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Chapter 2: Problem 11

Write the equation that results in the desired translation. Do not use a calculator. The square root function, shifted 3 units upward and 6 units to the left

Short Answer

Expert verified
\( f(x) = \sqrt{x + 6} + 3 \)

Step by step solution

01

Understanding the Parent Function

Start with the parent function, which is the square root function given by \( f(x) = \sqrt{x} \). This function represents the square root of \( x \).
02

Translation Left by 6 Units

To translate a function to the left, you add a number inside the function's argument. For a leftward shift of 6 units, replace \( x \) with \( x + 6 \). Thus, the equation becomes \( f(x) = \sqrt{x + 6} \).
03

Translation Upward by 3 Units

To translate a function upward, add a number outside of the radical, which shifts the graph vertically. Add 3 to the function, resulting in the equation \( f(x) = \sqrt{x + 6} + 3 \).

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Key Concepts

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

Square Root Function
The square root function is a basic and essential concept in mathematics, widely used in various fields. To express it algebraically, the square root function is written as \( f(x) = \sqrt{x} \). This expression simply indicates that for any value of \( x \), the value of the function \( f(x) \) is the square root of \( x \). For example, if \( x = 9 \), then \( f(x) = \sqrt{9} = 3 \).
  • Graphical Representation: The graph of the square root function produces a curve that starts at the origin (0,0) when there are no translations or modifications.
  • Domain and Range: The domain (possible values of \( x \)) of the square root function is \( x \geq 0 \), while the range (possible values of \( f(x) \)) is also \( f(x) \geq 0 \).
Understanding this function is crucial as it serves as a parent function in various transformations, helping to illustrate shifts and translations.
Parent Function
A parent function is the simplest form of a function without any transformations applied to it. In the case we're examining, the parent function is the square root function, \( f(x) = \sqrt{x} \). Parent functions establish the basic shape and behavior of graphs which is then altered through transformations to produce more complex functions.
  • Importance: Parent functions allow for easy visualization and recognition of function behavior, providing a template for graphing transformed functions.
  • Types: Alongside the square root function, other common parent functions include linear functions, quadratic functions, and cubic functions.
Recognizing parent functions will improve your understanding of how different changes affect graphs.
Horizontal Shift
A horizontal shift in a function occurs when the graph of the function moves left or right on the coordinate plane. This shift is applied by changing the input \( x \) within the function. For example, to shift the square root function \( f(x) = \sqrt{x} \) 6 units to the left, we modify it to \( f(x) = \sqrt{x + 6} \).
  • Left Shift: Add a positive number to \( x \) (e.g., \( x + k \)), which shifts the graph \( k \) units to the left.
  • Right Shift: Subtract a number from \( x \) (e.g., \( x - k \)), for a rightward shift of \( k \) units.
These shifts do not affect the function's range but alter its domain slightly as per the translation applied.
Vertical Shift
A vertical shift moves the graph of a function up or down. This type of transformation affects the entire function equally by adding or subtracting a constant outside the main function's expression. For example, applying a vertical shift by adding 3 to the square root function results in \( f(x) = \sqrt{x} + 3 \), moving the curve upward by 3 units.
  • Upward Shift: Add a constant to the function (e.g., \( f(x) + k \)), which raises the graph by \( k \) units.
  • Downward Shift: Subtract a constant from the function (e.g., \( f(x) - k \)), lowering the graph by \( k \) units.
Vertical shifts change the range of the function, but unlike horizontal shifts, they do not affect domain.

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Most popular questions from this chapter

Solve each equation graphically. $$|0.5 x+2|+|0.25 x+4|=9$$

Sketch a graph that depicts the amount of water in a 100 -gallon tank. The tank is initially empty and then filled at a rate of 5 gallons per minute. Immediately after it is full, a pump is used to empty the tank at 2 gallons per minute.

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Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-61.5| \leq 12.5,\) Buenos Aires, Argentina
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