Problem 73 - Algebra for College Students

Question

The equation \(x=y^{2}\) is equivalent to \(y=\pm \sqrt{x} .\) Graph both \(y=\sqrt{x}\) and \(y=-\sqrt{x}\) on a graphing calculator. How does the graph of \(x=y^{2}\) compare to the graph of \(y=x^{2} ?\)

Step-by-Step Solution

Answer

The graph of \(x = y^2\) opens to the right, while the graph of \(y = x^2\) opens upwards.

Step 1: Understand the given equations

The equation provided is \(x = y^2\), which gives us \(y\) in terms of \(x\) as \(y = \pm \sqrt{x}\).

Step 2: Graph the functions

Use a graphing calculator to plot the functions \(y = \sqrt{x}\) and \(y = -\sqrt{x}\). These are the upper and lower halves of the parabola, respectively, opening to the right.

Step 3: Graph the comparison function

Also, plot the function \(y = x^2\) on the same graph to see the comparison. This is a standard parabola opening upwards.

Step 4: Analyze the graphs

The graph of \(x = y^2\) (or \(y = \pm \sqrt{x}\)) creates a parabola that opens to the right. Conversely, the graph of \(y = x^2\) is a parabola that opens upwards.

Step 5: Compare the graphs

Note that \(x = y^2\) represents a parabola that is symmetrical about the \(x\)-axis, whereas \(y = x^2\) represents a parabola that is symmetrical about the \(y\)-axis.

Key Concepts

Graphing Parabolas

Graphing parabolas involves plotting quadratic equations on a coordinate plane. A parabola is a U-shaped curve that can open upward, downward, left, or right. The standard form of a quadratic equation for a parabola is either:

• y = ax^2 + bx + c (opens upward or downward)
• x = ay^2 + by + c (opens left or right)

In our exercise, we focused on the equations:

• y = sqrt{x}
• y = -sqrt{x}
• y = x^2.

The first two forms represent a parabola that opens rightward, while the last equation represents a parabola that opens upwards. It's essential to use a graphing calculator or graphing software to accurately plot these functions. By graphing the functions, you can better visualize the differences and similarities between the parabolas.

Equivalent Equations

Equivalent equations are different forms of the same equation that yield the same solutions. For example, the equation x = y^2 is equivalent to y = ±sqrt{x}. Both forms represent the same set of points on a graph.
To convert x = y^2 into y in terms of x, we take the square root of both sides:

• y = sqrt{x}
• y = -sqrt{x}

These equations indicate two branches of the same parabola. Equivalent equations help in graphing because they might be easier to understand or work with in one form over another. The ability to identify and convert equivalent equations is crucial for simplifying complex problems in algebra.

Symmetry of Parabolas

Symmetry in parabolas is all about understanding their balance around a central axis. For the parabola given by x = y^2, or equivalently y = ±sqrt{x}, the symmetry is about the x-axis. This means that if you fold the graph along the x-axis, both halves will align perfectly.
In contrast, the parabola given by y = x^2 is symmetrical about the y-axis. If you fold the graph along the y-axis, both halves will match.
Understanding symmetry helps in:

• Graphing by reducing the amount of work needed (you only need to plot one half).
• Analyzing root and intersection behavior on the graph.

Visualizing symmetry aids in telling whether a parabola opens upwards, downwards, left, or right, making it easier to understand and compare different quadratic functions.

Function Comparison

Comparing functions involves looking at their shapes, direction of opening, and points of intersection. In our exercise, we compared:

• x = y^2 (or y = ±sqrt{x})
• y = x^2

The first function (x = y^2) forms a parabola opening rightwards with symmetry about the x-axis. The second function (y = x^2) forms a parabola opening upwards with symmetry about the y-axis.
Key points for comparison:

• Axis of symmetry: x-axis for x = y^2 and y-axis for y = x^2.
• Direction of opening: right for x = y^2 and up for y = x^2.
• Vertex location: both functions have a vertex at (0,0).

Comparing these functions helps in understanding their geometric properties and how they behave relative to one another on a coordinate plane.