$, with the curve approaching $$ y = 1 $$ from below as $$ x $$ decreases and continuing downward as $$ x $$ increases.

---

### **5. Summary of Key Features**

| Feature               | $$ f(x) = -2x^{2} - 4x + 6 $$ | $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ |
|-----------------------|--------------------------------|-----------------------------------|
| **Vertex**            | $$ (-1, 8) $$                   | N/A                               |
| **X-Intercepts**      | $$ (-3, 0) $$ and $$ (1, 0) $$  | $$ (0, 0) $$                      |
| **Y-Intercept**       | $$ (0, 6) $$                     | $$ (0, 0) $$                      |
| **Additional Points** | N/A                            | $$ (1, -1) $$, $$ (2, -3) $$, $$ (-1, 0.5) $$ |
| **Asymptote**         | N/A                            | Horizontal asymptote at $$ y = 1 $$ |
| **Orientation**       | Downward-opening parabola      | Exponential decay (reflected & shifted) |

By plotting these features accurately, you can effectively sketch both graphs on the same set of axes, clearly indicating all intercepts, the turning point of the parabola, and the asymptote of the exponential function.
To sketch the graphs of $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ on the same axes: 1. **For $$ f(x) $$:** - **Vertex:** $$ (-1, 8) $$ - **X-Intercepts:** $$ (-3, 0) $$ and $$ (1, 0) $$ - **Y-Intercept:** $$ (0, 6) $$ - **Graph:** Downward-opening parabola passing through these points. 2. **For $$ g(x) $$:** - **Y-Intercept:** $$ (0, 0) $$ - **Additional Points:** $$ (1, -1) $$, $$ (2, -3) $$, $$ (-1, 0.5) $$ - **Asymptote:** Horizontal line $$ y = 1 $$ - **Graph:** Exponential curve approaching $$ y = 1 $$ from below as $$ x $$ decreases and continuing downward as $$ x $$ increases. Plot these points and draw the respective shapes to visualize both functions on the same set of axes.

QUESTION 4
Sketch on the same set of axes the graphs of and .
Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

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QUESTION 4 Sketch on the same set of axes the graphs of and . Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).