Let's determine the key characteristics of the function $f(x) = -2(x + 2)^2 + 1$.

This function is in the form $f(x) = a(x - h)^2 + k$, which is a standard form of a quadratic equation. It represents a parabola with its vertex at $(h, k)$ and opens upwards if $a > 0$ or downwards if $a < 0$. Let's analyze this specific function step by step.

1. Vertex:

The vertex of the function $f(x) = a(x - h)^2 + k$ is $(h, k)$.

• In this function, $a = -2$, $h = -2$, and $k = 1$.
• Therefore, the vertex is at $(-2, 1)$.

2. Axis of Symmetry:

The axis of symmetry for a quadratic function $f(x) = a(x - h)^2 + k$ is $x = h$.

• Here, $h = -2$.
• The axis of symmetry is $x = -2$.

3. Direction of Opening:

The direction in which the parabola opens is determined by the sign of $a$.

• Since $a = -2$, which is negative, the parabola opens downwards.

4. Y-intercept:

To find the y-intercept, set $x = 0$ and solve for $f(x)$:

$f(0) = -2(0 + 2)^2 + 1 = -2(4) + 1 = -8 + 1 = -7$

• The y-intercept is $(0, -7)$.

5. Range:

Since the parabola opens downward and the vertex is at $(-2, 1)$, the function has a maximum value of $y = 1$.

• Therefore, the range is $(-\infty, 1]$.

Summary:

• Vertex: $(-2, 1)$
• Axis of Symmetry: $x = -2$
• Direction of Opening: Downwards
• Y-intercept: $(0, -7)$
• Range: $(-\infty, 1]$

Would you like more details on any part of this analysis, or do you have any other questions?

Here are five related questions for further exploration:

1. How do you find the x-intercepts of this quadratic function?
2. What is the domain of the function $f(x)$?
3. How does changing the value of $a$ affect the shape of the parabola?
4. What is the significance of the vertex in the context of quadratic functions?
5. How do you determine the maximum or minimum values of quadratic functions?

Tip: For quadratic functions, the vertex can be found directly using the formula $x = -\frac{b}{2a}$ if given in the form $ax^2 + bx + c$.