Problem 4 - Algebra and Trigonometry

Question

(a) If a graph is symmetric with respect to the \(x\) -axis and \((a, b)\) is on the graph, then (_____,____) is also on the graph. (b) If a graph is symmetric with respect to the \(y\) -axis and \((a, b)\) is on the graph, then (_____,_____) is also on the graph. (c) If a graph is symmetric about the origin and \((a, b)\) is on the graph, then (_____,_____) is also on the graph.

Step-by-Step Solution

Answer

(a) (a, -b), (b) (-a, b), (c) (-a, -b)

Step 1: Understanding Symmetry with Respect to the x-axis

When a graph is symmetric with respect to the x-axis, the graph reflects across the x-axis. If a point \((a, b)\) is on the graph, then its reflection across the x-axis is \((a, -b)\). This means that for every point \((a, b)\), \((a, -b)\) is also on the graph.

Step 2: Understanding Symmetry with Respect to the y-axis

For a graph symmetric with respect to the y-axis, the graph reflects across the y-axis. This means if a point \((a, b)\) is located on the graph, then its reflection, \((-a, b)\), will also be present on the graph. Thus, every point \((a, b)\) has a corresponding point \((-a, b)\) on the graph.

Step 3: Understanding Symmetry About the Origin

Symmetry about the origin involves both the x-axis and y-axis inversions, which results in a rotation of 180 degrees around the origin. If the point \((a, b)\) is on the graph, then the point \((-a, -b)\) will also appear on the graph due to this rotational symmetry.

Key Concepts

x-axis symmetry

When we discuss a graph having **x-axis symmetry**, we mean that it mirrors itself across the horizontal axis, the x-axis. Imagine the x-axis as a line of reflection, like a mirror. For any point that you place on the graph, let's call it \((a, b)\), there's another point directly across the x-axis at \((a, -b)\). The x-coordinate remains the same, while the y-coordinate's sign is inverted. This ensures that the graph looks the same above and below the x-axis.
Here's a handy way to remember this:

• If \((a, b)\) is on the graph, then \((a, -b)\) must be there too.
• The graph's appearance remains unchanged regardless of flipping it over the x-axis.

Understanding this reflection can help you analyze whether a function or graph is genuinely symmetric with respect to the x-axis, which is often a requirement in solving equations and graphing functions.

y-axis symmetry

When a graph exhibits **y-axis symmetry**, it means the graph can be reflected over the vertical y-axis and still look identical. Similar to looking into a mirror placed along the y-axis, any point \((a, b)\) has a counterpart at \((-a, b)\). Here, the y-coordinate stays the same while the x-coordinate's sign changes. This symmetry is common in many even functions, such as a simple parabola \(y = x^2\).
To check for y-axis symmetry, remember:

• For every point \((a, b)\), there is a mirror point \((-a, b)\) on the graph.
• The graph appears the same from either side of the y-axis.

Grasping y-axis symmetry is especially useful when trying to determine the behavior of graphs, as it indicates that the graph doesn't change when you "flip" it over the y-axis.

origin symmetry

**Origin symmetry** is a bit more complex because it combines both the ideas of x-axis and y-axis symmetry. A graph with origin symmetry can be rotated 180 degrees around the origin (where the x-axis and y-axis meet) and still look the same. For a point \((a, b)\) on the graph, the corresponding point must be \((-a, -b)\). Both coordinates are inverted, which means it's moved diagonally through the origin.
To visualize and remember origin symmetry, consider:

• If \((a, b)\) is on the graph, then \((-a, -b)\) must also be on the graph.
• After a half-turn (180 degrees rotation), the graph remains unaltered.

Understanding origin symmetry is crucial when identifying certain types of functions like odd functions, for example, \(y = x^3\), where flipping both dimensions leads the graph to match its original structure.