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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Certainly! Let's analyze and sketch the graphs of the two functions $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ on the same coordinate system. We'll identify their key features, including intercepts, turning points, and asymptotes.

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

### **Type of Function**
- **Quadratic Function:** Represents a parabola.
- **Direction:** Opens **downward** since the coefficient of $$ x^{2} $$ is negative.

### **Key Features**

#### **a. Vertex (Turning Point)**
The vertex form of a quadratic function is $$ f(x) = a(x-h)^2 + k $$, where $$ (h, k) $$ is the vertex.

- **Formula to Find Vertex:**
  $$
  h = -\frac{b}{2a}, \quad k = f(h)
  $$
  
- **Calculations:**
  $$
  a = -2, \quad b = -4
  $$
  $$
  h = -\frac{-4}{2 \times -2} = -1
  $$
  $$
  k = f(-1) = -2(-1)^2 - 4(-1) + 6 = -2 + 4 + 6 = 8
  $$
  
- **Vertex:** $$ (-1, 8) $$

#### **b. Intercepts**

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

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

#### **c. Summary for $$ f(x) $$**
- **Vertex:** $$ (-1, 8) $$
- **Y-Intercept:** $$ (0, 6) $$
- **X-Intercepts:** $$ (1, 0) $$ and $$ (-3, 0) $$
- **Axis of Symmetry:** $$ x = -1 $$
- **No Asymptotes**

### **Graph Shape**
A downward-opening parabola with its highest point at $$ (-1, 8) $$, crossing the y-axis at $$ (0, 6) $$, and x-axis at $$ (1, 0) $$ and $$ (-3, 0) $$.

---

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

### **Type of Function**
- **Exponential Function**

### **Key Features**

#### **a. Simplifying the Function**
$$
g(x) = -2 \cdot 2^{x-1} + 1 = -2^{x} \cdot 2^{-1} \cdot 2 + 1 = -2^{x} + 1
$$

#### **b. Asymptote**

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

#### **c. Intercepts**

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

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

#### **d. Behavior**

- **As $$ x \to \infty $$:** $$ 2^{x} \to \infty $$, so $$ g(x) \to -\infty $$.
- **As $$ x \to -\infty $$:** $$ 2^{x} \to 0 $$, so $$ g(x) \to 1 $$.

#### **e. Summary for $$ g(x) $$**
- **Horizontal Asymptote:** $$ y = 1 $$
- **Y-Intercept:** $$ (0, 0) $$
- **X-Intercept:** $$ (0, 0) $$ (Same as Y-Intercept)
- **No Turning Points**

### **Graph Shape**
An exponential decay curve (reflected over the x-axis due to the negative coefficient) approaching $$ y = 1 $$ as a horizontal asymptote. It intersects the origin $$ (0, 0) $$ and decreases towards $$ -\infty $$ as $$ x $$ increases.

---

## 3. Combined Graph on the Same Axes

### **Plotting the Features**

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

2. **Exponential Function $$ g(x) $$:**
   - Plot the intercept at $$ (0, 0) $$.
   - Draw the horizontal asymptote $$ y = 1 $$ (a dashed line).
   - Sketch the exponential curve starting near $$ y = 1 $$ for $$ x $$ far left, passing through $$ (0, 0) $$, and decreasing without bound as $$ x $$ increases.

### **Visual Representation**

While I cannot provide a visual sketch, here's a description to visualize the combined graph:

- **Quadratic $$ f(x) $$:** A large "∩" shaped curve centered at $$ (-1, 8) $$, crossing the y-axis at $$ (0, 6) $$ and extending down to cross the x-axis at $$ (-3, 0) $$ and $$ (1, 0) $$.
  
- **Exponential $$ g(x) $$:** A decreasing curve starting at the origin $$ (0, 0) $$, approaching the horizontal line $$ y = 1 $$ as $$ x $$ moves to the left, and descending below the parabola as $$ x $$ increases.

### **Intersection Points**

To find if $$ f(x) $$ and $$ g(x) $$ intersect, set $$ f(x) = g(x) $$:

$$
-2x^{2} - 4x + 6 = -2^{x} + 1
$$
$$
-2x^{2} - 4x + 5 = -2^{x}
$$

This equation isn't straightforward to solve analytically. However, by inspection:

- **At $$ x = 0 $$:**
  $$
  f(0) = 6, \quad g(0) = 0 \quad \Rightarrow \quad f(0) > g(0)
  $$
  
- **At $$ x = 1 $$:**
  $$
  f(1) = -2(1)^2 -4(1) +6 = 0, \quad g(1) = -2^{1} +1 = -1 \quad \Rightarrow \quad f(1) > g(1)
  $$
  
- **At $$ x = 2 $$:**
  $$
  f(2) = -8 -8 +6 = -10, \quad g(2) = -4 +1 = -3 \quad \Rightarrow \quad f(2) < g(2)
  $$

By Intermediate Value Theorem, there's at least one intersection between $$ x =1 $$ and $$ x =2 $$. A numerical method (like Newton-Raphson) would be needed for an exact value.

### **Final Notes**

- **$$ f(x) $$:** Dominates for larger negative $$ x $$ due to the parabola's shape.
- **$$ g(x) $$:** Dominates for larger positive $$ x $$ as it decreases rapidly.
- **Asymptote $$ y = 1 $$:** Only affects $$ g(x) $$, not $$ f(x) $$.

By plotting these features accurately, both graphs can be visualized on the same set of axes, clearly showing their intercepts, vertex, and asymptote.
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:** $$ (-1, 8) $$ - **Y-Intercept:** $$ (0, 6) $$ - **X-Intercepts:** $$ (1, 0) $$ and $$ (-3, 0) $$ - **Asymptote:** None 2. **For $$ g(x) $$:** - **Y-Intercept:** $$ (0, 0) $$ - **Horizontal Asymptote:** $$ y = 1 $$ - **No Turning Points** Plot these features on the same set of axes to visualize both functions.

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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