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 $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ on the same set of axes, we'll analyze each function separately to determine their key features: intercepts, turning points, and asymptotes.

---

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

### **a. Vertex (Turning Point)**
The vertex of a quadratic function $$ ax^{2} + bx + c $$ is found at:
$$
x = -\frac{b}{2a}
$$
For $$ f(x) $$:
$$
a = -2, \quad b = -4
$$
$$
x = -\frac{-4}{2 \times -2} = -1
$$
$$
f(-1) = -2(-1)^{2} - 4(-1) + 6 = -2 + 4 + 6 = 8
$$
**Vertex:** $$ (-1, 8) $$

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

### **c. X-Intercepts**
Solve $$ f(x) = 0 $$:
$$
-2x^{2} - 4x + 6 = 0 \Rightarrow x^{2} + 2x - 3 = 0
$$
$$
x = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm 4}{2}
$$
$$
x = 1 \quad \text{and} \quad x = -3
$$
**X-Intercepts:** $$ (1, 0) $$ and $$ (-3, 0) $$

### **d. Direction of Opening**
Since $$ a = -2 < 0 $$, the parabola opens **downward**.

---

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

First, simplify the function:
$$
g(x) = -2 \cdot 2^{x-1} + 1 = -2^{x} + 1
$$

### **a. Horizontal Asymptote**
As $$ x \to -\infty $$:
$$
2^{x} \to 0 \Rightarrow g(x) \to 1
$$
**Horizontal Asymptote:** $$ y = 1 $$

### **b. Y-Intercept**
Set $$ x = 0 $$:
$$
g(0) = -2^{0} + 1 = -1 + 1 = 0
$$
**Y-Intercept:** $$ (0, 0) $$

### **c. X-Intercept**
Set $$ g(x) = 0 $$:
$$
-2^{x} + 1 = 0 \Rightarrow 2^{x} = 1 \Rightarrow x = 0
$$
**X-Intercept:** $$ (0, 0) $$

### **d. Direction of the Function**
The coefficient of $$ 2^{x} $$ is negative, so the function is **decreasing**.

### **e. Turning Points**
Exponential functions do not have turning points; they are always increasing or decreasing.

---

## 3. Combined Sketch Features

When plotting both functions on the same set of axes:

- **Quadratic $$ f(x) $$:**
  - **Vertex:** $$ (-1, 8) $$
  - **Y-Intercept:** $$ (0, 6) $$
  - **X-Intercepts:** $$ (-3, 0) $$ and $$ (1, 0) $$
  - **Shape:** Downward-opening parabola

- **Exponential $$ g(x) $$:**
  - **Horizontal Asymptote:** $$ y = 1 $$
  - **Y-Intercept and X-Intercept:** $$ (0, 0) $$
  - **Shape:** Decreasing exponential curve approaching $$ y = 1 $$ as $$ x \to -\infty $$ and decreasing without bound as $$ x \to +\infty $$

### **Intersection Points:**
- Both functions intersect at $$ (0, 0) $$.

### **Visual Description:**
1. **Draw the Axes:** Label the x-axis and y-axis appropriately.
2. **Plot $$ f(x) $$:**
   - Mark the vertex at $$ (-1, 8) $$.
   - Plot the y-intercept at $$ (0, 6) $$.
   - Plot the x-intercepts at $$ (-3, 0) $$ and $$ (1, 0) $$.
   - Draw a symmetric downward-opening parabola passing through these points.

3. **Plot $$ g(x) $$:**
   - Draw the horizontal asymptote $$ y = 1 $$ as a dashed line.
   - Plot the y-intercept and x-intercept at $$ (0, 0) $$.
   - Sketch a smooth, decreasing exponential curve that approaches $$ y = 1 $$ as $$ x \to -\infty $$ and descends below $$ y = 0 $$ as $$ x \to +\infty $$.

### **Final Notes:**
- Ensure that both graphs are clearly distinguishable, perhaps by using different colors or line styles.
- Label all intercepts, the vertex, and the asymptote for clarity.

---

**Illustrative Graph:**

While I cannot provide an actual sketch here, below is a conceptual representation of how the graphs would appear together:

```
y
|
|               f(x)
|                *
|               * \
|              *   \
|             *     \
|------------*-------\---------- g(x)
|           /         \
|          /           \
|         /             \
|        *               \
|
|---------------------------------- x
```

- The **parabola $$ f(x) $$** peaks at $$ (-1, 8) $$, crosses the y-axis at $$ (0, 6) $$, and the x-axis at $$ (-3,0) $$ and $$ (1,0) $$.
- The **exponential $$ g(x) $$** starts at $$ (0,0) $$, decreases towards negative infinity as $$ x $$ increases, and approaches the horizontal asymptote $$ y = 1 $$ as $$ x $$ decreases.

This description should help you accurately sketch both functions on the same set of axes, highlighting all required features.
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. **Quadratic Function $$ f(x) $$:** - **Vertex:** $$ (-1, 8) $$ - **Y-Intercept:** $$ (0, 6) $$ - **X-Intercepts:** $$ (-3, 0) $$ and $$ (1, 0) $$ - **Shape:** Downward-opening parabola 2. **Exponential Function $$ g(x) $$:** - **Horizontal Asymptote:** $$ y = 1 $$ - **Y-Intercept and X-Intercept:** $$ (0, 0) $$ - **Shape:** Decreasing exponential curve approaching $$ y = 1 $$ as $$ x \to -\infty $$ and decreasing without bound as $$ x \to +\infty $$ **Intersection Point:** Both functions intersect at $$ (0, 0) $$. **Sketching Tips:** - Plot the vertex, intercepts, and asymptote for clarity. - Draw the parabola symmetrically downward. - Sketch the exponential curve approaching the asymptote from below. Ensure both graphs are clearly labeled and distinguishable on the same set of axes.

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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