Problem 80 - College Algebra and Calculus: An Applied Approach

Question

Find the coordinates of a second point on the graph of a function \(f\) if the given point is on the graph and the function is (a) even and (b) odd. \(\left(-\frac{3}{2}, 4\right)\)

Step-by-Step Solution

Answer

The second point on the graph of an even function is \((3/2, 4)\) and on an odd function is \((3/2, -4)\)

Step 1: Identify a Second Point for an Even Function

For even functions, the graph is symmetric with respect to the y-axis. That is, if a point \((-a, b)\) is on the graph, then the point \((a, b)\) is also on the graph. In this case, the given point is \((-3/2, 4)\). So the second point on the graph of an even function is \((3/2, 4)\).

Step 2: Identify a Second Point for an Odd Function

For odd functions, the graph is symmetric with respect to the origin. That is, if a point \((-a, b)\) is on the graph, then the point \((a, -b)\) is also on the graph. In this case, the given point is \((-3/2, 4)\). So the second point on the graph of an odd function is \((3/2, -4)\).

Key Concepts

Even Functions

Even functions possess a unique quality of symmetry. They are symmetric with respect to the y-axis. This means that if you draw a vertical line along the y-axis, the shape of the graph on the left side is identical to the shape on the right side.

What's happening with even functions mathematically is that if you have a point \((-a, b)\) on the graph, there will also be a matching point \((a, b)\) located on the opposite side of the y-axis, at the same distance.

**Examples and Importance**

• The function \(f(x) = x^2\) is a classic example of an even function. Its graph confirms this y-axis symmetry, displaying perfect reflections on each side.
• Many physical phenomena, such as the distribution of pressure in certain contexts, adhere to this kind of symmetry, making even functions crucial in practical applications.

Identifying even functions involves checking if \(f(x) = f(-x)\) for all values of \(x\). If this is true, the function is even.

Odd Functions

Odd functions exhibit a different kind of symmetry, this time with respect to the origin. Imagine a graph where you can rotate it 180 degrees around the origin, and it will look exactly the same. This is the hallmark of odd function symmetry.

Mathematically, if a graph has a point \((-a, b)\), there will be another point \((a, -b)\). It flips over both the x-axis and y-axis, leading to an elegant rotational symmetry.

**Examples and Importance**

• A prime example is \(f(x) = x^3\), where the graph showcases this rotational symmetry around the origin, reflecting the principle of odd functions.
• Understanding odd functions helps in analyzing signals and waves in engineering, where certain symmetrical properties can simplify complex problems.

To determine if a function is odd, check if \(-f(x) = f(-x)\) for all values of \(x\). If true, the function is odd.

Function Graph Symmetry

When studying functions, one of the intriguing visual aspects is their symmetry. Recognizing symmetric graphs helps in predicting behavior, simplifying calculations, and understanding the broader dynamics of the function.

**Types of Symmetry**

• **y-axis symmetry:** Typical of even functions, where reflection across the vertical axis produces a graph that matches exactly with itself.
• **Origin symmetry:** Characteristic of odd functions, when flipping the graph about the origin (180-degree rotation) results in an unchanged appearance.

**Applications**

Symmetry in graphs isn't just a mathematician's fascination; it serves practical purposes. In physics, symmetry can simplify equations, creating models that better mimic natural phenomena. In design and engineering, understanding such symmetry allows for innovations in creating more efficient and balanced structures.

Recognizing these symmetrical relationships in functions provides a deeper understanding of their nature and facilitates quicker problem-solving techniques.