Problem 28 - 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 99 ) of \(y=x^{2}\) that occur and (b) how each parabola opens. (a) \(y=x^{2}\) (b) \(y=-x^{2}\) (c) \(y=-(x-2)^{2}\)

Step-by-Step Solution

Answer

(a) y = -(x-2)² is shifted right 2 units; (b) y = x² opens up, others open down.

Step 1: Understand the Basic Parabola

The basic parabola equation given is \( y = x^2 \). This parabola opens upwards, and its vertex is at the origin point, (0,0).

Step 2: Analyze the Parabola y = -x²

The equation \( y = -x^2 \) represents a vertical reflection of the basic parabola \( y = x^2 \). This parabola opens downwards, and its vertex remains at the origin, (0,0). The negative sign in front indicates the downward opening.

Step 3: Analyze the Parabola y = -(x-2)²

The equation \( y = -(x-2)^2 \) also opens downwards due to the negative sign. Moreover, it involves a horizontal shift to the right by 2 units, resulting in the vertex being at (2,0).

Step 4: Visualize the Graphs

Using the knowledge from the previous steps, graph all three parabolas on the same coordinate system. \( y = x^2 \) is a standard parabola opening upwards, \( y = -x^2 \) opens downwards, and \( y = -(x-2)^2 \) opens downwards starting from the vertex at (2,0). The shifts and openings display their transformations visually.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a particularly helpful way to understand parabolas and their transformations. It makes it easy to identify key features such as the vertex, direction, and width of the parabola. This form is structured as:
\[ y = a(x-h)^2 + k \]
Here,

• \( (h, k) \) is the vertex of the parabola,
• \( a \) determines the parabola's width and the direction it opens.

The vertex is the point where the parabola changes direction, and is either the highest or lowest point on the graph, depending on how the parabola opens. The value of \( h \) indicates how far the vertex is shifted horizontally from the origin. The value of \( k \) shows the vertical shift.
If \( a \) is positive, the parabola opens upwards, forming a "cup shape". If \( a \) is negative, the parabola opens downwards, like an upside-down cup. The greater the absolute value of \( a \), the narrower the parabola is. Conversely, smaller absolute values make the parabola wider.

Parabola Transformations

Parabola transformations involve changing the position and size of the standard parabola to fit different scenarios. These transformations help depict real-world phenomena or solve mathematical problems.
Here are the common types of transformations:

• Vertical Reflections: Changing the sign in front of the equation, like from \( y = x^2 \) to \( y = -x^2 \), reflects the parabola over the x-axis. This means the parabola will flip its direction from upwards to downwards or vice versa.
• Horizontal Shifts: The expression inside the squared term, \((x-h)\), determines horizontal movement. For instance, \( y = -(x-2)^2 \) indicates a shift of 2 units to the right, positioning the vertex at (2, 0).
• Vertical Shifts: The value \( k \) in the vertex form \( y = a(x-h)^2 + k \) shifts the parabola up or down. There is no vertical shift in the given equations, but it would adjust the vertex's vertical position.

Understanding these transformations helps in predicting how a parabola will look on a graph based on changes made to its equation.

Coordinate Systems

A coordinate system is a crucial tool in graphing parabolas and other shapes in mathematics. It helps us locate points in a space using numbers. Typically presented as a grid on the x and y axes, it gives us a visual way to interpret equations like a parabola.
The coordinate system consists of:

• X-Axis: The horizontal line in the graph. It shows whether a point is to the left or right of the origin (0,0).
• Y-Axis: The vertical line. It displays whether a point is above or below the origin.
• Quadrants: The axes divide the space into four quadrants. Starting from the top right and moving counterclockwise, each holds distinct sign conventions for coordinates (positives, negatives).

Plotting on a coordinated system allows us to visualize different equations, such as the parabolas discussed earlier. You can see how alterations in the parabola's equation make it move or change shape within this system. For example, shifts in the vertex made by equations like \( y = (x-2)^2 \) result in visible displacements across the grid. Recognizing how these shifts work helps interpret the parabolas' behavior effectively.