The question sheet you uploaded is a mathematics test for 10th grade, focused on analyzing the function $f(x) = |(x + 2)^3| - 1$. Let’s address each question one by one.

Given:

$f(x) = |(x + 2)^3| - 1$

Step-by-Step Solution:

1. Find the zeros of the function:

   • To find the zeros, we set $f(x) = 0$: $|(x + 2)^3| - 1 = 0$ $|(x + 2)^3| = 1$
   • This implies: $(x + 2)^3 = 1 \quad \text{or} \quad (x + 2)^3 = -1$
   • Solving these two cases:
     • For $(x + 2)^3 = 1$: $x + 2 = 1 \implies x = -1$
     • For $(x + 2)^3 = -1$: $x + 2 = -1 \implies x = -3$
   • Thus, the zeros are at $x = -1$ and $x = -3$.

2. Find the intersection of the function with the y-axis:

   • The y-axis intersection occurs when $x = 0$: $f(0) = |(0 + 2)^3| - 1 = |8| - 1 = 7$
   • So, the function intersects the y-axis at $(0, 7)$.

3. List the maximum and minimum points of the function in the interval $[-4, 1]$:

   • To identify the maximum and minimum points, we evaluate the function at key points within the interval $[-4, 1]$, including endpoints and critical points. We’ll also need to check if the function changes its behavior at $x = -3$ and $x = -1$ due to absolute value impacts.

   Calculations:

   • At $x = -4$: $f(-4) = |(-4 + 2)^3| - 1 = |-8| - 1 = 7$
   • At $x = -3$: $f(-3) = |(-3 + 2)^3| - 1 = |-1| - 1 = -2$
   • At $x = -1$: $f(-1) = |(-1 + 2)^3| - 1 = |1| - 1 = 0$
   • At $x = 1$: $f(1) = |(1 + 2)^3| - 1 = |27| - 1 = 26$
   • The maximum value in this interval $[-4, 1]$ is 26 at $x = 1$, and the minimum value is -2 at $x = -3$.

4. Specify the intervals in $[-4, 1]$ where the function is decreasing and increasing:

   • Decreasing Interval: The function decreases on the interval $[-4, -3]$ and $[-2, -1]$.
   • Increasing Interval: The function increases on the intervals $[-3, -2]$ and $[-1, 1]$.

5. Evaluate and complete the graph of the function for $x \leq 0$:

   • For $x \leq 0$, we analyze $f(x)$ by plotting points and noting behavior changes at zeros and inflection points. The key pairs in this region are $(-4, 7)$, $(-3, -2)$, $(-2, -1)$, and $(-1, 0)$.

6. Evaluate and complete the graph of the function for $x > 0$:

   • For $x > 0$, we have previously calculated $f(1) = 26$ at $x = 1$. You would continue by plotting values between 0 and 1, identifying the rapid increase in $f(x)$.

Let me know if you would like further details on any part of these solutions.

Related Questions for Deeper Understanding:

1. How do we determine the intervals of increase and decrease for functions with absolute values?
2. What is the impact of absolute values on the zeros of functions?
3. Why do we only consider certain intervals when finding maximum and minimum points?
4. How would the function change if it were $f(x) = (x + 2)^3 - 1$ (without the absolute value)?
5. How can we find points of inflection in absolute value functions?

Tip:

When dealing with functions involving absolute values, always split the function into cases to handle the positive and negative scenarios separately. This helps in finding accurate intervals and behavior changes.