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Creating a table of values is the foundational step in graphing functions like \( y = \sqrt{4 - x^2} \). By choosing specific \( x \) values from the function's domain and calculating the corresponding \( y \) values, we can plot these points on a coordinate grid. This helps create a visual representation of the equation.

For \( y = \sqrt{4 - x^2} \), we need to ensure that the expression under the square root is non-negative. This results in the condition \( 4 - x^2 \geq 0 \), meaning \( -2 \leq x \leq 2 \), because the square root of a negative number is not defined in the real numbers.

After choosing integer values \(-2, -1, 0, 1, 2\), calculate each \( y \) by substituting \( x \) into \( y = \sqrt{4 - x^2} \). For example:

• \( x = -2 \), \( y = \sqrt{0} = 0 \)
• \( x = -1 \), \( y = \sqrt{3} \)
• \( x = 0 \), \( y = 2 \)
• \( x = 1 \), \( y = \sqrt{3} \)
• \( x = 2 \), \( y = \sqrt{0} = 0 \)

These points provide a framework for sketching the curve.

Intercepts are key points where the graph intersects the axes; understanding them is crucial for graphing. For the equation \( y = \sqrt{4 - x^2} \), identifying the intercepts assists in outlining the graph's placement on the coordinate plane.

**Finding the \( x \)-Intercepts**
The \( x \)-intercepts occur where \( y = 0 \). From the function, \( y = \sqrt{4 - x^2} = 0 \) when \( x = -2 \) and \( x = 2 \). Therefore, the \( x \)-intercepts are the points \,(-2, 0)\, and \, (2, 0)\.

**Finding the \( y \)-Intercept**
The \( y \)-intercept is the point where the graph crosses the \( y \)-axis, which occurs when \( x = 0 \). Substitute this into the equation to find \( y \):

• \( y = \sqrt{4 - (0)^2} = 2 \)

The \( y \)-intercept is thus \( (0, 2) \). These intercepts provide critical reference points to start drawing the graph.

Symmetry in graphing helps simplify complex functions and can reveal innate characteristics of graphs. For the function \( y = \sqrt{4 - x^2} \), we test for symmetry with respect to the \( y \)-axis to see if the graph is mirrored evenly across this axis.

A function is symmetric with respect to the \( y \)-axis if substituting \(-x\) for \(x\) yields the original function. Let's test this notion:

Substitute \(-x\) in place of \(x\):
\( y = \sqrt{4 - (-x)^2} = \sqrt{4 - x^2} \)

This new expression equals the original function, proving that \( y = \sqrt{4 - x^2} \) is symmetric regarding the \( y \)-axis.

Recognizing this symmetry assists in graphing: once we sketch one half, we can reflect these points across the \( y \)-axis to construct the other half.

Understanding the domain and range is crucial in determining where the function exists in the Cartesian plane. For the function \( y = \sqrt{4 - x^2} \), both these concepts help set the boundaries of its graph.

**Domain**
The domain refers to all possible \( x \) values for which the function is defined. Since the expression under the square root, \( 4 - x^2 \), must be non-negative:

• \( 4 - x^2 \geq 0 \)

This simplifies to \( -2 \leq x \leq 2 \). Thus, the domain comprises all real numbers from -2 to 2 inclusive.

**Range**
The range is all possible \( y \) values the function can produce. Observing \( y = \sqrt{4 - x^2} \), we see:

• The minimum value of \( y \) is 0 (when \( x = 2 \) or \( x = -2 \))
• The maximum value of \( y \) is 2 (when \( x = 0 \))

Thus, the range is \( 0 \leq y \leq 2 \).

Knowing the domain and range confines the graph's horizontal and vertical extents, allowing for accurate plotting.