Graphs and Graphing


Graph the curves in Exercises 33–42.
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y = 𝓍√4 ― 𝓍²

Verified step by step guidance

1. Identify the function to be graphed: \( y = x \sqrt{4 - x^2} \). This is a function involving a square root, which suggests that the domain is restricted to values where the expression under the square root is non-negative.
2. Determine the domain of the function. The expression under the square root, \( 4 - x^2 \), must be greater than or equal to zero. Solve the inequality \( 4 - x^2 \geq 0 \) to find the domain.
3. Solve \( 4 - x^2 \geq 0 \) by rearranging it to \( x^2 \leq 4 \). This implies \( -2 \leq x \leq 2 \). Therefore, the domain of the function is \([-2, 2]\).
4. Analyze the behavior of the function at the endpoints and critical points. Check the values of the function at \( x = -2, 0, \) and \( 2 \). Also, consider the symmetry of the function, as it involves \( x^2 \), which is an even function.
5. Sketch the graph using the information obtained. Plot the points calculated, consider the symmetry, and ensure the graph is within the domain \([-2, 2]\). The function will have a shape similar to a semicircle, as it is derived from the equation of a circle \( x^2 + y^2 = 4 \).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function y = x√(4 - x²), the expression under the square root, 4 - x², must be non-negative. This means the domain is determined by solving the inequality 4 - x² ≥ 0, which results in the interval [-2, 2].

Square Root Function

The square root function, √x, is defined only for non-negative values of x. It represents the principal (non-negative) square root of x. In the context of y = x√(4 - x²), the square root affects the range and behavior of the function, ensuring that the output is real and non-negative for the domain where 4 - x² is non-negative.

Graphing Techniques

Graphing techniques involve plotting points, identifying key features like intercepts, symmetry, and asymptotes, and understanding the behavior of the function. For y = x√(4 - x²), it's important to consider symmetry about the y-axis and the endpoints of the domain. Analyzing these aspects helps in sketching an accurate graph of the function within its domain.