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 the functions $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ on the same set of axes, we'll analyze each function separately, identifying their key features such as intercepts, turning points, and asymptotes. Here's a comprehensive breakdown:

---

### **1. Quadratic Function: $$ f(x) = -2x^{2} - 4x + 6 $$**

**a. Graph Shape:**
- **Parabola Orientation:** Downward-opening (since the coefficient of $$ x^{2} $$ is negative).

**b. Intercepts:**
- **Y-Intercept:** 
  - Set $$ x = 0 $$:
    $$
    f(0) = -2(0)^2 - 4(0) + 6 = 6
    $$
  - **Point:** $$ (0, 6) $$
  
- **X-Intercepts:** 
  - Set $$ f(x) = 0 $$:
    $$
    -2x^{2} - 4x + 6 = 0 \
    \Rightarrow x^{2} + 2x - 3 = 0 \quad (\text{Divide both sides by } -2)
    $$
    $$
    x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-3)}}{2(1)} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}
    $$
    $$
    \Rightarrow x = 1 \quad \text{and} \quad x = -3
    $$
  - **Points:** $$ (1, 0) $$ and $$ (-3, 0) $$

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

**d. Summary of Key Features for $$ f(x) $$:**
- **Parabola opens downward.**
- **Y-Intercept:** $$ (0, 6) $$
- **X-Intercepts:** $$ (1, 0) $$ and $$ (-3, 0) $$
- **Vertex:** $$ (-1, 8) $$
- **No Asymptotes**

---

### **2. Exponential Function: $$ g(x) = -2 \cdot 2^{x-1} + 1 $$**

**a. Graph Shape:**
- **Exponential Decay:** The negative coefficient indicates the graph decreases as $$ x $$ increases.

**b. Asymptote:**
- **Horizontal Asymptote:** $$ y = 1 $$ (since the function approaches this value as $$ x $$ approaches infinity).

**c. Intercepts:**
- **Y-Intercept:** 
  - Set $$ x = 0 $$:
    $$
    g(0) = -2 \cdot 2^{-1} + 1 = -2 \cdot \frac{1}{2} + 1 = -1 + 1 = 0
    $$
  - **Point:** $$ (0, 0) $$
  
- **X-Intercept:** 
  - Set $$ g(x) = 0 $$:
    $$
    -2 \cdot 2^{x-1} + 1 = 0 \
    \Rightarrow 2^{x-1} = \frac{1}{2} \
    \Rightarrow x - 1 = -1 \quad (\text{Since } 2^{-1} = \frac{1}{2}) \
    \Rightarrow x = 0
    $$
  - **Point:** $$ (0, 0) $$ *(Same as Y-Intercept)*

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

**e. Summary of Key Features for $$ g(x) $$:**
- **Exponential decay curve.**
- **Horizontal Asymptote:** $$ y = 1 $$
- **Intercept:** $$ (0, 0) $$
- **Additional Points:** $$ (-1, 0.5) $$, $$ (1, -1) $$, etc.
- **No Turning Points:** The function is always decreasing.

---

### **3. Combined Graph Features**

To plot both functions on the same set of axes:

1. **Draw Coordinate Axes:** Label the $$ x $$-axis and $$ y $$-axis with appropriate scales.

2. **Plot $$ f(x) $$:**
   - **Vertex:** $$ (-1, 8) $$
   - **Y-Intercept:** $$ (0, 6) $$
   - **X-Intercepts:** $$ (1, 0) $$ and $$ (-3, 0) $$
   - **Symmetric Point:** Since the vertex is at $$ x = -1 $$, a symmetric point across the vertex for $$ f(x) $$ is $$ (-2, 6) $$ (mirroring the Y-intercept across the vertex).

3. **Plot $$ g(x) $$:**
   - **Intercept:** $$ (0, 0) $$
   - **Additional Points:** $$ (-1, 0.5) $$, $$ (1, -1) $$, etc.
   - **Asymptote:** Draw a dashed horizontal line at $$ y = 1 $$.

4. **Draw the Parabola ($$ f(x) $$):** Connect the vertex and intercepts smoothly, ensuring the parabola opens downward.

5. **Draw the Exponential Curve ($$ g(x) $$):** Connect the plotted points, approaching the asymptote $$ y = 1 $$ as $$ x $$ decreases and extending downward as $$ x $$ increases.

6. **Final Touches:**
   - Clearly label each curve with its respective function.
   - Highlight intercepts, vertex, and asymptote for clarity.

---

### **Visual Representation Guidelines**

While I can't provide an actual sketch, here's how the two functions relate on the graph:

- **Parabola $$ f(x) $$:**
  - Peaks at $$ (-1, 8) $$.
  - Crosses the $$ y $$-axis at $$ (0, 6) $$.
  - Crosses the $$ x $$-axis at $$ (-3, 0) $$ and $$ (1, 0) $$.
  
- **Exponential Curve $$ g(x) $$:**
  - Passes through the origin $$ (0, 0) $$.
  - Decreases through $$ (1, -1) $$.
  - Approaches the horizontal asymptote $$ y = 1 $$ as $$ x $$ decreases.
  - Continues to decrease without bound as $$ x $$ increases.

By plotting these features accurately, you'll achieve a clear and informative graph displaying both functions simultaneously.
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 $$ f(x) $$:** - **Parabola opening downward.** - **Y-intercept at (0, 6).** - **X-intercepts at (1, 0) and (-3, 0).** - **Vertex at (-1, 8).** - **No asymptotes.** 2. **For $$ g(x) $$:** - **Exponential decay curve.** - **Y-intercept at (0, 0).** - **Horizontal asymptote at y = 1.** - **Additional points at (-1, 0.5) and (1, -1).** - **No turning points.** Plot these features on the same set of axes to visualize both functions accurately.

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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