For the quadratic function $f(x) = -x^2 + 6x + 7$, let's break down each part:

a) Vertex

The vertex of a parabola in the form $f(x) = ax^2 + bx + c$ can be found using the formula: $x = \frac{-b}{2a}$ Here, $a = -1$, $b = 6$, and $c = 7$. Plugging in the values: $x = \frac{-6}{2(-1)} = 3$ Now, substitute $x = 3$ into the original function to find the y-coordinate of the vertex: $f(3) = -(3)^2 + 6(3) + 7 = -9 + 18 + 7 = 16$ So, the vertex is: $\boxed{(3, 16)}$

b) Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. Since the x-coordinate of the vertex is $x = 3$, the axis of symmetry is: $\boxed{x = 3}$

c) X-intercepts

The x-intercepts are the points where $f(x) = 0$, so we solve: $-x^2 + 6x + 7 = 0$ Multiply the equation by $-1$ to simplify: $x^2 - 6x - 7 = 0$ Now, solve this quadratic using the quadratic formula: $x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-7)}}{2(1)} = \frac{6 \pm \sqrt{36 + 28}}{2} = \frac{6 \pm \sqrt{64}}{2}$ $x = \frac{6 \pm 8}{2}$ Thus, the solutions are: $x = \frac{6 + 8}{2} = 7 \quad \text{and} \quad x = \frac{6 - 8}{2} = -1$ So, the x-intercepts are: $\boxed{(-1, 0) \text{ and } (7, 0)}$

d) Y-intercept

The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the function: $f(0) = -(0)^2 + 6(0) + 7 = 7$ So, the y-intercept is: $\boxed{(0, 7)}$

e) Domain

The domain of a quadratic function is always all real numbers because the function is defined for every $x$. Thus, the domain is: $\boxed{(-\infty, \infty)}$

f) Range

Since the parabola opens downwards (because $a = -1$ is negative), the maximum value of the function occurs at the vertex. The maximum value is the y-coordinate of the vertex, which is 16. Therefore, the range is: $\boxed{(-\infty, 16]}$

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Would you like a more detailed explanation of any part?

Here are 5 related questions to deepen your understanding:

1. How do you find the vertex form of a quadratic function?
2. How can you determine whether a parabola opens upwards or downwards?
3. How do you find the discriminant of a quadratic equation and what does it tell you?
4. How is the quadratic formula derived?
5. How can the concept of symmetry help in solving quadratic equations graphically?

Tip: Always check the sign of the quadratic coefficient (a) to determine the direction of the parabola (upward or downward).