The given function is:

$f(x) = -(x + 1)^2 + 4$

This is a quadratic function, and the negative sign in front of $(x + 1)^2$ indicates that the parabola opens downward. Let's analyze the function step by step:

1. Vertex Form of a Quadratic Function

The function is already in vertex form, which is generally written as:

$f(x) = a(x - h)^2 + k$

Where $(h, k)$ is the vertex of the parabola. For your function:

$f(x) = -(x + 1)^2 + 4$

This is equivalent to:

$f(x) = -1(x - (-1))^2 + 4$

Thus, the vertex is at $(-1, 4)$.

2. Opening Direction

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

3. Y-intercept

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

$f(0) = -(0 + 1)^2 + 4 = -(1)^2 + 4 = -1 + 4 = 3$

Thus, the y-intercept is $f(0) = 3$, or the point $(0, 3)$.

4. Axis of Symmetry

The axis of symmetry is the vertical line passing through the vertex. Since the vertex is at $x = -1$, the axis of symmetry is:

$x = -1$

5. X-intercepts (Roots)

To find the x-intercepts, set $f(x) = 0$:

$0 = -(x + 1)^2 + 4$

Solving for $x$:

$(x + 1)^2 = 4$

$x + 1 = \pm 2$

$x = -1 \pm 2$

Thus, the x-intercepts are:

$x = -1 + 2 = 1 \quad \text{and} \quad x = -1 - 2 = -3$

So, the x-intercepts are $x = 1$ and $x = -3$, or the points $(1, 0)$ and $(-3, 0)$.

Summary

• Vertex: $(-1, 4)$
• Direction: Opens downward
• Y-intercept: $(0, 3)$
• Axis of symmetry: $x = -1$
• X-intercepts: $(1, 0)$ and $(-3, 0)$

Would you like more detailed explanations on any of these steps? Or perhaps visualizations of the function?

Related Questions:

1. How would the graph change if the equation were $f(x) = (x + 1)^2 + 4$?
2. How to solve for the vertex of a quadratic function in general?
3. What is the importance of the axis of symmetry in quadratic functions?
4. How do you find the maximum or minimum value of a quadratic function?
5. What are the different forms of quadratic functions, and how are they useful?

Tip: To sketch a quadratic function easily, always start by identifying the vertex, axis of symmetry, and x- and y-intercepts.