Problem 10 - 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 4 units upward and 1 unit to the left

Step-by-Step Solution

Answer

\( y = (x + 1)^2 + 4 \)

Step 1: Identify the Parent Function

The parent function given is the squaring function, which is represented as \( y = x^2 \). Our task is to adjust this function for the specified translations.

Step 2: Determine the Vertical Shift

A vertical shift of 4 units upwards means we need to add 4 to the function. Thus, if our function was \( y = x^2 \), the vertical shift modifies it to \( y = x^2 + 4 \).

Step 3: Determine the Horizontal Shift

A horizontal shift to the left involves modifying the input variable \( x \). A shift of 1 unit to the left is represented by replacing \( x \) with \( x + 1 \). So our function becomes \( y = (x + 1)^2 \).

Step 4: Combine Vertical and Horizontal Shifts

Combine both the vertical and horizontal shifts to modify the parent function. Replace \( x \) with \( x + 1 \) and add 4 to the entire function: \( y = (x + 1)^2 + 4 \).

Step 5: Final Equation

The equation \( y = (x + 1)^2 + 4 \) represents the squaring function shifted 4 units upward and 1 unit to the left.

Key Concepts

Squaring Function

The squaring function is a fundamental concept in algebra. It's represented by the equation \( y = x^2 \). This function forms a parabola when graphed and opens upwards. The shape of this graph is symmetrical around the y-axis, which means that each point on the left is mirrored by a corresponding point on the right.

In simpler terms, the squaring function takes any number "x" and multiplies it by itself. This operation results in all positive outputs or zero, never negative values. It is often used to model real-world situations like physics problems or financial calculations because of its predictable nature.

Vertical Shift

A vertical shift changes the position of a graph up or down on a coordinate plane. This transformation is quite simple; you either add or subtract a number from the function. Adding a number shifts the graph upwards, while subtracting a number moves it downwards.

For our function \( y = x^2 \), shifting it 4 units upwards means we add 4 to every output value. This changes the function to \( y = x^2 + 4 \). This adjustment means that every point on the graph moves 4 units higher than it originally was, without altering the shape of the parabola itself.

Think of the parabola as a slider up and down the y-axis, maintaining its form while its position modifies.

Horizontal Shift

A horizontal shift changes the position of a graph left or right along the x-axis. This transformation involves adjusting the input variable "x" to affect how the function appears on a graph.

For horizontal shifts, the direction can be a bit counterintuitive. If you want the function to move to the left, you will replace "x" with "x + h" where \(h\) is a positive number. Conversely, moving it to the right involves replacing "x" with "x - h".

In our example, to shift the squaring function \( y = x^2 \) 1 unit to the left, we replace "x" with "x + 1" resulting in \( y = (x + 1)^2 \). This transformation modifies the x-coordinates, moving the graph one step left while keeping its recognizable U-shape intact.