Question

WRITING Two quadratic functions have graphs with vertices (2, 4) and (2, ?3). Explain why you can not use the axes of symmetry to distinguish between the two functions.

Short answer

The axes of symmetry for both the quadratic functions are the same (x = 2). Therefore, we cannot use them to differentiate between the two functions. This is because the axis of symmetry only expresses information about the x-coordinate of the vertex and doesn't take into account the y-coordinate.

Step-by-step solution

Understand the Axes of Symmetry

In a quadratic function, the axis of symmetry is the vertical line x = h, where (h, k) is the vertex of the parabola. This line divides the parabola into two identical halves. Therefore, the axis of symmetry is solely determined by the x-coordinate of the vertex.

Identify the X-Coordinate of the Vertices

From the problem statement, the vertices for the two quadratic functions are given as (2, 4) and (2, -3). The x-coordinates for both these vertices are the same and both equal to 2.

Determining the Axes of Symmetry

Given that the x-coordinates of both the vertices are the same (x = 2), the axes of symmetry for both the quadratic functions will be the same vertical line x = 2.

Conclude

Since the axes of symmetry for both the quadratic functions are the same (x = 2), they can not be used to distinguish between these two functions. The axis of symmetry only gives information about the 'horizontal position' of the parabola and doesn't express anything about its 'vertical position' or the y-value of the vertex, which in this case is different for the two functions (4 and -3).

Key concepts

Axes of Symmetry

Understanding the axis of symmetry in quadratic functions is essential in visualizing and analyzing the graph of a parabola. In simple terms, the axis of symmetry is an imaginary vertical line that splits the parabola into two mirror images, each reflecting the other. For any quadratic function expressed in the standard form, which is y = ax^2 + bx + c, the axis of symmetry can be found using the formula x = -b/2a.

For example, if we consider the quadratic functions from our exercise, which have vertices at (2, 4) and (2, -3), the x-coordinate of these vertices, being 2, points to the fact that the axis of symmetry for both functions is the line x = 2. This means that if you were to fold the graph of the parabola along the line x = 2, each half would match the other perfectly. This symmetry enables the parabola to have a single maximum or minimum point, which is none other than the vertex itself. However, because this characteristic does not take into account the y-coordinate of the vertex, it fails to provide a distinction between parabolas that may differ in their vertical positions—it is like the line of reflection in a mirror, highlighting the balance of the form rather than its placement.

Vertex of a Parabola

Delving into the vertex of a parabola, we uncover the point on the graph where the curve turns; it is either the highest or lowest point on the parabola, depending on whether the parabola opens upwards or downwards. The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form illustrates how the vertex serves as a cornerstone, determining the location of the entire parabola on the coordinate plane.

The vertices from the exercise, (2, 4) and (2, -3), are critical in understanding that while the x-coordinates are identical—indicating a shared axis of symmetry—the differing y-coordinates signify that the parabolas are at distinct vertical positions. In other words, even though the axes of symmetry tell us the 'horizontal' placement, they provide no insight into the 'vertical' aspect where the actual vertex resides. Thus, only focusing on the x-coordinate would be like getting a partial picture of the scenario — knowing where the center of the picture is but not knowing whether it is high up in a sunny sky or closer to the ground in a different scenery.

Parabola

When exploring the concept of a parabola, it is essential to comprehend that it is the graph of a quadratic function, which has a characteristic 'U' shape, either opening upwards or downwards. The general form of a quadratic function is y = ax^2 + bx + c, where the sign of the coefficient a determines the direction of the opening—if a is positive, the parabola opens upwards, and if negative, it opens downwards.

In relation to our exercise, we have two parabolas with vertices at (2, 4) and (2, -3). The fact that they both have the same x-coordinate for their vertices implies they share the same axis of symmetry, which is reflected in the way a parabola mirrors along this axis. However, the distinct y-coordinates highlight that these are two separate parabolas, possibly opening in the same or opposite directions, and positioned at different levels vertically. The parabola as a geometric shape represents a myriad of real-world phenomena, from trajectories of projectiles to the focus in satellite dishes, making its study practical as well as theoretical. It emphasizes the influence of symmetry and vertex position in quadratic functions, providing a visual representation of the mathematical relationships at play.