Problem 25 - Basic Technical Mathematics with Calculus

Question

Use a calculator to graph all three parabolas on the same coordinate system. Describe (a) the shifts (see page 105 ) of \(y=x^{2}\) that occur and \((b)\) how each parabola opens. (a) \(y=x^{2}\) (c) \(y=x^{2}-3\) (b) \(y=x^{2}+3\)

Step-by-Step Solution

Answer

All three parabolas open upwards, with shifts of 3 units down (\(y=x^2-3\)) and 3 units up (\(y=x^2+3\)).

Step 1: Understand the Basic Parabola

The basic parabola given is \( y = x^2 \). This graph opens upwards and has a vertex at the origin (0,0). It is symmetric about the y-axis.

Step 2: Identify the Shift in y=x^2-3

For the function \( y = x^2 - 3 \), the entire graph of \( y = x^2 \) is shifted downward by 3 units. The vertex of this parabola moves from the origin to (0, -3), but it still opens upwards.

Step 3: Identify the Shift in y=x^2+3

For the function \( y = x^2 + 3 \), the graph of \( y = x^2 \) is shifted upward by 3 units. The vertex moves from the origin to (0, 3), while the parabola continues to open upwards.

Key Concepts

Vertex

The vertex of a parabola is a key point where the curve changes direction. For a standard parabola of the form \( y = ax^2 + bx + c \), the vertex is at the point \( (h, k) \). In simpler cases like \( y = x^2 \), the vertex is located at the origin, \( (0,0) \). This point represents both the minimum and maximum value of the function, depending on whether the parabola opens upwards or downwards.
In our example for the equation \( y = x^2 \), the vertex exists at \( (0,0) \). When transformations occur, such as adding or subtracting a constant, the vertex shifts vertically. For instance, \( y = x^2 - 3 \) moves the vertex to \( (0, -3) \), while in \( y = x^2 + 3 \), it moves to \( (0,3) \). Despite these shifts, the fundamental structure of the parabola remains unchanged.

Graph transformations

Graph transformations involve shifting, stretching, or compressing graphs of functions. These modifications help us understand how equations change. For parabolas, transformations can be easy to identify by their impact on the vertex and the direction of the opening.
Consider the basic parabola \( y = x^2 \). Any changes in the equation, such as adding or subtracting constants (\( y = x^2 + c \)), results in a vertical shift. This means each point on the graph, including the vertex, moves up or down along the y-axis. In our explored equations:

• \( y = x^2 - 3 \) shifts the original parabola downward by 3 units.
• \( y = x^2 + 3 \) shifts it upward by 3 units.

These are simple vertical transformations that do not affect the width or the direction of opening of the parabola.

Symmetry of parabolas

Symmetry is a fascinating aspect of parabolas, making them predictable and easy to analyze. A standard parabola, \( y = ax^2 \), is always symmetric with respect to its axis. This axis is a vertical line that passes through the vertex.
For the equation \( y = x^2 \), the axis of symmetry is the y-axis (\( x = 0 \)). This means if you were to fold the graph along the y-axis, both halves would align perfectly.
Even when transformations occur, such as shifting the parabola up or down, the symmetry remains on the same axis. In our examples for \( y = x^2 \), \( y = x^2 - 3 \), and \( y = x^2 + 3 \), the axis of symmetry does not change despite the vertical shift. Therefore, symmetry acts as a useful tool in graph analysis, ensuring the graph's form and character remain, regardless of some transformations.