Draw the graph of function. `y=|4-x^(2)|,-3lexle3`.

Answer

To draw the graph of the function \( y = |4 - x^2| \) for the range \( -3 \leq x \leq 3 \), we will break down the problem into manageable steps.

### Step 1: Understand the function
The function \( y = |4 - x^2| \) involves an absolute value, which means we need to consider two cases:
1. When \( 4 - x^2 \geq 0 \) (i.e., \( x^2 \leq 4 \) or \( -2 \leq x \leq 2 \)), then \( y = 4 - x^2 \).
2. When \( 4 - x^2 < 0 \) (i.e., \( x^2 > 4 \) or \( x < -2 \) or \( x > 2 \)), then \( y = x^2 - 4 \).

### Step 2: Determine the intervals
From the above analysis, we can break down the intervals for \( x \):
- For \( -3 \leq x < -2 \): \( y = x^2 - 4 \)
- For \( -2 \leq x \leq 2 \): \( y = 4 - x^2 \)
- For \( 2 < x \leq 3 \): \( y = x^2 - 4 \)

### Step 3: Calculate key points
Now we will calculate the values of \( y \) at key points in each interval.

1. **Interval \( -3 \leq x < -2 \)**:
- At \( x = -3 \):
\[ y = (-3)^2 - 4 = 9 - 4 = 5 \quad \Rightarrow \quad (-3, 5) \]
- At \( x = -2 \):
\[ y = (-2)^2 - 4 = 4 - 4 = 0 \quad \Rightarrow \quad (-2, 0) \]

2. **Interval \( -2 \leq x \leq 2 \)**:
- At \( x = -2 \):
\[ y = 4 - (-2)^2 = 4 - 4 = 0 \quad \Rightarrow \quad (-2, 0) \]
- At \( x = 0 \):
\[ y = 4 - 0^2 = 4 \quad \Rightarrow \quad (0, 4) \]
- At \( x = 2 \):
\[ y = 4 - 2^2 = 4 - 4 = 0 \quad \Rightarrow \quad (2, 0) \]

3. **Interval \( 2 < x \leq 3 \)**:
- At \( x = 2 \):
\[ y = 2^2 - 4 = 4 - 4 = 0 \quad \Rightarrow \quad (2, 0) \]
- At \( x = 3 \):
\[ y = 3^2 - 4 = 9 - 4 = 5 \quad \Rightarrow \quad (3, 5) \]

### Step 4: Plot the points
Now we have the following points to plot:
- \( (-3, 5) \)
- \( (-2, 0) \)
- \( (0, 4) \)
- \( (2, 0) \)
- \( (3, 5) \)

### Step 5: Draw the graph
1. For the interval \( -3 \leq x < -2 \), plot the points and draw the curve of \( y = x^2 - 4 \) which opens upwards.
2. For the interval \( -2 \leq x \leq 2 \), plot the points and draw the curve of \( y = 4 - x^2 \) which opens downwards, reaching a maximum at \( (0, 4) \).
3. For the interval \( 2 < x \leq 3 \), plot the points and draw the curve of \( y = x^2 - 4 \) which again opens upwards.

### Final Graph
The graph will consist of two upward-opening parabolas on either side of a downward-opening parabola in the middle.

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