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Chapter 1: Problem 9

Sketch a graph of an even function and give the function's defining property.

Short Answer

Expert verified
Answer: The defining property of even functions is that they satisfy the equation \(f(-x) = f(x)\) for all x in their domain. The graph of the function \(f(x) = x^2\) demonstrates this property by being symmetric about the y-axis, meaning that for every x value to the left of the y-axis, there is a corresponding x value to the right with the same y value. This confirms that the function \(f(x) = x^2\) is even and satisfies its defining property \(f(-x)=f(x)\).

Step by step solution

01

Recall the Defining Property of Even Functions

An even function is a function with the property \(f(-x) = f(x)\) for all x in the domain. In other words, even functions have symmetry with respect to the y-axis.
02

Choose an Example of an Even Function

Let's consider the function \(f(x) = x^2\). We will show that this function is even and sketch its graph.
03

Check if the Example is an Even Function

We'll now verify that the chosen function is even by checking whether \(f(x) = f(-x)\) for \(f(x) = x^2\). Compute \(f(-x)\): \(f(-x) = (-x)^2 = x^2\) Since \(f(-x) = f(x)\), \(f(x) = x^2\) is an even function.
04

Sketch the Graph of the Even Function

To sketch the graph of the even function \(f(x) = x^2\), first plot some points on the graph for different x values: - For \(x=-4\), \(f(x)=16\) - For \(x=-3\), \(f(x)=9\) - For \(x=-2\), \(f(x)=4\) - For \(x=-1\), \(f(x)=1\) - For \(x=0\), \(f(x)=0\) - For \(x=1\), \(f(x)=1\) - For \(x=2\), \(f(x)=4\) - For \(x=3\), \(f(x)=9\) - For \(x=4\), \(f(x)=16\) Now connect these points with a smooth curve to sketch the graph of the function.
05

Explain the Graph's Symmetry

As we can see from the graph of \(f(x) = x^2\), the function is symmetric about the y-axis, meaning that for every x value to the left of the y-axis, there is a corresponding x value to the right with the same y value. This confirms that our function is even, which is consistent with its defining property \(f(-x)=f(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Symmetry
Function symmetry is an important concept in mathematics, which helps us understand how a graph behaves with respect to certain axes. A function is said to be even if it exhibits symmetry about the y-axis. This means that if you "fold" the graph along the y-axis, both sides will match perfectly. Mathematically, an even function satisfies the condition \( f(-x) = f(x) \) for all points \( x \) in its domain.

To see symmetry in action, consider the function \( f(x) = x^2 \). For any positive or negative value of \( x \), the y-value remains the same due to squaring. For instance, \( f(2) = 4 \) and \( f(-2) = 4 \). This reflects the fact that the graph's right side mirrors the left across the y-axis. Recognizing such symmetry can simplify graph sketching and analysis.
Graph Sketching
Graph sketching is the process of drawing a rough graph using mathematical insights rather than plotting numerous points. For an even function like \( f(x) = x^2 \), understanding its symmetry can greatly streamline this process. By identifying key points and patterns, you can sketch the entire graph accurately.

Start by choosing a few points on one side of the y-axis. Calculate their corresponding y-values:
  • For \( x = -2 \), \( f(x) = 4 \)
  • For \( x = 0 \), \( f(x) = 0 \)
  • For \( x = 2 \), \( f(x) = 4 \)
Due to symmetry, these points will have matching y-values on the opposite side of the y-axis. Connect these points smoothly to sketch the graph. This method demonstrates the elegance and efficiency of recognizing and utilizing symmetry.
Quadratic Function
A quadratic function is a type of polynomial function characterized by the highest degree of the variable being two. The general form is \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. Quadratics often form a u-shaped curve called a parabola when graphed.

Quadratic functions are versatile and appear in everything from projectile motion to optimization problems. They have distinctive features:
  • Vertex: The highest or lowest point of the parabola, depending on the direction of the "arms." This occurs at \( x = -\frac{b}{2a} \).
  • Axis of Symmetry: A vertical line through the vertex, dividing the parabola into two mirror-image halves. For an even function like \( f(x) = x^2 \), the axis of symmetry is the y-axis.
  • Roots/Zeros: The x-values where the parabola crosses the x-axis, found using the quadratic formula or factoring.
By understanding these features and their implications, students can quickly identify how quadratics will behave graphically.

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