Problem 14 - 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 1000 units to the left and 255 units downward

Step-by-Step Solution

Answer

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

Step 1: Understanding the Parent Function

The parent function here is the squaring function, which is typically expressed as \( y = x^2 \). This is a simple quadratic function graphically represented as a parabola.

Step 2: Determine the Horizontal Shift

The function is shifted 1000 units to the left. We adjust the function by replacing \(x\) with \(x + 1000\) in the equation. Thus, the modified function becomes \( y = (x + 1000)^2 \).

Step 3: Determine the Vertical Shift

The problem states the function is shifted 255 units downward. To account for this, subtract 255 from the \(y\) value of the function. Therefore, the equation transforms to \( y = (x + 1000)^2 - 255 \).

Step 4: Write the Final Translated Function

Combining both translations, the equation for the shifted function is \( y = (x + 1000)^2 - 255 \). This represents the squaring function translated 1000 units left and 255 units down.

Key Concepts

Parabola

A parabola is a U-shaped curve that is the graphical representation of a quadratic function. Typically, the simplest form of a quadratic function is the squaring function, expressed as \( y = x^2 \). In this form, the vertex of the parabola is at the origin \((0,0)\). The parabola opens upwards if the coefficient of \( x^2 \) is positive. If the coefficient is negative, it points downwards.Key features of a parabola include:

• The vertex, which is the highest or lowest point depending on the orientation.
• The axis of symmetry, a vertical line that divides the parabola into two mirror-image halves.
• The direction in which the parabola opens, either upwards or downwards.

Understanding these properties is crucial for graph transformations.

Function Translation

Function translation moves the entire graph of a function without altering its shape. It is a transformation that shifts the graph horizontally and/or vertically in a plane.For quadratic functions like \( y = x^2 \), translations involve modifying the \( x \) and/or \( y \) variables. The modified equation results from shifting the function in the coordinate system:

• A horizontal translation involves replacing \( x \) with \( x + h \), where \( h \) represents the shift direction and magnitude.
• A vertical translation changes the \( y \) value by adding or subtracting a constant \( k \).

These shifts help position the parabola exactly where needed.

Horizontal and Vertical Shifts

Horizontal and vertical shifts are specific types of function translations. These involve translating the graph of the function left, right, up, or down in the coordinate plane:- **Horizontal Shift**: This occurs when you shift the graph left or right. It's achieved by changing the \( x \)-variable in the function: - Replacing \( x \) with \( x + h \) shifts the graph to the left by \( h \) units. - Replacing \( x \) with \( x - h \) shifts the graph to the right by \( h \) units.- **Vertical Shift**: This is executed by altering the \( y \) value of the function: - Adding a constant \( k \) shifts the graph upwards by \( k \) units. - Subtracting a constant \( k \) shifts it downwards by \( k \) units.In our case, replacing \( x \) with \( x + 1000 \) shifts the parabola 1000 units left. Subtracting 255 from \( y \) translates it 255 units down.