To sketch the graph of the quadratic function $y = -x^2 + 6x - 8$, let's follow these steps:

1. Identify the form of the quadratic function:

The equation is in the standard form $y = ax^2 + bx + c$, where:

• $a = -1$ (coefficient of $x^2$)
• $b = 6$ (coefficient of $x$)
• $c = -8$ (constant term)

2. Determine the vertex:

The vertex of a parabola given by $y = ax^2 + bx + c$ can be found using the formula: $x = \frac{-b}{2a}$ Substituting the values of $a$ and $b$: $x = \frac{-6}{2(-1)} = \frac{-6}{-2} = 3$ To find the $y$-coordinate of the vertex, substitute $x = 3$ into the equation: $y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1$ So, the vertex is at $(3, 1)$.

3. Determine the axis of symmetry:

The axis of symmetry is the vertical line that passes through the vertex: $x = 3$

4. Identify the direction of the parabola:

Since the coefficient of $x^2$ (i.e., $a = -1$) is negative, the parabola opens downward.

5. Find the y-intercept:

The y-intercept occurs where $x = 0$: $y = -(0)^2 + 6(0) - 8 = -8$ So, the y-intercept is at $(0, -8)$.

6. Find the x-intercepts (roots):

To find the x-intercepts, solve the quadratic equation $-x^2 + 6x - 8 = 0$: $-x^2 + 6x - 8 = 0$ Multiply by -1 to simplify: $x^2 - 6x + 8 = 0$ Factor the quadratic equation: $(x - 2)(x - 4) = 0$ So, the x-intercepts are $x = 2$ and $x = 4$.

7. Sketch the graph:

• Plot the vertex at $(3, 1)$.
• Draw the axis of symmetry at $x = 3$.
• Mark the y-intercept at $(0, -8)$.
• Plot the x-intercepts at $(2, 0)$ and $(4, 0)$.
• Draw the parabola opening downward, passing through these points.

Would you like a visual sketch of the graph as well?

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Here are 5 relative questions that build on this:

1. How would the graph change if the coefficient of $x^2$ were positive?
2. What is the significance of the vertex in a quadratic function?
3. How would you solve a quadratic equation if it couldn't be factored?
4. Can you determine the maximum value of the function $y = -x^2 + 6x - 8$?
5. How does the discriminant of the quadratic equation relate to the number of x-intercepts?

Tip: The vertex of a quadratic function is either the maximum or minimum point on its graph, depending on the sign of the coefficient of $x^2$.