Problem 8 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ y=\sqrt{4-x^{2}} \text { (Hint: See Example 7.) } $$

Step-by-Step Solution

Answer

Graph the upper half of a circle centered at the origin with radius 2.

Step 1: Recognize the Shape of the Graph

The given function is \( y = \sqrt{4 - x^2} \). This resembles the standard equation of a circle, \( x^2 + y^2 = r^2 \). In this modified form, it represents the upper half of a circle centered at the origin with a radius of 2.

Step 2: Determine Domain and Range

To ensure the expression under the square root is non-negative, set \( 4 - x^2 \geq 0 \). Solving this inequality gives \( -2 \leq x \leq 2 \). The range of \( y \) is from 0 to 2, as \( y \) will be non-negative and at its maximum when \( x = 0 \).

Step 3: Sketch the Circle

Graph the full circle with the equation \( x^2 + y^2 = 4 \). This is a circle with center (0,0) and radius 2. However, we only consider the upper half of this circle due to the square root function, which only returns non-negative values.

Step 4: Plot Key Points on the Graph

Identify and plot key points: \((-2,0)\), \((0,2)\), \((2,0)\). These represent the leftmost, topmost, and rightmost points on the semicircle.

Step 5: Draw the Upper Semicircle

Connect the points with a smooth, curve which is part of a circle spanning from \((-2,0)\) to \((2,0)\), representing the upper half of a circle. This is the graph of the function \( y = \sqrt{4 - x^2} \).

Key Concepts

Domain and Range

The domain and range of a function are fundamental concepts when sketching graphs. For the function \( y = \sqrt{4 - x^2} \), understanding these concepts helps visualize its shape correctly.

• Domain: The domain refers to all possible input values for \( x \) that keep the expression under the square root non-negative. For the given function, the expression is \( 4 - x^2 \). We solve the inequality \( 4 - x^2 \geq 0 \) to find the domain. Rearranging gives \( -2 \leq x \leq 2 \), meaning \( x \) can be any value between -2 and 2.
• Range: The range of the function consists of all possible output values for \( y \). Since \( y \) represents a square root, it will always be non-negative. Here, \( y \) achieves a maximum value of 2 when \( x = 0 \), resulting in a range of \( 0 \leq y \leq 2 \).

Understanding the domain and range allows you to plot the function accurately, ensuring that all parts of the possible graph are considered.

Circle Equations

Circle equations are essential to understanding the shape formed by quadratic equations. The standard form of a circle's equation is \( x^2 + y^2 = r^2 \), where \( r \) is the radius.

In our exercise, \( y = \sqrt{4 - x^2} \) is related to the equation \( x^2 + y^2 = 4 \). This is a circle with:

• Center: Located at (0,0), the origin.
• Radius: The radius, \( r \), is 2 since \( 4 \) is the square of the radius.

When graphing functions that connect to a circle, like the upper semicircle in this problem, it's helpful to recognize the full circle first. Know that the root symbol indicates we only consider the non-negative \( y \) values, describing half of the circle.

Semicircle

A semicircle is simply half of a circle. For the function \( y = \sqrt{4 - x^2} \), this specifically refers to the upper half of the circle with equation \( x^2 + y^2 = 4 \).

A semicircle maintains the same:

• Center: The midpoint of the diameter; here, it is (0,0).
• Radius: Stays constant at 2 in this exercise.
• Shape: A smooth curve from one point on the horizontal axis to another, but only covering the top half.

Due to the square root in \( y = \sqrt{4 - x^2} \), only non-negative \( y \) values are depicted. Hence, the semicircle extends from \((-2,0)\) to \((2,0)\), passing smoothly through the topmost point \((0,2)\). This is the graphical representation of the function as an upper semicircle.