Problem 28 - 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), and the y-intercept is (0,-2). The graph is a downward semi-circle.

Step 1: Understand the Equation

The given equation is a function, \(y = -\sqrt{4-x^2}\), which represents a semi-circle in the negative half of the y-plane because the negative sign in front of the square root flips the circle downward. The expression \(4 - x^2\) suggests that if we set \(x^2\), the maximum value under the square root is 4, meaning \(x\) ranges from \(-2\) to \(2\).

Step 2: Create the Table of Values

Choose several values for \(x\) between \(-2\) and \(2\) to calculate corresponding \(y\) values. For example, if \(x = 0\), then \(y = -\sqrt{4-0^2} = -\sqrt{4} = -2\). Similarly, calculate for \(x = -2, -1, 1, 2.\)- When \(x = -2\) or \(x = 2\), \(y = -\sqrt{4-4} = 0\).- When \(x = -1\) or \(x = 1\), \(y = -\sqrt{4-1} = -\sqrt{3}\).

Step 3: Identify the Intercepts

To find the \(x\)-intercepts, set \(y=0\) and solve the equation: \[-\sqrt{4 - x^2} = 0\] \ \[\sqrt{4 - x^2} = 0\] \thus, \(x^2 = 4\), so \(x = \pm 2\).The \(y\)-intercept is found by setting \(x=0\): \[y = -\sqrt{4-0^2} = -2\], so the \(y\)-intercept is \((0, -2)\).

Step 4: Sketch the Graph

Use the calculated points \((-2,0), (-1, -\sqrt{3}), (0,-2), (1, -\sqrt{3}), (2,0)\) to sketch the graph. These points form a semi-circle centered on the x-axis from \(-2\) to \(2\) with the bottom of the semi-circle being \((0, -2)\).

Step 5: Conclusion

The graph is a semicircle opening downward with endpoints at \((-2,0)\) and \((2,0)\), with the lowest point at \((0,-2)\). The \(x\)-intercepts are \((-2,0)\) and \((2,0)\), and the \(y\)-intercept is \((0,-2)\).

Key Concepts

Semi-Circle

In geometry, a semi-circle is half of a circle. When a function graphically represents a semi-circle, it typically involves the equation of a circle. The equation \(y = -\sqrt{4 - x^2}\) describes the upper half of a circle flipped upside down. This flipping occurs because of the negative sign in front of the square root. The equation \(4 - x^2\) under the square root implies that the semi-circle is centered on the x-axis with a range for \(x\) that goes from \(-2\) to \(2\).

• Compared to a full circle, a semi-circle has its boundary stopping at the diameter.
• Here, the diameter extends from \(-2, 0\) to \(2, 0\) on the x-axis.

Understanding this shape is key to sketching and analyzing the graph of the function.

X-Intercept

An x-intercept is where a graph crosses the x-axis. To find the x-intercept of the function \(y = -\sqrt{4 - x^2}\), we set \(y = 0\) and solve for \(x\). The equation becomes:\[\sqrt{4 - x^2} = 0\]Solving it gives:\[x^2 = 4\] Hence, the solutions are \(x = 2\) and \(x = -2\). Therefore, the x-intercepts are at the points \((-2, 0)\) and \((2, 0)\).

• The x-intercept indicates where the function has no height or vertical value.
• These points are significant because they mark the endpoints of the semi-circle on the x-axis.

Recognizing these intercepts helps in accurately drawing and understanding the graph's shape.

Y-Intercept

A y-intercept is where a graph crosses the y-axis. The y-intercept is found by setting \(x = 0\) in the equation \(y = -\sqrt{4 - x^2}\). Doing this calculation gives:\[y = -\sqrt{4 - 0^2} = -\sqrt{4} = -2\]Thus, the y-intercept is at the point \((0, -2)\).

• This is the lowest point on the semi-circle, indicating where the curve reaches its maximum depth below the x-axis.
• Knowing the y-intercept allows for better understanding of the semi-circle's vertical span.

Identifying the y-intercept is crucial for plotting the graph accurately and understanding the function's behavior at \(x = 0\).

Table of Values

A table of values is a handy tool to understand how input values (x) affect output values (y) in a function. For the function \(y = -\sqrt{4-x^2}\), we select different values of \(x\) between \(-2\) and \(2\), because this is the range where the function is defined. As you compute the values:

• When \(x = 0\), \(y = -2\).
• When \(x = -2\) or \(x = 2\), \(y = 0\).
• When \(x = -1\) or \(x = 1\), \(y = -\sqrt{3}\).

By plotting these points,

• \((-2, 0)\)
• \((0, -2)\)
• \((2, 0)\)
• \((-1, -\sqrt{3})\)
• \((1, -\sqrt{3})\)

you can see how the graph forms a semi-circle. The table of values guides you in creating an accurate sketch and ensuring that the characteristics of the function are correctly represented.