lence plot the graph of $$ y=2(x+4)(x+2) $$ on the same set of axes.

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UpStudy AI · Accepted Answer

Certainly! Let's analyze and plot the graph of the quadratic function:

$$ y = 2(x + 4)(x + 2) $$

### Step 1: Expand the Equation
First, expand the equation to standard form for easier analysis.

$$
\begin{align*}
y &= 2(x + 4)(x + 2) \
&= 2(x^2 + 6x + 8) \
&= 2x^2 + 12x + 16
\end{align*}
$$

### Step 2: Identify Key Features

1. **Leading Coefficient (a):**  
   $$ a = 2 $$  
   - Since $$ a > 0 $$, the parabola opens **upwards**.

2. **Roots (x-intercepts):**  
   Set $$ y = 0 $$:  
   $$
   2(x + 4)(x + 2) = 0 \implies x = -4 \text{ and } x = -2
   $$
   - **Roots:** $$ (-4, 0) $$ and $$ (-2, 0) $$

3. **Axis of Symmetry:**  
   The axis of symmetry is the average of the roots:  
   $$
   x = \frac{-4 + (-2)}{2} = -3
   $$

4. **Vertex:**  
   Plug $$ x = -3 $$ into the equation to find $$ y $$:  
   $$
   y = 2(-3 + 4)(-3 + 2) = 2(1)(-1) = -2
   $$
   - **Vertex:** $$ (-3, -2) $$

5. **Y-intercept:**  
   Set $$ x = 0 $$:  
   $$
   y = 2(0 + 4)(0 + 2) = 2(4)(2) = 16
   $$
   - **Y-intercept:** $$ (0, 16) $$

### Step 3: Plotting the Graph

Here's a rough sketch of the graph with the identified key points:

```
 y
 ^
 |                        
16|                  * (0,16)
 |                        
 |                        
 |                        
8|                        
 |                        
 |                        
 |          * (-4,0)    * (-2,0)
 |                        
 |               *
4|                        
 |                        
 |                        
 |                        
0|-------------------------> x
 |                        
 |                        
 |                        
-2|               * (-3,-2)
 |                        
 |                        
```

### Step 4: Graph Description

- **Vertex:** The lowest point of the parabola is at $$ (-3, -2) $$.
- **Opening:** Since the leading coefficient is positive, the parabola opens upwards.
- **Intercepts:**
  - **X-intercepts:** The graph crosses the x-axis at $$ x = -4 $$ and $$ x = -2 $$.
  - **Y-intercept:** The graph crosses the y-axis at $$ y = 16 $$.
- **Symmetry:** The graph is symmetric about the line $$ x = -3 $$.

### Step 5: Additional Points (Optional)

To make the graph more accurate, you can plot additional points. For example:

- **When $$ x = -5 $$:**
  $$
  y = 2(-5 + 4)(-5 + 2) = 2(-1)(-3) = 6 \quad \Rightarrow (-5, 6)
  $$

- **When $$ x = -1 $$:**
  $$
  y = 2(-1 + 4)(-1 + 2) = 2(3)(1) = 6 \quad \Rightarrow (-1, 6)
  $$

Adding these points helps in sketching a more precise parabola.

### Final Sketch

Combining all the information, the parabola will have its vertex at $$ (-3, -2) $$, intersect the x-axis at $$ (-4, 0) $$ and $$ (-2, 0) $$, and the y-axis at $$ (0, 16) $$. It will be symmetric about the line $$ x = -3 $$ and open upwards.

---

If you have access to graphing tools or software (like Desmos, GeoGebra, or even graphing calculators), you can input the equation $$ y = 2(x + 4)(x + 2) $$ to visualize the graph more accurately.
To plot the graph of $$ y = 2(x + 4)(x + 2) $$: 1. **Identify the roots:** $$ x = -4 $$ and $$ x = -2 $$. 2. **Find the vertex:** $$ (-3, -2) $$. 3. **Determine the y-intercept:** $$ (0, 16) $$. 4. **Plot the points and sketch the parabola** opening upwards with symmetry at $$ x = -3 $$. The graph will have its lowest point at $$ (-3, -2) $$, cross the x-axis at $$ (-4, 0) $$ and $$ (-2, 0) $$, and the y-axis at $$ (0, 16) $$.

lence plot the graph of \( y=2(x+4)(x+2) \) on the same set of axes.

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