Problem 69 - College Algebra

Question

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose y-coordinate is the same as the given point. \(f(x)=(x-3)^{2}+2 ; \quad(6,11)\)

Step-by-Step Solution

Answer

The axis of symmetry for the given parabola is \(x = 3\). However, there is no other point on the parabola with the same y-coordinate as the given point (6, 11).

Step 1: Identify the Axis of Symmetry

The axis of symmetry of a parabola is the vertical line \(x = h\), where \(h\) is the x-coordinate of the vertex. For the given equation \(f(x) = (x-3)^2 + 2\), the axis of symmetry is \(x = 3\).

Step 2: Use the Axis of Symmetry to Find a Second Point

We are given a point (6, 11) and we know that another point lies on the axis of symmetry (\(x = 3\)) with the same y-coordinate. So, the required point is (3, 11) because the y-coordinate remains constant.

Step 3: Verify the Point

To ensure (3, 11) is on the parabola, plug it into the original equation for verification. \(f(3) = (3-3)^2 + 2 = 2\). As \(f(3)\) does not equal 11, there must be a mistake. So the assumption that there exists a second point with same y-coordinate on the given parabola is not right. Thus, no such point exists.

Key Concepts

Parabola

A parabola is a symmetrical, curved shape, which you often find in mathematics, as well as in the real world, like in satellite dishes or bridges. In terms of graphing, a parabola is the graphical representation of a quadratic function. One of the important features of a parabola is that it is U-shaped, and it can either open upwards or downwards. This shape is characterized by its vertex, the highest or lowest point on the curve, and the axis of symmetry, a line that perfectly divides the parabola into two matching halves.

• The axis of symmetry is often expressed as a vertical line of the format "x = constant".
• The direction in which the parabola opens depends on the sign of the quadratic term. If the leading coefficient is positive, the parabola opens upwards, and if negative, it opens downwards.

These unique characteristics not only help in identifying parabolas but also allow you to better understand the properties of quadratic functions.

Vertex

The vertex of a parabola serves as a crucial point in understanding its geometry. As mentioned, it represents the highest or lowest point on the curve. In the standard vertex form of a quadratic function, which is expressed as
\[ f(x) = a(x - h)^2 + k \]
the vertex is denoted by the point \( (h, k) \). It effectively serves as a pivot point, highlighting where the parabola changes direction.

• The vertex offers valuable information about the extremum of the parabola; for an upward-opening parabola, the vertex is the minimum point, and for a downward-opening parabola, the vertex is the maximum point.
• In practical terms, knowing the vertex allows you to quickly sketch or interpret the parabola on a graph.

Understanding the vertex is indispensable when solving problems involving parabolas as it provides foundational insight into the parabola's geometric structure.

Quadratic Functions

Quadratic functions form the foundation for understanding parabolas. In algebra, a quadratic function is any function that can be expressed in the following standard form:
\[ ax^2 + bx + c \]
where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These functions are essential in mathematics as they model various phenomena, such as projectile motion, and solve practical problems across different fields.

• The term "quadratic" comes from "quadrate" meaning square since the variable "x" is squared.
• One of the significant features of quadratic functions is that they always produce a parabola when plotted on a Cartesian plane.
• Analyzing a quadratic equation in its vertex form—which highlights the vertex—simplifies finding its key features, including the axis of symmetry and direction of opening.

Learning how to manipulate and understand quadratic functions paves the way for advanced study in algebra and calculus.

Symmetry in Mathematics

Symmetry is a fundamental concept in mathematics, and it plays a significant role in understanding and analyzing parabolas. Essentially, symmetry refers to the balanced and proportional arrangement of parts in a shape or object. For a parabola, symmetry signifies that one side of the parabola is the mirror image of the other.

• The axis of symmetry is a line that divides the parabola into two equal halves that are mirror images.
• For a quadratic function, the vertex and the axis of symmetry are closely linked since the vertex lies on the axis of symmetry.
• The concept of symmetry extends beyond mathematics and finds applications in art, design, and nature, enhancing the comprehension and beauty of these fields.

Grasping the concept of symmetry helps students not only in solving problems related to parabolas but also enriches their understanding of mathematical structures and the natural world.