/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Graph each function "by hand." [... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 25

Graph each function "by hand." [Note: Even if you have a graphing calculator, it is important to be able to sketch simple curves by finding a few important points.] $$ f(x)=-x+1 $$

Short Answer

Expert verified
A straight line with slope -1, crossing y-axis at (0, 1).

Step by step solution

01

Identify the type of function

The function given is in the form of a linear equation, which is typically written as \( y = mx + b \). In this case, the function is \( f(x) = -x + 1 \), where \( m = -1 \) and \( b = 1 \). This tells us the slope is \(-1\) and the y-intercept is \(1\).
02

Plotting the y-intercept

The y-intercept of the function is the point where the graph crosses the y-axis. This occurs at \( x = 0 \). Substitute \( x = 0 \) into the function: \( f(0) = -0 + 1 = 1 \). So plot the point \( (0, 1) \).
03

Use the slope to find another point

The slope \( m = -1 \) tells us that for every unit movement to the right along the x-axis, the y-value decreases by 1 unit. From the y-intercept \( (0, 1) \), move 1 unit to the right to \( x = 1 \) and 1 unit down to \( y = 0 \). This gives us the point \( (1, 0) \). Plot this point as well.
04

Draw the line

Using the points determined, \( (0, 1) \) and \( (1, 0) \), draw a straight line through these points. This line represents the graph of the function \( f(x) = -x + 1 \).
05

Verify with another point

To ensure accuracy, test another point. Choose \( x = 2 \): \( f(2) = -2 + 1 = -1 \). Plot \( (2, -1) \) on the graph. It should fall on the line, verifying its correctness.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Slope-Intercept Form
When graphing linear functions, one of the most straightforward forms to use is the slope-intercept form. This form is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.
  • Slope \( (m) \): The slope indicates the steepness and direction of the line. For example, a positive slope moves upward, while a negative slope goes downward as you move from left to right.
  • Y-Intercept \( (b) \): The y-intercept is the point where the line crosses the y-axis. This happens when \( x = 0 \).
In the function \( f(x) = -x + 1 \), the slope \( m \) is \(-1\) and the y-intercept \( b \) is \(1\). Knowing these values simplifies the process of graphing because they direct us on where to start (the y-intercept) and how to proceed (using the slope).
Y-Intercept
The y-intercept of a linear function is crucial for sketching its graph. It is the point where the line touches the y-axis. In any linear equation given in slope-intercept form, the y-intercept \( b \) is simply the constant term. For \( f(x) = -x + 1 \), the y-intercept is \( 1 \), meaning that the line crosses the y-axis at the point \( (0, 1) \).To find the y-intercept quickly:
  • Set \( x = 0 \) in the equation. The y-value you get will be the intercept. For instance, substituting \( 0 \) in \( f(x) = -x + 1 \) gives \( f(0) = 1 \).
This step is always your first plotting point, marking the start of your line on a graph.
Plotting Points
Once you've identified the slope and y-intercept, plotting points for the line becomes straightforward. Plotting points refers to marking specific points on the graph that the line passes through. These points help guide the drawing of the accurate line.
  • Start with the y-intercept: Locate it at \( (0, b) \). For \( f(x) = -x + 1 \), start at \( (0, 1) \).
  • Use the slope: From the y-intercept, apply the slope \( m \) to find more points. Here, a slope of \(-1\) indicates moving one unit right and one unit down. Thus, from \( (0, 1) \), move to \( (1, 0) \).
Continuing this method, you can determine more points such as \( (2, -1) \). After plotting, draw a straight line through these points, ensuring they align perfectly. This graphical method verifies the behavior and correctness of the linear function.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

GENERAL: Stopping Distance A car traveling at speed \(v\) miles per hour on a dry road should be able to come to a full stop in a distance of $$ D(v)=0.055 v^{2}+1.1 v \text { feet } $$ Find the stopping distance required for a car traveling at: \(60 \mathrm{mph}\).

Use the TABLE feature of your graphing calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for values of \(x\) such as \(100,10,000,1,000,000\), and higher values. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter \(4 .\)

How will the graph of \(y=(x+3)^{3}+6\) differ from the graph of \(y=x^{3} ?\) Check by graphing both functions together.

Find, rounding to five decimal places: a. \(\left(1+\frac{1}{100}\right)^{100}\) b. \(\left(1+\frac{1}{10,000}\right)^{10,000}\) c. \(\left(1+\frac{1}{1,000,000}\right)^{1,000,000}\) d. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4 .

GENERAL: Boiling Point At higher altitudes, water boils at lower temperatures. This is why at high altitudes foods must be boiled for longer times - the lower boiling point imparts less heat to the food. At an altitude of \(h\) thousand feet above sea level, water boils at a temperature of \(B(h)=-1.8 h+212\) degrees Fahrenheit. Find the altitude at which water boils at \(98.6\) degrees Fahrenheit. (Your answer will show that at a high enough altitude, water boils at normal body temperature. This is why airplane cabins must be pressurized - at high enough altitudes one's blood would boil.)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.