Answer
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 \( (0, 16) \).
Solution
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.
Reviewed and approved by the UpStudy tutoring team
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