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.