https://file.aiquickdraw.com/mathbot/file/1736954245675udgoqbls.jpg

Let’s analyze the two problems from the uploaded image and solve them step by step:

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### **Problem 5**
The quadratic function is given as:
\[
f(x) = (x + 2)(x - 4)
\]

#### 1. Expand and Rewrite:
Expanding the equation:
\[
f(x) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8
\]

#### 2. Identify Key Features:
- **X-Intercepts:** Set \(f(x) = 0\):
  \[
  (x + 2)(x - 4) = 0 \quad \Rightarrow \quad x = -2 \text{ and } x = 4
  \]
  The x-intercepts are: \((-2, 0)\) and \((4, 0)\).

- **Y-Intercept:** Set \(x = 0\):
  \[
  f(0) = (0 + 2)(0 - 4) = -8
  \]
  The y-intercept is: \((0, -8)\).

- **Vertex:** The vertex lies at the midpoint of the x-intercepts:
  \[
  x_{\text{vertex}} = \frac{-2 + 4}{2} = 1
  \]
  Substitute \(x = 1\) into \(f(x)\):
  \[
  f(1) = (1 + 2)(1 - 4) = 3(-3) = -9
  \]
  The vertex is: \((1, -9)\).

#### 3. Graph:
- Plot the x-intercepts \((-2, 0)\) and \((4, 0)\), the y-intercept \((0, -8)\), and the vertex \((1, -9)\).
- The parabola opens upwards as the coefficient of \(x^2\) is positive.

---

### **Problem 6**
The quadratic function is given as:
\[
g(x) = (x - 2)^2 - 1
\]

#### 1. Expand and Rewrite:
Expanding the equation:
\[
g(x) = x^2 - 4x + 4 - 1 = x^2 - 4x + 3
\]

#### 2. Identify Key Features:
- **Y-Intercept:** Set \(x = 0\):
  \[
  g(0) = (0 - 2)^2 - 1 = 4 - 1 = 3
  \]
  The y-intercept is: \((0, 3)\).

- **Vertex:** The vertex form directly gives the vertex as:
  \((2, -1)\).

- **Another Point:** For \(x = 1\):
  \[
  g(1) = (1 - 2)^2 - 1 = 1 - 1 = 0
  \]
  Another point is: \((1, 0)\).

- The parabola opens upwards because the coefficient of \(x^2\) is positive.

#### 3. Graph:
- Plot the vertex \((2, -1)\), the y-intercept \((0, 3)\), and the point \((1, 0)\).
- Reflect points to the right of the vertex to the left (e.g., reflect \((1, 0)\) to \((3, 0)\)).

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Would you like me to graph these functions for clarity? Let me know! 😊

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### Related Questions:
1. How can we identify whether a parabola opens upwards or downwards?
2. What are the differences between the factored form and vertex form of a quadratic function?
3. How can you determine the axis of symmetry of a parabola from its equation?
4. How do transformations like shifting and stretching affect the graph of a parabola?
5. How do we find the roots of a quadratic equation if it does not factor easily?

### Tip:
The vertex form of a quadratic function is very useful for quickly identifying the vertex and transformations of the parent function \(y = x^2\).

Graph the functions f(x) = (x + 2)(x - 4) and g(x) = (x - 2)^2 - 1 on the coordinate plane, label the x-intercepts, y-intercepts, vertex, and any other coordinates.

Explore step-by-step solutions for graphing quadratic functions f(x) = (x + 2)(x - 4) and g(x) = (x - 2)^2 - 1. Learn to identify x-intercepts, y-intercepts, vertices, and other coordinates. Perfect for students in Grades 9-11.

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