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Chapter 2: Problem 13

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

Short Answer

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

Step by step solution

01

Identify the Basic Function

The basic squaring function is given by \( y = x^2 \). This is the function that we will shift in order to achieve the desired translation.
02

Translate the Function Horizontally

To shift the function 2000 units to the right, we replace \( x \) with \( x - 2000 \) in the function. This gives us the new function \( y = (x - 2000)^2 \).
03

Translate the Function Vertically

To shift the function 500 units upward, we add 500 to the entire function. This transforms the equation \( y = (x - 2000)^2 \) into \( y = (x - 2000)^2 + 500 \).
04

Write the Final Equation

The final equation that represents the squaring function with the specified translations is \( y = (x - 2000)^2 + 500 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Squaring Function
The squaring function is one of the most fundamental functions in mathematics. It is defined by the equation \( y = x^2 \). This function graphs a perfect parabola that opens upwards with its vertex located at the origin, \( (0,0) \). It is symmetric about the y-axis, meaning it looks the same on both sides of this axis.
Some important characteristics of the squaring function include:
  • It is an even function, since \( f(x) = f(-x) \).
  • Its graph passes through the point \( (0,0) \).
  • As \( x \) increases positively or negatively, \( y \) values are positive and increase.
Understanding this basic function is crucial as it serves as a starting point for applying transformations such as shifts and stretches to create more complex functions.
Horizontal Shift
In function transformations, a horizontal shift involves moving the graph of a function left or right along the x-axis. For the squaring function \( y = x^2 \), this shift is achieved by replacing \( x \) with \( x - h \).
When \( h > 0 \), the graph is shifted to the right by \( h \) units. Conversely, if \( h < 0 \), the graph is shifted left.
In our example, to shift the function 2000 units to the right, we replace \( x \) with \( x - 2000 \). This modification changes the function to \( y = (x - 2000)^2 \).
A few points to note about horizontal shifts:
  • They do not affect the shape of the graph, so the parabola remains the same size and direction.
  • Only the graph鈥檚 location along the x-axis changes.
Vertical Shift
A vertical shift moves the graph of a function up or down along the y-axis. For the squaring function, a vertical shift is achieved by adding or subtracting a constant from the squared term.
To shift \( y = x^2 \) upward, we add a positive number \( k \) to the entire equation. If we wanted to move it down, we would subtract \( k \).
In the given exercise, we need to shift the function 500 units upward, resulting in the equation \( y = (x - 2000)^2 + 500 \).
Key points about vertical shifts include:
  • The shape of the graph does not change.
  • The shift only affects where the graph is positioned on the y-axis.
  • Vertical shifts are straightforward, often simply adding or subtracting from the function.
By combining horizontal and vertical shifts, we can translate functions around the plane to meet specific graphing needs.

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Most popular questions from this chapter

Use the table to evaluate each expression, if possible. (a) \((f+g)(2)\), (b) \((f-g)(4)\), (c) \((f g)(-2)\), (d) \(\left(\frac{f}{g}\right)(0)\). $$\begin{array}{r|c|c}\boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) & \boldsymbol{g}(\boldsymbol{x}) \\ \hline-2 & -4 & 2 \\\\\hline 0 & 8 & -1 \\\\\hline 2 & 5 & 4 \\\\\hline 4 & 0 & 0\end{array}$$

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Solve each equation graphically. $$|x|+|x-4|=8$$
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