Sketching the Graph of y x - sqrt(1 - x): A Step-by-Step Guide

In this article, we will explore how to sketch the graph of the function y x - sqrt(1 - x). To achieve this, we will break down the expression based on the definition of the square root function. We will then provide a detailed explanation of the steps involved in sketching the graph, including identifying key points and determining the behavior of each piece.

Function Breakdown

We can break down the function based on the properties of the square root function:

For x geq 0:

When ( x ) is non-negative, ( sqrt(1 - x) ) is defined and we have:

y x - sqrt(1 - x)

For x

When ( x ) is negative, ( 1 - x ) is positive, and we have:

y x - sqrt(1 - x) x - (-sqrt(1 - x)) x sqrt(1 - x)

Steps to Sketch the Graph

Identify Key Points

At ( x 0 ):

From the first case, ( y 0 - sqrt(1 - 0) 0 - 1 -1 )

At ( x 1 ):

From the first case, ( y 1 - sqrt(1 - 1) 1 - 0 1 )

At ( x -1 ):

From the second case, ( y -1 sqrt(1 - (-1)) -1 sqrt(2) approx -1 1.414 0.414 )

Determine the Behavior of Each Piece

For ( x geq 0 ): The graph is a line that starts at the point ( (0, -1) ) and increases linearly as ( x ) increases. It will cross the x-axis at ( (1, 0) ).

For ( x

Plot the Points and Draw the Lines

Plot the points ( (0, -1) ) and ( (1, 0) ). Draw a line connecting these points from ( (0, -1) ) to ( (1, 0) ). For ( x

Final Graph

The graph of ( y x - sqrt(1 - x) ) consists of a line that starts at ( (0, -1) ) and increases to ( (1, 0) ) for ( x geq 0 ). For ( x

Visual Representation

To visualize the graph directly, you can use the following R script:

library(ggplot2)library(cowplot)# Define the functionf - function(x) {  ifelse(x > 0, x - sqrt(1 - x), x   sqrt(1 - x))}# Generate x valuesx - seq(-1, 1.5, by  0.01)# Compute corresponding y valuesy - f(x)# Create the plotgg - ggplot((x, y), aes(x  x, y  y))    geom_line()    geom_point(data  (x  0, y  -1),   FALSE, color  'red')    geom_point(data  (x  1, y  0),   FALSE, color  'red')    geom_point(data  (x  -1, y  0.414),   FALSE, color  'blue')    labs(title  "Graph of y  x - sqrt(1 - x)", x  "x", y  "y")    theme_cowplot()# Print the plotprint(gg)

By following these steps, you can accurately sketch the graph of the function ( y x - sqrt(1 - x) ).

Summary

The graph of ( y x - sqrt(1 - x) ) consists of two distinct parts:

A line that starts at ( (0, -1) ) and increases linearly to ( (1, 0) ) for ( x geq 0 ). A curve that starts at ( (-1, 0.414) ) and decreases as it moves to the left for ( x

If you have any further questions or need additional assistance, feel free to ask!