https://file.aiquickdraw.com/mathbot/file/17301290848001eli002e.jpg

Let's go through this problem step-by-step.

### Step 1: Analyze the functions
We have two functions defined as:
1. \( f(x) = x(x + 4) \)
2. \( g(x) = x(x - 4) \)

### Step 2: Expand each function
1. **For \( f(x) \):**
   \[
   f(x) = x(x + 4) = x^2 + 4x
   \]
   This is a quadratic function in the form \( f(x) = ax^2 + bx + c \) with \( a = 1 \), \( b = 4 \), and \( c = 0 \).

2. **For \( g(x) \):**
   \[
   g(x) = x(x - 4) = x^2 - 4x
   \]
   This is another quadratic function in the form \( g(x) = ax^2 + bx + c \) with \( a = 1 \), \( b = -4 \), and \( c = 0 \).

### Step 3: Complete the tables
Let's evaluate each function at the given values of \( x \).

#### For \( f(x) = x^2 + 4x \):
- \( f(-5) = (-5)^2 + 4(-5) = 25 - 20 = 5 \)
- \( f(-4) = (-4)^2 + 4(-4) = 16 - 16 = 0 \)
- \( f(-3) = (-3)^2 + 4(-3) = 9 - 12 = -3 \)
- \( f(-2) = (-2)^2 + 4(-2) = 4 - 8 = -4 \)
- \( f(-1) = (-1)^2 + 4(-1) = 1 - 4 = -3 \)
- \( f(0) = (0)^2 + 4(0) = 0 \)
- \( f(1) = (1)^2 + 4(1) = 1 + 4 = 5 \)
- \( f(2) = (2)^2 + 4(2) = 4 + 8 = 12 \)
- \( f(3) = (3)^2 + 4(3) = 9 + 12 = 21 \)
- \( f(4) = (4)^2 + 4(4) = 16 + 16 = 32 \)
- \( f(5) = (5)^2 + 4(5) = 25 + 20 = 45 \)

#### For \( g(x) = x^2 - 4x \):
- \( g(-5) = (-5)^2 - 4(-5) = 25 + 20 = 45 \)
- \( g(-4) = (-4)^2 - 4(-4) = 16 + 16 = 32 \)
- \( g(-3) = (-3)^2 - 4(-3) = 9 + 12 = 21 \)
- \( g(-2) = (-2)^2 - 4(-2) = 4 + 8 = 12 \)
- \( g(-1) = (-1)^2 - 4(-1) = 1 + 4 = 5 \)
- \( g(0) = (0)^2 - 4(0) = 0 \)
- \( g(1) = (1)^2 - 4(1) = 1 - 4 = -3 \)
- \( g(2) = (2)^2 - 4(2) = 4 - 8 = -4 \)
- \( g(3) = (3)^2 - 4(3) = 9 - 12 = -3 \)
- \( g(4) = (4)^2 - 4(4) = 16 - 16 = 0 \)
- \( g(5) = (5)^2 - 4(5) = 25 - 20 = 5 \)

### Step 4: Identify intercepts and vertices
1. **Intercepts with the x-axis** (where \( f(x) = 0 \) or \( g(x) = 0 \)):
   - For \( f(x) = x(x + 4) \): The x-intercepts are \( x = 0 \) and \( x = -4 \).
   - For \( g(x) = x(x - 4) \): The x-intercepts are \( x = 0 \) and \( x = 4 \).

2. **Vertices**:
   - For \( f(x) = x^2 + 4x \): The vertex is at \( x = -\frac{b}{2a} = -\frac{4}{2} = -2 \). Substituting \( x = -2 \) into \( f(x) \):
     \[
     f(-2) = (-2)^2 + 4(-2) = 4 - 8 = -4
     \]
     So, the vertex for \( f(x) \) is at \( (-2, -4) \).

   - For \( g(x) = x^2 - 4x \): The vertex is at \( x = -\frac{b}{2a} = -\frac{-4}{2} = 2 \). Substituting \( x = 2 \) into \( g(x) \):
     \[
     g(2) = (2)^2 - 4(2) = 4 - 8 = -4
     \]
     So, the vertex for \( g(x) \) is at \( (2, -4) \).

### Final Answers

- **For \( f(x) \):**
  - **Table values:** Calculated above.
  - **X-intercepts:** \( x = 0 \) and \( x = -4 \).
  - **Vertex:** \( (-2, -4) \).

- **For \( g(x) \):**
  - **Table values:** Calculated above.
  - **X-intercepts:** \( x = 0 \) and \( x = 4 \).
  - **Vertex:** \( (2, -4) \).

Would you like more detailed explanations for any specific part?

### Additional Questions:
1. What would the y-intercept be for each function?
2. How does the vertex form of a quadratic function help in identifying the vertex quickly?
3. How would the graphs of \( f(x) \) and \( g(x) \) compare visually?
4. How do the x-intercepts relate to the factors of each function?
5. Can you find the axis of symmetry for each function?

### Tip:
When finding the vertex of a quadratic function in standard form \( ax^2 + bx + c \), use the formula \( x = -\frac{b}{2a} \) for the x-coordinate of the vertex.

Consider two functions defined by f(x) = x(x + 4) and g(x) = x(x - 4). Complete the value table for each function, then determine the x-axis intercepts and the vertex of each graph.

Learn how to analyze quadratic functions by completing a value table for f(x) = x(x + 4) and g(x) = x(x - 4), identifying intercepts, and calculating vertices. Perfect for students studying algebra in Grades 8-10.

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