Problem 14 - A Graphical Approach to College Algebra

Question

Write the equation that results in the desired translation. Do not use a calculator. The squaring function, shifted 1000 units to the left and 255 units downward

Step-by-Step Solution

Answer

The equation is \( h(x) = (x + 1000)^2 - 255 \).

Step 1: Identify the Original Function

The original squaring function is given by \( f(x) = x^2 \). This is a standard parabola centered at the origin.

Step 2: Translate Left by 1000 Units

To translate the function 1000 units to the left, replace \( x \) with \( x + 1000 \) in the original function. This results in the function \( g(x) = (x + 1000)^2 \).

Step 3: Translate Downward by 255 Units

To shift the function downward by 255 units, subtract 255 from the function obtained after the leftward translation. This gives \( h(x) = (x + 1000)^2 - 255 \).

Step 4: Write the Final Equation

The final equation after performing both translations is \( h(x) = (x + 1000)^2 - 255 \). This represents the parabola shifted left 1000 units and downward 255 units.

Key Concepts

Squaring Function

The squaring function is one of the simplest and most common mathematical functions. It is defined as \( f(x) = x^2 \). This function graphs as a U-shaped curve known as a parabola, with its vertex at the origin, \((0, 0)\), and is symmetric around the y-axis.
The squaring function is continuous, meaning there are no breaks or holes in its graph. As you move away from the vertex in either direction along the x-axis, the outputs increase, resulting in the upward-opening shape of the parabola.
Mathematically:

• When \( x = 0 \), \( f(x) = 0^2 = 0 \).
• When \( x = 1 \), \( f(x) = 1^2 = 1 \).
• When \( x = -1 \), \( f(x) = (-1)^2 = 1 \).

These properties make the squaring function a fundamental model for understanding quadratic transformations.

Horizontal Translation

A horizontal translation involves shifting a function left or right on a graph. This transformation doesn't affect the shape of the graph, only its position along the x-axis.
For the squaring function \( f(x) = x^2 \), translating it 1000 units to the left involves changing the variable \( x \) in the equation by adding 1000 to it.
This results in the new function \( g(x) = (x + 1000)^2 \).

• If we add a constant to \( x \), it translates the graph to the left by that numerical value.
• If we subtract, the graph moves to the right.

Understanding this concept is crucial for moving functions along a plane without altering their inherent structure.

Vertical Translation

A vertical translation involves shifting a function up or down on a graph. Like horizontal translations, vertical translations maintain the shape of the graph.
After translating the squaring function left and obtaining \( g(x) = (x + 1000)^2 \), we perform a vertical translation downward by 255 units by subtracting 255 from this new function.
Thus, the equation becomes \( h(x) = (x + 1000)^2 - 255 \).

• To move a graph downward, we subtract a constant from the entire function.
• To move it upward, we add a constant.

Vertical translations help in adjusting the height of the function on the graph, allowing for the function's peak or trough to move up or down without affecting its width or basic shape.

Parabola

A parabola is a symmetrical, open plane curve formed by the graph of a quadratic function, such as the squaring function \( f(x) = x^2 \).
The parabola's fundamental characteristic is its U-shape, with the vertex being its lowest point (for functions that open upward) or its highest point (for functions that open downward).
To grasp parabola transformations:

• The vertex form of a parabola, \( f(x) = a(x - h)^2 + k \), reveals how the graph shifts.
• In this form, \( a \) determines the width and direction, \( h \) dictates horizontal shift, and \( k \) decides vertical placement.

Understanding the basic design of a parabola is essential for interpreting quadratic graphs and their transformations. Moving parabolas around a graph through translations can help model real-world scenarios and enhance problem-solving skills in mathematics.