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.