Problem 35 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ y=\sqrt{4-x^{2}} $$

Step-by-Step Solution

Answer

The graph is the upper semicircle of a circle with radius 2, with x-intercepts at (2, 0) and (-2, 0) and y-intercept at (0, 2). It is symmetric about the y-axis.

Step 1: Understand the Equation

The given equation is \( y = \sqrt{4 - x^2} \). This represents the upper half of a circle centered at the origin \((0,0)\) with a radius of 2. Our task is to find x- and y-intercepts, test for symmetry, make a table of values, and sketch the graph.

Step 2: Finding the x-intercepts

To find the x-intercepts, set \( y = 0 \). The equation becomes \( 0 = \sqrt{4 - x^2} \), which gives \( x^2 = 4 \). Solving for \( x \), we get \( x = \pm 2 \). Thus, the x-intercepts are \((2, 0)\) and \((-2, 0)\).

Step 3: Finding the y-intercept

To find the y-intercept, set \( x = 0 \). Plugging this into the equation, we get \( y = \sqrt{4 - 0^2} = \sqrt{4} = 2 \). So, the y-intercept is \((0, 2)\).

Step 4: Test for Symmetry

This graph is symmetric about the y-axis. For a given point \((x, y)\), if \((x, y)\) is on the graph, then \((-x, y)\) is also on the graph because \(y = \sqrt{4 - x^2}\) has \(x^2\), which is even. This confirms y-axis symmetry.

Step 5: Make a Table of Values

Choose some values for \(x\) and compute corresponding \(y\) values: - \( x = -2, y = \sqrt{4 - (-2)^2} = 0 \) - \( x = -1, y = \sqrt{4 - (-1)^2} = \sqrt{3} \approx 1.73 \) - \( x = 0, y = \sqrt{4 - 0^2} = 2 \) - \( x = 1, y = \sqrt{4 - 1^2} = \sqrt{3} \approx 1.73 \) - \( x = 2, y = \sqrt{4 - 2^2} = 0 \)

Step 6: Sketch the Graph

The graph is the upper semicircle with radius 2, centered at the origin. Plot the points you found: \((-2, 0)\), \((-1, \sqrt{3})\), \((0, 2)\), \((1, \sqrt{3})\), and \((2, 0)\). Connect these with a smooth curve to form the semicircle.

Key Concepts

x-intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. These points are found by setting the equation equal to zero. For the equation \( y = \sqrt{4 - x^2} \), to find the x-intercepts, we set \( y = 0 \). This gives us the equation \( \sqrt{4 - x^2} = 0 \). Solving it leads us to \( x^2 = 4 \), which implies that \( x = \pm 2 \). Therefore, the x-intercepts are the points \((2, 0)\) and \((-2, 0)\).

• x-intercepts occur when \( y = 0 \).
• Solve \( \sqrt{4 - x^2} = 0 \) to find x-intercepts.
• Result: \( x = 2 \) and \( x = -2 \).

y-intercepts

To find the y-intercepts of a graph, we check where the graph crosses the y-axis. This happens when \( x = 0 \). For the given equation \( y = \sqrt{4 - x^2} \), substitute \( x = 0 \), resulting in \( y = \sqrt{4 - 0^2} = \sqrt{4} = 2 \). Therefore, the y-intercept of the graph is the point \((0, 2)\).

• y-intercepts occur when \( x = 0 \).
• Plug in \( x = 0 \) for the equation: \( y = \sqrt{4} \).
• Result: \( y = 2 \), so intercept is \((0, 2)\).

symmetry

Testing a graph for symmetry helps understand its shape and structure. For the equation \( y = \sqrt{4 - x^2} \), we test for symmetry about the y-axis. A graph is symmetric about the y-axis if whenever \((x, y)\) is on the graph, then \((-x, y)\) must also be on the graph. The equation contains \( x^2 \), which is an even function, indicating it is symmetric about the y-axis. This means the graph on the right mirrors that on the left, creating a balanced visual appearance.

• Check the equation for even powers (like \( x^2 \)) for symmetry.
• Symmetry can simplify graph analysis and sketching.
• For our equation, the graph is y-axis symmetric.

table of values

When sketching a graph, creating a table of values is a helpful step. It involves choosing values for \( x \) and then calculating the corresponding \( y \) values using the given equation. For \( y = \sqrt{4 - x^2} \), reasonable \( x \) choices include those within the domain where \( 4 - x^2 \geq 0 \). Some chosen \( x \) values and calculated \( y \) values are:

• \( x = -2 \), \( y = \sqrt{4 - (-2)^2} = 0 \)
• \( x = -1 \), \( y = \sqrt{4 - (-1)^2} = \sqrt{3} \approx 1.73 \)
• \( x = 0 \), \( y = \sqrt{4 - 0^2} = 2 \)
• \( x = 1 \), \( y = \sqrt{4 - 1^2} = \sqrt{3} \approx 1.73 \)
• \( x = 2 \), \( y = \sqrt{4 - 2^2} = 0 \)

These points are then plotted on a graph to form the upper semicircle of a circle with radius 2.