Problem 7 - Precalculus: Mathematics for Calculus

Question

(a) If a graph is symmetric with respect to the x-axis and 1a, b2 is on the graph, then (_____ , _______) is also on the graph. (b) If a graph is symmetric with respect to the y-axis and 1a, b2 is on the graph, then (______ , _____) is also on the graph. (c) If a graph is symmetric about the origin and 1a, b2 is on the graph, then (______ , _____) is also on the graph.

Step-by-Step Solution

Answer

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

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

A graph is symmetric with respect to the x-axis if for every point \((a, b)\) on the graph, the point \((a, -b)\) is also on the graph. This means that if you have a point \((a, b)\), its mirror image across the x-axis will be \((a, -b)\).

Step 2: Identifying Points for x-axis Symmetry

Given that point \((a, b)\) is on the graph, and knowing the symmetry property, the point \((a, -b)\) is also on the graph.

Step 3: Understanding Symmetry with Respect to the y-axis

A graph is symmetric with respect to the y-axis if for every point \((a, b)\) on the graph, the point \((-a, b)\) is also on the graph. This means that if you have a point \((a, b)\), its mirror image across the y-axis will be \((-a, b)\).

Step 4: Identifying Points for y-axis Symmetry

Given that point \((a, b)\) is on the graph, and knowing the symmetry property, the point \((-a, b)\) is also on the graph.

Step 5: Understanding Symmetry about the Origin

A graph is symmetric about the origin if for every point \((a, b)\) on the graph, the point \((-a, -b)\) is also on the graph. This means that if you have a point \((a, b)\), its mirror image through the origin will be \((-a, -b)\).

Step 6: Identifying Points for Origin Symmetry

Given that point \((a, b)\) is on the graph, and knowing the symmetry property, the point \((-a, -b)\) is also on the graph.

Key Concepts

x-axis symmetry

Symmetry with respect to the x-axis occurs when a graph looks exactly the same above and below the x-axis. This means that each point on the graph has a corresponding point that is a mirror image across the x-axis. In simpler terms, if you draw a horizontal line along the x-axis, the top and bottom halves of the graph should reflect each other.

For example, if there's a point

• \((a, b)\)

on the graph, its corresponding point in x-axis symmetry is

• \((a, -b)\)

What's happening here is that the x-coordinate remains the same, but the y-coordinate changes its sign. This is what creates the mirror effect. Such symmetry can often be seen in graphs of functions like circles or even-parity polynomials.

y-axis symmetry

Symmetry about the y-axis implies that for every point on one side of the y-axis, there is an identical point on the opposite side. Essentially, if you imagine placing a mirror vertically along the y-axis, the graph should appear unchanged.

With y-axis symmetry, if you have a point

• \((a, b)\)

the corresponding mirrored point would be

• \((-a, b)\)

This time, the y-coordinate stays the same, while the x-coordinate flips sign. This kind of symmetry is common in functions like parabolas, which open upwards or downwards, as well as even functions where \(f(x) = f(-x)\).

origin symmetry

Origin symmetry, also known as point symmetry, occurs when a graph can be rotated 180 degrees around the origin and appear unchanged. Another way to visualize this is: every point on the graph reflects through the origin to another point directly opposite.

For origin symmetry, if there’s a point

• \((a, b)\)

on the graph, the reflected point would be

• \((-a, -b)\)

Both the x and y coordinates flip signs. This creates a true rotational symmetry through the center point (0,0). You’ll often find this symmetry in odd functions, where \(f(x) = -f(-x)\). An example is the cubic function which often shows origin symmetry.