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).

Ask by Haynes Ball. in South Africa

Jan 16,2025

Question

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 and
Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

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