We are analyzing the function $y = 2|x + 2| + 1$ to understand how the y-values behave as x increases and as x decreases.

Breakdown of the function:

• $|x + 2|$ represents the absolute value of $x + 2$.
• The absolute value function produces two behaviors:
  • If $x + 2 \geq 0$ (i.e., $x \geq -2$), the function simplifies to $x + 2$.
  • If $x + 2 < 0$ (i.e., $x < -2$), the function becomes $-(x + 2)$.
• The absolute value function causes the graph to have a V-shape.

The term 2 scales the output of the absolute value by a factor of 2, and the +1 shifts the graph upward by 1 unit.

As $x$ increases:

• When $x \geq -2$, the expression inside the absolute value becomes positive, so $|x + 2| = x + 2$.
• The function simplifies to $y = 2(x + 2) + 1 = 2x + 5$.
• As $x$ increases, the y-value increases linearly because of the positive slope $2$.

As $x$ decreases:

• When $x < -2$, the expression inside the absolute value is negative, so $|x + 2| = -(x + 2)$.
• The function simplifies to $y = 2(-(x + 2)) + 1 = -2x - 4 + 1 = -2x - 3$.
• As $x$ decreases, the y-value also increases linearly because the slope is now $-2$, but it increases as $x$ becomes more negative.

Summary:

• For $x \geq -2$, the y-values increase as $x$ increases.
• For $x < -2$, the y-values increase as $x$ decreases (because the absolute value flips the negative slope).

Would you like more details on this behavior?

Follow-up Questions:

1. How does shifting the graph of an absolute value function affect its y-values?
2. What is the significance of the vertex in the graph of an absolute value function?
3. How do transformations (scaling, shifting) affect the rate of change of y-values in functions?
4. Can you express the vertex form of a general absolute value function?
5. What would happen to the function if the coefficient of the absolute value were negative?

Tip: When analyzing functions with absolute values, focus on how the function behaves at the points where the inside of the absolute value is zero, as this often reveals key changes in the function's behavior.