Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Write an equation in \(x\) and \(y\) that results in the desired translation. Do not use a calculator. The squaring function, shifted 2000 units to the right and 500 units upward

Step-by-Step Solution

Answer

The equation is \( y = (x - 2000)^2 + 500 \).

Step 1: Understand the function transformation

The basic squaring function is given by the equation \( y = x^2 \). Shifting this function horizontally and vertically requires knowledge of translation transformations.

Step 2: Apply the horizontal shift

To shift the squaring function 2000 units to the right, we replace \( x \) in \( y = x^2 \) with \( x - 2000 \). This means the new equation becomes \( y = (x - 2000)^2 \).

Step 3: Apply the vertical shift

To shift the function 500 units upward, we add 500 to the previous step's equation. Thus, the new equation is \( y = (x - 2000)^2 + 500 \).

Step 4: Construct the final equation

Combine the transformations to get the final equation in terms of \( x \) and \( y \): \( y = (x - 2000)^2 + 500 \).

Key Concepts

Horizontal Shift

Shifting a function horizontally means moving it left or right along the x-axis. To achieve a horizontal shift, we adjust the variable inside the function. For instance, with the squaring function, which is expressed as \( y = x^2 \), a horizontal shift will involve changing \( x \) to \( x - k \) for a shift to the right, or \( x + k \) for a shift to the left. Here, \( k \) represents the number of units we want to shift.

In our example, we need to shift the squaring function 2000 units to the right. This modifies the equation to \( y = (x - 2000)^2 \). This is because we substitute \( x \) with \( x - 2000 \), effectively moving every point on the graph 2000 units in the positive x-direction.

Key points to remember about horizontal shifts:

• \( x - k \) shifts the graph \( k \) units to the right.
• \( x + k \) shifts the graph \( k \) units to the left.
• Shifts do not change the shape of the graph, only its position.

Vertical Shift

Vertical shifts adjust the position of a graph up or down along the y-axis. They are applied by adding or subtracting a constant from the function's output. In essence, you modify the \( y \) value directly. For any function \( f(x) \), to move it vertically by \( c \) units, you will write it as \( y = f(x) + c \) to shift upwards or \( y = f(x) - c \) to shift downwards.

In our specific case, we are transforming the already horizontally shifted squaring function \( y = (x - 2000)^2 \). We want to move it 500 units upward. Therefore, we add 500 to the function, resulting in \( y = (x - 2000)^2 + 500 \). This moves every point on the graph up by 500 units.

Important points about vertical shifts include:

• Adding a positive number shifts the graph upwards.
• Adding a negative number (or subtracting a positive) shifts the graph downwards.
• Vertical shifts change the starting point but not the shape of the function.

Squaring Function

The squaring function is one of the simplest and most fundamental mathematical functions. Its basic form is \( y = x^2 \), graphically represented as a parabola opening upwards and centered at the origin (0,0).

The squaring function has unique properties because every output value is the square of its input, making it always non-negative. Each side of the graph is symmetrical with respect to the y-axis, which means it looks the same on both sides of this axis.

When we apply transformations, such as horizontal and vertical shifts to the squaring function, the vertex of the parabola moves but its shape remains intact. For example, the equation \( y = (x - 2000)^2 + 500 \) represents a squaring function shifted 2000 units right and 500 units up. These shifts move the vertex from the origin to the point (2000, 500) while maintaining its classic shape.

Key characteristics of the squaring function are:

• The graph is a U-shaped parabola.
• The vertex is the lowest point for the function's standard form.
• It is symmetric across the y-axis.