Problem 66 - Algebra for College Students

Question

Graph the two equations \(y=\pm \sqrt{x}\) (Example 3 ) on the same set of axes, using the following boundaries. (Let \(Y_{1}=\sqrt{x}\) and \(Y_{2}=-\sqrt{x}\) ) (a) \(-15 \leq x \leq 15\) and \(-10 \leq y \leq 10\) (b) \(-1 \leq x \leq 15\) and \(-10 \leq y \leq 10\) (c) \(-1 \leq x \leq 15\) and \(-5 \leq y \leq 5\)

Step-by-Step Solution

Answer

Graph each pair of curves, considering only non-negative \(x\) values and their corresponding \(y\) range within given boundaries.

Step 1: Understanding the Equations

We are given two equations to graph: \(Y_{1} = \sqrt{x}\) and \(Y_{2} = -\sqrt{x}\). These represent the two branches of a parabola that opens vertically. \(Y_{1}\) is the upper half and \(Y_{2}\) is the lower half, as they are reflections of each other across the x-axis.

Step 2: Setting Up the Axes Boundaries for Scenario (a)

For part (a), we need to consider the graph in the range \(-15 \leq x \leq 15\) and \(-10 \leq y \leq 10\). However, the square root function only applies to non-negative values of \(x\). Thus, we only use \(0 \leq x \leq 15\) for this part. The allowable range for \(y\) will be \(\pm \sqrt{15}\).

Step 3: Graphing for Scenario (a)

For \(x\) values between 0 and 15, compute \(\sqrt{x}\) and \(-\sqrt{x}\). Plot these points on a graph with the axes ranging from \(-15\) to \(15\) in \(x\) and from \(-10\) to \(10\) in \(y\). The resulting graph consists of the top half-curve from \(y=\sqrt{x}\) and the bottom half-curve from \(y=-\sqrt{x}\).

Step 4: Setting Up the Axes Boundaries for Scenario (b)

For part (b), the graph needs to cover \(-1 \leq x \leq 15\) and \(-10 \leq y \leq 10\). The square root is only real for \(0 \leq x \leq 15\). The \(y\) values will range between \(-\sqrt{15}\) and \(\sqrt{15}\).

Step 5: Graphing for Scenario (b)

Within the valid \(x\) domain of \(0 \leq x \leq 15\), calculate \(\sqrt{x}\) and \(-\sqrt{x}\), and plot these values on the graph. The axes will display the curves within the range of \(-1 \leq x \leq 15\) and \(-10 \leq y \leq 10\).

Step 6: Setting Up the Axes Boundaries for Scenario (c)

For part (c), limit the graphing range from \(-1 \leq x \leq 15\) and \(-5 \leq y \leq 5\). \(\sqrt{x}\) is only applicable from 0 to 15 for \(x\), with resulting \(y\) values constrained to \(-\sqrt{15}\) and \(\sqrt{15}\), but only \(-5 \leq y \leq 5\) will be visible.

Step 7: Graphing for Scenario (c)

Graph \(y=\sqrt{x}\) and \(y=-\sqrt{x}\) within the given bounds. For \(0 \leq x \leq 15\), calculate and plot values of \(\sqrt{x}\) and \(-\sqrt{x}\) that fit within \(-5 \leq y \leq 5\). The axes will reflect these constraints, showing only a portion of the full parabolic curves.

Key Concepts

Square Root Function

The square root function is one of the basic functions in mathematics. It is represented by the equation \(y = \pm \sqrt{x}\). This function creates two branches of a parabola, one in the positive y-direction and one in the negative y-direction. The upper branch, \(y = \sqrt{x}\), forms a curve that rises to the right, whereas the lower branch, \(y = -\sqrt{x}\), forms a curve that extends downward. This type of function can only take non-negative values of \(x\) because we cannot take the square root of a negative number without involving complex numbers.

• **Positive Branch:** \(y = \sqrt{x}\) creates the top half of a parabola.
• **Negative Branch:** \(y = -\sqrt{x}\) forms the lower half, mirroring the positive branch over the x-axis.

Each branch starts at the origin (0,0), showing that the smallest number under the square root is zero. The graph of these functions displays a symmetrical shape, illustrating how these two expressions are mirror images of one another.

Coordinate Axes

When graphing equations like the square root function, understanding the coordinate axes is essential. The coordinate axes consist of two perpendicular lines, usually called the x-axis (horizontal) and the y-axis (vertical). They intersect at a point called the origin.

• **X-axis:** This is the horizontal axis along which the value of \(x\) is plotted. Increasing rightwards and decreasing leftwards.
• **Y-axis:** This is the vertical axis where \(y\) values are plotted, increasing upwards and decreasing downwards.
• **Origin:** The point where the x-axis and the y-axis intersect (\(0,0\)).

In the context of our problem, knowing where to limit these axes using specific x and y boundaries is crucial to accurately depict the behavior of the square root functions. The axes serve as the framework that helps us place our graph in the correct quadrant or coordinate system. Setting boundaries on these axes, such as from -15 to 15 for x and -10 to 10 for y, helps in visualizing a more focused part of the graph, which can reveal more about the function's behavior within specified ranges.

Domain and Range

Understanding domain and range is crucial when graphing functions like \(y = \sqrt{x}\). The domain is the set of all possible input values (x-values) for which the function is defined. Meanwhile, the range is the set of all possible output values (y-values) that the function can produce. For the square root function, the domain is constrained to non-negative values because square roots of negative numbers are not defined in the set of real numbers. In the problem, this is why we only consider values where \(0 \leq x \leq 15\).

• **Domain:** For \(y = \sqrt{x}\), the domain is \(x \geq 0\).
• **Range:** For \(y = \sqrt{x}\), the range is \(y \geq 0\).

In contrast, for \(y = -\sqrt{x}\), although the domain stays the same, the range is \(y \leq 0\). This reflects the graph across the x-axis. In this exercise, depending on the boundaries set, the visible portions of the graph will shift. Adjusting the domains and ranges can also show how functions behave differently when various portions are isolated or focused upon, such as when you limit y-values from -5 to 5 to zoom into a specific part of the graph.