Problem 36 - 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 a semi-circle in the lower half-plane, x-intercepts are (-2, 0) and (2, 0), y-intercept is (0, -2), and it's symmetric about the y-axis.

Step 1: Identify the Domain

To ensure that the expression under the square root is non-negative, we need to satisfy the inequality \(4 - x^2 \geq 0\). This implies \(-2 \leq x \leq 2\). Therefore, the domain of \(y = -\sqrt{4-x^2}\) is \([-2, 2]\).

Step 2: Create a Table of Values

Choose a set of x-values within the domain \([-2, 2]\), calculate corresponding y-values using the equation. For example: - If \(x = -2\), \(y = -\sqrt{4 - (-2)^2} = -\sqrt{0} = 0\)- If \(x = 0\), \(y = -\sqrt{4 - 0^2} = -2\)- If \(x = 2\), \(y = -\sqrt{4 - 2^2} = 0\)Construct a table listing these x and y values.

Step 3: Sketch the Graph

Plot the points from the table on a coordinate plane. Connect the points with a smooth curve. The graph is expected to be a semi-circle in the lower half of the plane, with endpoints at \((-2, 0)\) and \((2, 0)\) and the lowest point at \((0, -2)\).

Step 4: Find the Intercepts

To find the y-intercept, set \(x = 0\): \[ y = -\sqrt{4 - 0^2} = -2 \]So, the y-intercept is \((0, -2)\).To find x-intercepts, set \(y = 0\): \[ 0 = -\sqrt{4 - x^2} \Rightarrow \sqrt{4 - x^2} = 0 \Rightarrow 4 - x^2 = 0 \Rightarrow x = \pm 2 \]So, the x-intercepts are \((-2, 0)\) and \((2, 0)\).

Step 5: Test for Symmetry

1. **Symmetry with respect to the y-axis**: If \(y = -\sqrt{4-x^2}\), replacing \(x\) with \(-x\) gives the same equation, indicating symmetry about the y-axis.2. **Symmetry with respect to the x-axis**: Not possible as y is negative.3. **Symmetry with respect to the origin**: Neither possible, as replacing both \(x\) and \(y\) with negatives gives a different equation.Thus, this graph is symmetric only about the y-axis.

Key Concepts

Domain of a Function

The domain of a function consists of all the input values (or x-values) for which a function is defined. In simpler terms, it's the set of x-values that you can plug into the equation without causing any mathematical mishaps, like division by zero or square roots of negative numbers. For our equation, \( y = -\sqrt{4-x^2} \), we need to be cautious about the square root.
Square roots are defined for non-negative numbers only, meaning the expression inside the square root, \( 4-x^2 \), must be greater than or equal to zero.
- Solve the inequality \( 4-x^2 \geq 0 \): - Rearrange the inequality to \( -x^2 \geq -4 \) - Flip the inequality sign (since we divide by a negative number) to get \( x^2 \leq 4 \) - Find the values of \( x \) that satisfy this, resulting in \( -2 \leq x \leq 2 \) Thus, the domain of our function is the interval \( [-2, 2] \). This means you can plug any number between -2 and 2 into the function without running into problems.

X- and Y-Intercepts

The intercepts of a graph are the points where the graph crosses the axes.These points give insight into the behavior and position of a graph relative to the axis. - **Y-Intercept**: - This is where the graph crosses the y-axis. - To find the y-intercept, we set \( x = 0 \) and solve for \( y \). - Substituting, we get: \( y = -\sqrt{4 - 0^2} = -2 \). - Therefore, the y-intercept is at the point \( (0, -2) \).- **X-Intercepts**: - These are points where the graph crosses the x-axis. - We'll find these by setting \( y = 0 \) and solving for \( x \). - So, \( 0 = -\sqrt{4 - x^2} \) leading to \( \sqrt{4 - x^2} = 0 \). - Solving gives \( 4 - x^2 = 0 \) or \( x^2 = 4 \), hence, \( x = \pm 2 \). - The x-intercepts are \( (-2, 0) \) and \( (2, 0) \).These intercepts help in sketching the general layout of the graph on the coordinate plane.

Symmetry in Graphs

Symmetry in graphs provides a shortcut to understanding graph behavior without plotting numerous points. A graph can show three types of symmetry: with respect to the y-axis, x-axis, or the origin.
For our function, \(y = -\sqrt{4-x^2}\), we will test these symmetries:

• **Y-axis Symmetry**: If replacing \( x \) with \( -x \) in the function gives us the same equation, the graph is symmetric with respect to the y-axis. For our equation, substituting \(-x\) gives \(y = -\sqrt{4-(-x)^2}\), which simplifies back to \(y = -\sqrt{4-x^2}\). Thus, the graph is symmetric about the y-axis.
• **X-axis Symmetry**: A graph is symmetric about the x-axis if replacing \( y \) with \( -y \) results in the same equation, but this cannot hold for \(y = -\sqrt{4-x^2}\) since all \( y \) values would become positive.
• **Origin Symmetry**: Checking for origin symmetry involves replacing both \( x \) and \( y \) with \(-x\) and \(-y\). In our case, this does not yield the original equation, ruling out origin symmetry.

Understanding symmetry can greatly simplify the graphing process by reducing the number of points needed to represent the function adequately.