Problem 27 - Algebra and Trigonometry

Question

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=\sqrt{4-x^{2}} $$

Step-by-Step Solution

Answer

The x-intercepts are (-2, 0) and (2, 0); the y-intercept is (0, 2).

Step 1: Determine the Domain of the Function

The equation given is \( y = \sqrt{4-x^2} \). Since the expression inside the square root, \( 4-x^2 \), must be non-negative, we set up the inequality \( 4-x^2 \geq 0 \). Solving this gives \(-2 \leq x \leq 2\). Thus, the function is defined for \( x \) values in this range.

Step 2: Create a Table of Values

With the domain in mind, select several \( x \) values within the interval \([-2, 2]\) to calculate corresponding \( y \) values. For example:- \( x = -2 \), \( y = \sqrt{4 - (-2)^2} = \sqrt{0} = 0 \)- \( x = -1 \), \( y = \sqrt{4 - (-1)^2} = \sqrt{3} \approx 1.732 \)- \( x = 0 \), \( y = \sqrt{4 - 0^2} = \sqrt{4} = 2 \)- \( x = 1 \), \( y = \sqrt{4 - 1^2} = \sqrt{3} \approx 1.732 \)- \( x = 2 \), \( y = \sqrt{4 - 2^2} = \sqrt{0} = 0 \).

Step 3: Graph the Equation Using the Table of Values

Using the table of values, plot the points \((-2, 0)\), \((-1, 1.732)\), \((0, 2)\), \((1, 1.732)\), and \((2, 0)\) on a coordinate grid. Draw a smooth curve connecting these points to visualize the graph. The graph is a semi-circle above the \( x \)-axis, centered at the origin \((0,0)\) with a radius of 2.

Step 4: Identify the Intercepts

- The \( y \)-intercept occurs where the graph crosses the \( y \)-axis (\( x=0\)). From the table, at \( x=0 \), \( y=2 \), so the \( y \)-intercept is \((0, 2)\).- The \( x \)-intercepts occur where \( y=0 \). From the table, this is at \( x=-2 \) and \( x=2 \), so the \( x \)-intercepts are \((-2, 0)\) and \((2, 0)\).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values, commonly known as the "x-values," for which the function is defined and produces real outputs. In the context of square root functions, the expression under the square root must be non-negative because the square root of a negative number is not a real number.

Considering the given function, \( y = \sqrt{4-x^2} \), we need to ensure that the expression inside the square root, \( 4-x^2 \), is greater than or equal to zero. This requirement leads to the inequality \( 4 - x^2 \geq 0 \). Solving for \( x \) here will show us which x-values make the expression non-negative:

• Rearrange the inequality: \( x^2 \leq 4 \)
• Take the square root: \(-2 \leq x \leq 2 \)

So, the domain of this function is the set \([-2, 2]\), meaning the function is only defined for x-values between -2 and 2, inclusive, where it results in real (non-imaginary) y-values.

Intercepts in Graphs

Intercepts are key features in understanding graphs. They reveal crucial information about where the graph crosses the axes.

Y-intercept: This occurs where the graph meets the y-axis, which is when \( x=0 \). To find it, substitute \( x=0 \) into the equation and solve for \( y \):

• \( y = \sqrt{4-0^2} = \sqrt{4} = 2 \)

Therefore, the y-intercept is at the point \((0, 2)\). It is where the graph touches the y-axis.X-intercepts: These happen where the graph hits the x-axis and occur when \( y=0 \). For our function, set \( y=0 \) and solve for \( x \):

• \( 0 = \sqrt{4-x^2} \)
• Squaring both sides gives \( 0 = 4 - x^2 \)
• Solving the equation leads to \( x^2 = 4 \)
• Therefore, \( x = -2 \) or \( x = 2 \)

Thus, the x-intercepts are at \((-2, 0)\) and \((2, 0)\). These are the points where the graph intersects the x-axis.

Table of Values for Graphing

Creating a table of values is a practical step for sketching the graph of a function. This involves choosing specific input values (x-values) within the domain, calculating their corresponding output values (y-values), and using these to plot points on the graph. For the function \( y = \sqrt{4-x^2} \), use values within the domain \([-2, 2]\).Here's how to build such a table:

• Select x-values within the domain such as -2, -1, 0, 1, 2.
• Use the function to find y corresponding to each x.
  • For \( x = -2 \), \( y = \sqrt{4 - (-2)^2} = \sqrt{0} = 0 \)
  • For \( x = -1 \), \( y = \sqrt{4 - (-1)^2} = \sqrt{3} \approx 1.732 \)
  • For \( x = 0 \), \( y = \sqrt{4 - 0^2} = \sqrt{4} = 2 \)
  • For \( x = 1 \), \( y = \sqrt{4 - 1^2} = \sqrt{3} \approx 1.732 \)
  • For \( x = 2 \), \( y = \sqrt{4 - 2^2} = \sqrt{0} = 0 \)

Using these points, you can place them on a coordinate grid to visually represent the function as a plot. Join the points smoothly to reveal the characteristic shape of the graph. In this case, it forms a semi-circle above the x-axis, centered at the origin with a radius of 2.