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Chapter 2: Problem 67

Use transformations to graph the functions. $$ r(x)=-\sqrt{-x}+1 $$

Short Answer

Expert verified
Reflect \( \sqrt{x} \) across the y-axis, x-axis, and shift up 1 unit.

Step by step solution

01

Identify the base function

The base function for this exercise is the square root function: \( \sqrt{x} \).
02

Reflect across the y-axis

The given function involves \( \sqrt{-x} \). This transformation reflects the base function \( \sqrt{x} \) across the y-axis.
03

Reflect across the x-axis

The given function involves \( -\sqrt{-x} \). This transformation reflects the previously reflected function \( \sqrt{-x} \) across the x-axis.
04

Vertical shift

The given function includes a \( +1 \) outside of the square root. This transformation shifts the function 1 unit upwards.
05

Graph the function

Start with the graph of \( \sqrt{x} \). Apply the transformations step by step: reflect across the y-axis, then the x-axis, and finally shift up by 1 unit. The final graph should show the reflected and shifted square root function.

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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 one of the most fundamental functions in algebra. It is written as \( \sqrt{x} \), which means taking the non-negative value that, when squared, gives 'x'. Its graph is a curved line that starts at the origin (0,0) and rises to the right. Understanding this basic shape will help you apply various transformations easily.
Graph transformations
Graph transformations are processes that change the position or shape of the graph. Key types include:
  • Reflections: Turning the graph over a specific axis.
  • Shifts: Moving the graph up, down, left, or right.
  • Stretches and compressions: Changing the graph's width or height.
For our exercise, we will focus on reflections and vertical shifts.
Vertical shift
A vertical shift moves a graph up or down without changing its shape. For the function \( r(x) = -\sqrt{-x} + 1 \), the +1 outside the square root indicates a vertical shift. This means every point on the graph moves one unit up. Imagine sliding the entire curve upwards, keeping its shape intact.
Reflection across axes
Reflections turn the graph over a specific axis. For the function \( r(x) = -\sqrt{-x} + 1 \):
  • Y-axis Reflection: The negative inside the square root (\sqrt{-x}) reflects the graph over the y-axis.
  • X-axis Reflection: The negative sign outside the square root (-\sqrt{-x}) reflects the graph over the x-axis.
By performing these reflections in sequence, you get a new graph that is flipped across both axes.

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Most popular questions from this chapter

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Suppose that a function \(H\) gives the high temperature \(H(x)\) (in \({ }^{\circ} \mathrm{F}\) ) for day \(x\). Suppose that a function \(L\) gives the low temperature \(L(x)\) (in \({ }^{\circ} \mathrm{F}\) ) for day \(x\). What \(\operatorname{does}\left(\frac{H+L}{2}\right)(x)\) represent?

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