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Chapter 2: Problem 32

Use translations to graph the given functions. $$ d(x)=\sqrt{x+4}-1 $$

Short Answer

Expert verified
Shift the graph of \(\sqrt{x}\) left by 4 units and down by 1 unit.

Step by step solution

01

Identify the base function

The base function is the simplest form of the given function. Here, the base function is the square root function: \(f(x) = \sqrt{x}\).
02

Identify horizontal translation

Observe the term inside the square root. The expression \(x + 4\) indicates a horizontal translation to the left by 4 units. So, transform the base function to \(g(x) = \sqrt{x + 4}\).
03

Identify vertical translation

The term outside the square root, -1, indicates a vertical translation down by 1 unit. Therefore, the function becomes \(d(x) = \sqrt{x + 4} - 1\).
04

Graph the base function

Draw the graph for the base function \(f(x)=\sqrt{x}\). This function starts at the origin (0, 0) and increases gradually.
05

Apply the horizontal translation

Shift the entire graph of \(f(x) = \sqrt{x}\) to the left by 4 units to obtain the graph of \(g(x) = \sqrt{x+4}\).
06

Apply the vertical translation

Move the graph of \(g(x) = \sqrt{x+4}\) downwards by 1 unit to get the final graph of \(d(x) = \sqrt{x + 4} - 1\).
07

Plot key points

Identify and plot the key points of the function \(d(x) = \sqrt{x + 4} - 1\) for precise graphing. For example, \(d(-4)= -1\) and \(d(0)=\sqrt{4}-1=1\).

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Key Concepts

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

Horizontal Translation
In functions, a horizontal translation refers to shifting the graph of a function left or right on the coordinate plane.
To determine the direction, look at the term inside the function's argument.
For example, in the function \(d(x) = \sqrt{x + 4} - 1\), the term \(x + 4\) indicates a shift.
  • When a positive number is added inside the function, like \(x + 4\), the graph moves to the left by that number of units.
  • Conversely, a negative number like \(x - 4\) would shift the graph to the right by 4 units.
So, in our function, the graph of \(\sqrt{x}\) is moved 4 units to the left because of the \(x + 4\).
Vertical Translation
Vertical translation involves moving the graph of a function up or down.
This movement is determined by the term outside the function.
For instance, in \(d(x) = \sqrt{x + 4} - 1\), the term \(-1\) dictates the vertical shift.
  • If the term is negative, the function graph shifts down by that number of units.
  • So for \(d(x)\), the \(-1\) means our graph will move 1 unit downward.
  • If the term was positive, such as \(+3\), the graph would move 3 units upwards.
Combining this with the horizontal shift, \(\sqrt{x + 4}\) moves 4 units left first, then 1 unit down to form \(d(x) = \sqrt{x + 4} - 1\).
Square Root Function
The square root function, represented as \(f(x) = \sqrt{x}\), is essential in algebra.
  • It starts at the origin (0, 0) and only includes non-negative values of x, since you can't have a real number square root of a negative number.
  • Its graph is a smooth curve that gradually increases as x increases, creating a gentle upward arc.
  • This basic shape helps in understanding graph translations.

Start by sketching the basic curve of \(\sqrt{x}\).
Apply the horizontal and vertical translations to get the desired function graph.
For \(d(x) = \sqrt{x + 4} - 1\), the process involves first moving the \(\sqrt{x}\) graph left by 4 units and then down by 1 unit.
Plotting points like \(d(-4)=-1\) and \(d(0)=1\) helps ensure accuracy.

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