/*! 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 87 Graph function. \(f(x)=x\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 87

Graph function. \(f(x)=x\)

Short Answer

Expert verified
The graph of \(f(x) = x\) is a straight line through the origin with a slope of 1.

Step by step solution

01

Understanding the Function

The function given is \(f(x) = x\). This is a linear function with a slope of 1 and a y-intercept at 0. It represents a straight line.
02

Identify the Slope and Intercept

Since this is a linear equation in the form \(f(x) = mx + b\), we identify the slope \(m = 1\) and y-intercept \(b = 0\). This means the line increases by 1 unit up for every 1 unit it moves to the right on the graph.
03

Plot Key Points

To graph the function, start by plotting the y-intercept. At \(x = 0\), \(f(x) = 0\), so one point is (0, 0). Then, using the slope, another point would be (1, 1). Continue plotting points such as (2, 2), (3, 3), and (-1, -1).
04

Draw the Line

With the points (0, 0), (1, 1), (2, 2), and (-1, -1) plotted, draw a straight line through these points. This line represents the graph of \(f(x) = x\). It is perfectly diagonal, passing through the origin and rising at a 45-degree angle.

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
The slope-intercept form of a linear equation is a vital tool in graphing functions. It is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. Let's break it down further:

\(
  • Slope \(m\): This is the rate of change of the line. It tells us how steep the line is and in which direction it is going. A positive slope means the line ascends from left to right, while a negative slope means it descends.

  • Y-intercept \(b\): This is the point where the line crosses the y-axis. It is the value of \(y\) when \(x = 0\).
\)
In our example, the equation \(f(x) = x\) can be viewed as \(y = 1x + 0\), showing that the slope \(m\) is 1 and the y-intercept \(b\) is 0.
Understanding this form is important for quickly sketching and analyzing the linear equation on a graph.
Plotting Points
Plotting points is a fundamental step in graphing linear functions. By identifying key points from the equation, we can accurately draw the graph. Here's how you do it:

First, start with the y-intercept, which is the point where the function crosses the y-axis. Then, use the slope to find additional points on the line.

Consider the function \(f(x) = x\):

  • The y-intercept is (0,0). It tells us that the line passes through the origin.

  • Using the slope of 1, move from the y-intercept. For each "1" unit moved to the right on the x-axis, go "1" unit up on the y-axis. This gives the next point: (1,1).

  • Repeat these steps to find more points. For example, (2,2), (3,3), (-1,-1), etc.
Every plotted point serves as a guide when you draw your line, ensuring it reflects the linear function accurately.
Y-Intercept
The y-intercept is a crucial element in graphing and comprehending linear functions. It's essentially where your line lands on the y-axis.

The value of the y-intercept directly affects where the entire line sits with respect to the origin. In the function \(f(x) = x\), our y-intercept \(b\) is 0, indicating that the line crosses the y-axis exactly at the origin (0,0).

Understanding the y-intercept allows you to quickly locate the starting point for graphing the function on a coordinate plane:
  • If \(b = 0\), like in \(f(x) = x\), the origin (0,0) is your starting point.

  • If \(b\) had been, say, 3, the line would cross the y-axis at (0,3).
Recognizing the y-intercept not only helps you begin graphing but also provides insights into the function's behavior and its graphical representation.

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

Write an equation for a linear function whose graph has the given characteristics. Slope \(5, y\) -intercept \((0,-3)\)

Find \(g(2)\) and \(g(3)\). \(g(x)=5 x^{2}+2 x+2\)

Graph function. \(g(x)=0.75 x\)

Write an equation for a linear function whose graph has the given characteristics. Horizontal, passes through \((9,-32)\)

Concessionaires. \(\quad\) A baseball club pays a vendor \(\$ 125\) per game for selling bags of peanuts for \(\$ 4.75\) each. a. Write a linear function that describes the profit the vendor makes for the baseball club during a game if she sells \(b\) bags of peanuts. b. Find the profit the baseball club will make if the vendor sells 110 bags of peanuts during a game.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.