Let's analyze the quadratic function:

$y = -x^2 + 8x - 10$

1. Axis of Symmetry

The general form of a quadratic function is:

$y = ax^2 + bx + c$

For the axis of symmetry, we use the formula:

$x = -\frac{b}{2a}$

Here, $a = -1$, $b = 8$, and $c = -10$. Substituting into the formula:

$x = -\frac{8}{2(-1)} = \frac{8}{2} = 4$

So, the axis of symmetry is $x = 4$.

2. Vertex

The vertex lies on the axis of symmetry, and its $x$-coordinate is $4$. To find the $y$-coordinate, substitute $x = 4$ into the original equation:

$y = -(4)^2 + 8(4) - 10$ $y = -16 + 32 - 10 = 6$

Thus, the vertex is $(4, 6)$.

3. Y-Intercept

The $y$-intercept occurs when $x = 0$. Substituting $x = 0$ into the equation:

$y = -(0)^2 + 8(0) - 10 = -10$

So, the y-intercept is $(0, -10)$.

4. Range

Since the coefficient of $x^2$ is negative ($a = -1$), the parabola opens downwards. The vertex represents the maximum point. The $y$-coordinate of the vertex is 6, so the range is:

$y \leq 6$

Thus, the range is $(-\infty, 6]$.

Summary:

• Axis of symmetry: $x = 4$
• Vertex: $(4, 6)$
• Y-intercept: $(0, -10)$
• Range: $y \leq 6$ or $(-\infty, 6]$

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How do you find the x-intercepts of a quadratic function?
2. How does the sign of the coefficient $a$ affect the direction of the parabola?
3. What is the significance of the discriminant in a quadratic function?
4. Can the axis of symmetry ever be negative?
5. How does changing the constant $c$ affect the graph of a quadratic function?

Tip: The vertex form of a quadratic function can make it easier to identify the vertex directly!