Problem 69 - Precalculus : Building Concepts and Connections

Question

Suppose the vertex of the parabola associated with a certain quadratic function is \((2,1),\) and another point on this parabola is (3,-1) (a) Find the equation of the axis of symmetry of the parabola. (b) Use symmetry to find a third point on the parabola. (c) Sketch the parabola.

Step-by-Step Solution

Answer

The equation of the axis of symmetry is \(x = 2\). The third point on the parabola is \((1, -1)\).

Step 1: Find the equation of the axis of symmetry

The axis of symmetry goes through the vertex of the parabola. The x-coordinate of the vertex is the equation of the axis of symmetry. Given that the vertex of the parabola is \((2,1)\), the equation of the axis of symmetry is \(x=2.\)

Step 2: Use symmetry to find a third point on the parabola

By property of symmetry in a parabola, any point \(P\) from the axis of symmetry has its reflection \(P'\) at the same distance but on the other side of the axis. Thus, a third point on the parabola can be found. Given a point \((3,-1)\) on the parabola, its distance from the axis of symmetry \(x=2\) is 1 unit to the right. Therefore, its symmetrical point is 1 unit to the left of the axis of symmetry. Hence, a third point is \((1, -1)\)

Step 3: Sketch the parabola

Mark the points \((2,1), (3,-1), (1,-1)\) and the line \(x=2\) which is our axis of symmetry on the Cartesian plane. Next, draw a smooth curve through these points making sure that it is mirrored about the line \(x=2\). Thus, completing the sketch of the parabola.

Key Concepts

Axis of Symmetry

The axis of symmetry in the context of a parabola is a vertical line that divides the parabola into two mirror-image halves. This concept is essential because it helps us understand the inherent symmetry that characterizes all parabolic shapes.

For a parabola in standard form given by the equation \(y = ax^2 + bx + c\), the axis of symmetry can be calculated using the formula \(x = -\frac{b}{2a}\). In our exercise, we were informed that the vertex was at \((2,1)\), from which we can deduce that the axis of symmetry is the vertical line passing through the x-coordinate of the vertex, hence \(x=2\).

Understanding the axis of symmetry is not only critical for sketching the parabola but also for solving a variety of problems involving quadratic functions, such as finding the maximum or minimum values of the function, and solving optimization problems.

Quadratic Function

A quadratic function is a type of polynomial that can be described by an equation of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The graph of a quadratic function is a parabola, which either opens upwards if \(a > 0\) or downwards if \(a < 0\).

The values of \(a\), \(b\), and \(c\) affect the shape and position of the parabola on the Cartesian plane. For instance, the coefficient \(a\) determines how 'wide' or 'narrow' the parabola is, while \(b\) and \(c\) influence the location of its axis of symmetry and vertex.

Quadratic functions are fundamental in various mathematical computations and real-world applications, including physics (projectile motion), economics (profit functions), and engineering (structural designs). Understanding how to manipulate and graph these functions is crucial for students tackling algebra and calculus.

Vertex of a Parabola

The vertex of a parabola is a specific point where the parabola changes direction. It can be thought of as the 'tip' or the highest or lowest point of the parabola, depending on whether the parabola opens upwards or downwards.

In the quadratic function \(y = ax^2 + bx + c\), the vertex's coordinates can be found using the axis of symmetry and the formula \(y = a(-\frac{b}{2a})^2 + b(-\frac{b}{2a}) + c\). Alternatively, if you have the vertex form of the quadratic equation \(y = a(x - h)^2 + k\), the vertex is straightforwardly at the point \((h, k)\).

In the exercise, we used the given vertex \((2,1)\) to determine the axis of symmetry. The vertex, being the pivotal point, helps in graphing the entire parabola since it's usually the first point to plot on the Cartesian plane. It's significant in problems requiring optimization as well, because the vertex represents the maximum or minimum value of the quadratic function.