To sketch the graph of the function \( y = -x^2 + 6x - 8 \), we'll follow a similar step-by-step approach to analyze its key features, including the vertex, axis of symmetry, intercepts, and overall shape.

### Step-by-Step Analysis

1. **Identify the Type of Function**:  
   The function \( y = -x^2 + 6x - 8 \) is a quadratic function. Its general form is \( y = ax^2 + bx + c \), where \( a = -1 \), \( b = 6 \), and \( c = -8 \).

2. **Determine the Shape**:  
   Since \( a = -1 < 0 \), the parabola opens downwards.

3. **Find the Vertex**:  
   The vertex form of a parabola is given by \( x = -\frac{b}{2a} \). For this function:
   \[
   x = -\frac{6}{2 \times (-1)} = \frac{-6}{-2} = 3.
   \]
   To find the y-coordinate of the vertex, substitute \( x = 3 \) into the function:
   \[
   y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1.
   \]
   So, the vertex is at \( (3, 1) \).

4. **Axis of Symmetry**:  
   The axis of symmetry for a parabola \( y = ax^2 + bx + c \) is the vertical line passing through the vertex. In this case, it is \( x = 3 \).

5. **Find the Y-intercept**:  
   The y-intercept occurs when \( x = 0 \):
   \[
   y = -(0)^2 + 6(0) - 8 = -8.
   \]
   So, the y-intercept is at \( (0, -8) \).

6. **Find the X-intercepts**:  
   The x-intercepts occur when \( y = 0 \). We solve the quadratic equation:
   \[
   -x^2 + 6x - 8 = 0.
   \]
   Multiply the equation by -1 to simplify:
   \[
   x^2 - 6x + 8 = 0.
   \]
   Factor the quadratic:
   \[
   (x - 2)(x - 4) = 0.
   \]
   So, \( x = 2 \) and \( x = 4 \). The x-intercepts are at \( (2, 0) \) and \( (4, 0) \).

7. **Sketch the Graph**:  
   - Plot the vertex at \( (3, 1) \).
   - The parabola opens downwards.
   - The axis of symmetry is the vertical line \( x = 3 \).
   - Plot the y-intercept at \( (0, -8) \).
   - Plot the x-intercepts at \( (2, 0) \) and \( (4, 0) \).

### Graph of the Function
Based on the analysis above, the graph is a downward-opening parabola with the vertex at \( (3, 1) \), x-intercepts at \( (2, 0) \) and \( (4, 0) \), and a y-intercept at \( (0, -8) \).

If you'd like a visual representation, here is a sketch of the graph:

```
  y
  |
  |           *
  |        *     *
  |     *           *
  |  *                 *
  |                         *      
--+--------------------------- x
  |                     *
  |                *
  |          *
  |     *
  |  *
  |
```

Would you like more details on any step, or have any specific questions?

### Related Questions:

1. How would the graph change if the equation were \( y = x^2 - 6x + 8 \)?
2. What is the importance of finding the vertex in a quadratic function?
3. How do you determine the direction in which a parabola opens?
4. What is the role of the discriminant in finding the x-intercepts of a quadratic function?
5. How can you convert a quadratic function from standard form to vertex form?

**Tip:** Knowing how to factor quadratic expressions can greatly simplify finding the x-intercepts of a quadratic function.

Learn how to sketch the graph of the quadratic function Y = -x^2 + 6x - 8 with detailed steps. Understand the vertex, axis of symmetry, intercepts, and overall shape of the parabola. Suitable for Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation