Question

describe the graph of the function given below.
\\(f(x) = -2(x + 4)^2 + 12\\)
as \\(x \to 2\\), the graph decreases / increases to the y-value.
as \\(x \to -4\\), the graph decreases / increases to the y-value.
as \\(x \to 12\\), the graph decreases / increases to the y-value.

Explanation:

🆕 New Concept Discovered: Piecewise Functions
Different rules for different x-values

Step 1: Understand the function definition

A piecewise function uses different formulas depending on the input value \( x \). The given function is:

\[
f(x) =

$$\begin{cases} -2 & \text{for } x \le -2 \\ x & \text{for } -2 < x \le 2 \\ 2 & \text{for } x > 2 \end{cases}$$

\]

This means:

• If \( x \) is less than or equal to \(-2\), the output is always the constant value \(-2\).
• If \( x \) is between \(-2\) and \( 2 \) (including \( 2 \)), the output is equal to \( x \) itself.
• If \( x \) is greater than \( 2 \), the output is always the constant value \( 2 \).

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Step 2: Analyze the behavior for \( x \le -2 \)

For the interval \( x \le -2 \), the rule is \( f(x) = -2 \).

• This is a constant value, which means the graph is a flat, horizontal line.
• Since it applies to all values of \( x \) less than or equal to \(-2\), this horizontal line extends infinitely to the left.

Therefore, for \( x \le -2 \), the graph is a horizontal line.

---

Step 3: Analyze the behavior for \( -2 < x \le 2 \)

For the interval \( -2 < x \le 2 \), the rule is \( f(x) = x \).

• This is a linear equation with a slope of \( 1 \) and a y-intercept of \( 0 \).
• Because the slope is positive (\( 1 \)), the line goes upward from left to right.

Therefore, for \( -2 < x \le 2 \), the graph is a line with a positive slope.

---

Step 4: Analyze the behavior for \( x > 2 \)

For the interval \( x > 2 \), the rule is \( f(x) = 2 \).

• This is another constant value, meaning the graph is a flat, horizontal line.
• Since it applies to all values of \( x \) greater than \( 2 \), this horizontal line extends infinitely to the right.

Therefore, for \( x > 2 \), the graph is a horizontal line.

Answer:

• For \( x \le -2 \), the graph is a horizontal line.
• For \( -2 < x \le 2 \), the graph is a line with a positive slope.
• For \( x > 2 \), the graph is a horizontal line.