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

Question

UpStudy AI · Accepted Answer

To graph 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, identify their key features, and then describe how they coexist on the graph.

---

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

### **a. Nature of the Function**
- **Type:** Quadratic Function (Parabola)
- **Orientation:** Downward-opening (since the coefficient of $$ x^2 $$ is negative)

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

- **Calculating the x-coordinate of the vertex:**
  $$
  x = -\frac{b}{2a} = -\frac{-4}{2 \times (-2)} = -\frac{4}{-4} = -1
  $$
  
- **Calculating the y-coordinate of the vertex:**
  $$
  f(-1) = -2(-1)^2 - 4(-1) + 6 = -2(1) + 4 + 6 = -2 + 4 + 6 = 8
  $$
  
- **Vertex:** $$ (-1, 8) $$

### **c. Intercepts**

- **Y-intercept:** Set $$ x = 0 $$
  $$
  f(0) = -2(0)^2 - 4(0) + 6 = 6
  $$
  $$
  \text{Y-intercept at } (0, 6)
  $$
  
- **X-intercepts (Roots):** Solve $$ f(x) = 0 $$
  $$
  -2x^2 - 4x + 6 = 0 \quad \Rightarrow \quad x^2 + 2x - 3 = 0
  $$
  $$
  x = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}
  $$
  $$
  x = 1 \quad \text{and} \quad x = -3
  $$
  $$
  \text{X-intercepts at } (1, 0) \text{ and } (-3, 0)
  $$

### **d. Summary of Key Features for $$ f(x) $$**

- **Vertex:** $$ (-1, 8) $$
- **Y-intercept:** $$ (0, 6) $$
- **X-intercepts:** $$ (1, 0) $$ and $$ (-3, 0) $$
- **No Asymptotes**

---

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

### **a. Nature of the Function**
- **Type:** Exponential Function
- **Base:** $$ 2 $$ (Exponential growth/decay)
- **Transformation:** 
  - Horizontal shift right by 1 unit
  - Vertical stretch by a factor of 2
  - Reflection over the x-axis (due to the negative coefficient)
  - Vertical shift up by 1 unit

### **b. Asymptote**

- **Horizontal Asymptote:** $$ y = 1 $$

### **c. Intercepts**

- **Y-intercept:** Set $$ x = 0 $$
  $$
  g(0) = -2 \cdot 2^{0-1} + 1 = -2 \cdot \frac{1}{2} + 1 = -1 + 1 = 0
  $$
  $$
  \text{Y-intercept at } (0, 0)
  $$
  
- **X-intercept:** Set $$ g(x) = 0 $$
  $$
  -2 \cdot 2^{x-1} + 1 = 0 \quad \Rightarrow \quad -2 \cdot 2^{x-1} = -1 \quad \Rightarrow \quad 2^{x-1} = \frac{1}{2}
  $$
  $$
  2^{x-1} = 2^{-1} \quad \Rightarrow \quad x - 1 = -1 \quad \Rightarrow \quad x = 0
  $$
  $$
  \text{X-intercept at } (0, 0)
  $$
  
### **d. Behavior Analysis**

- **As $$ x \to \infty $$:** 
  $$
  2^{x-1} \to \infty \quad \Rightarrow \quad g(x) \to -\infty
  $$
  
- **As $$ x \to -\infty $$:** 
  $$
  2^{x-1} \to 0 \quad \Rightarrow \quad g(x) \to 1
  $$
  
### **e. Summary of Key Features for $$ g(x) $$**

- **Horizontal Asymptote:** $$ y = 1 $$
- **Y-intercept:** $$ (0, 0) $$
- **X-intercept:** $$ (0, 0) $$
- **No Vertical Asymptotes**

---

## 3. Combined Graph Description

### **a. Axes Setup**
- **Coordinate Plane:** Standard $$ xy $$-plane with clearly marked scales.

### **b. Plotting $$ f(x) $$**
- **Plot Points:**
  - Vertex at $$ (-1, 8) $$
  - Y-intercept at $$ (0, 6) $$
  - X-intercepts at $$ (1, 0) $$ and $$ (-3, 0) $$
  
- **Shape:** Downward-opening parabola passing through the plotted points.

### **c. Plotting $$ g(x) $$**
- **Plot Points:**
  - Y-intercept and X-intercept coincide at $$ (0, 0) $$
  
- **Asymptote:** Draw a dashed horizontal line at $$ y = 1 $$ to represent the asymptote.
  
- **Behavior:**
  - For $$ x > 0 $$, the graph decreases rapidly towards negative infinity.
  - For $$ x < 0 $$, the graph approaches the horizontal asymptote $$ y = 1 $$.

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

### **e. Final Graph Features**

- **$$ f(x) $$:**
  - Parabola with vertex $$ (-1, 8) $$, crossing the y-axis at $$ (0, 6) $$, and x-axis at $$ (1, 0) $$ and $$ (-3, 0) $$.
  
- **$$ g(x) $$:**
  - Exponential curve passing through $$ (0, 0) $$, approaching $$ y = 1 $$ as $$ x $$ decreases, and descending towards negative infinity as $$ x $$ increases.
  
- **Asymptote:** $$ g(x) $$ has a horizontal asymptote at $$ y = 1 $$.

### **f. Visualization**

While I can't provide a visual sketch here, you can visualize the combined graph as follows:

1. **Draw the Parabola $$ f(x) $$:**
   - Start at the vertex $$ (-1, 8) $$.
   - Plot the y-intercept $$ (0, 6) $$.
   - Plot the x-intercepts $$ (-3, 0) $$ and $$ (1, 0) $$.
   - Sketch the downward-opening curve passing through these points.

2. **Draw the Exponential Curve $$ g(x) $$:**
   - Plot the point $$ (0, 0) $$, which is both the y-intercept and x-intercept.
   - Draw a dashed horizontal line at $$ y = 1 $$ for the asymptote.
   - Sketch the curve approaching $$ y = 1 $$ as $$ x $$ moves to the left and descending sharply below the x-axis as $$ x $$ moves to the right.

3. **Highlight Key Features:**
   - Mark the vertex, intercepts, and asymptote clearly.
   - Ensure both graphs are distinguishable, perhaps using different colors or line styles.

---

## 4. Summary of Key Features

| Feature           | $$ f(x) = -2x^2 - 4x + 6 $$ | $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ |
|-------------------|-----------------------------|-----------------------------------|
| **Type**          | Parabola                    | Exponential Function              |
| **Vertex**        | $$ (-1, 8) $$                | N/A                               |
| **Y-intercept**   | $$ (0, 6) $$                 | $$ (0, 0) $$                       |
| **X-intercepts**  | $$ (-3, 0) $$ and $$ (1, 0) $$ | $$ (0, 0) $$                     |
| **Asymptote**     | None                        | Horizontal asymptote at $$ y = 1 $$ |
| **Direction**     | Downward-opening            | Decreasing for $$ x > 1 $$, Approaches $$ y = 1 $$ as $$ x \to -\infty $$ |

---

By following this analysis, you can accurately sketch both functions on the same set of axes, clearly indicating all intercepts, the vertex of the parabola, and the asymptote of the exponential function.
To graph $$ f(x) = -2x^2 - 4x + 6 $$ 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:** $$ (-3, 0) $$ and $$ (1, 0) $$ - **Asymptote:** None 2. **For $$ g(x) $$:** - **Y-intercept and X-intercept:** $$ (0, 0) $$ - **Asymptote:** Horizontal line at $$ y = 1 $$ Plot both functions on the same graph, marking the key points and asymptote clearly.

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

Real Tutor Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Calculus Questions