/*! 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 3 Exer. \(1-20\) : Sketch the grap... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 3

Exer. \(1-20\) : Sketch the graph of the equation, and label the \(x\) - and \(y\) -intercepts. $$y=-x+1$$

Short Answer

Expert verified
Graph is a line through points (0, 1) and (1, 0). Intercepts are labeled accordingly.

Step by step solution

01

Understand the Equation

The equation given is in the slope-intercept form, which is \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. Here, \( m = -1 \) and \( b = 1 \).
02

Find the Y-intercept

The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation: \( y = -0 + 1 \), so \( y = 1 \). The y-intercept is the point \( (0, 1) \).
03

Find the X-intercept

The x-intercept occurs when \( y = 0 \). Set \( y = 0 \) in the equation and solve for \( x \): \( 0 = -x + 1 \). Rearrange to find \( x = 1 \). So, the x-intercept is the point \( (1, 0) \).
04

Plot the Intercepts

On a graph, plot the y-intercept at \( (0, 1) \) and the x-intercept at \( (1, 0) \).
05

Draw the Line

Draw a straight line through the points \( (0, 1) \) and \( (1, 0) \) to represent the equation \( y = -x + 1 \). This line shows the graph of the equation.

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.

Graphing Linear Equations
Graphing linear equations is a fundamental concept in algebra that involves creating a visual representation of a linear relationship between two variables. The graph of a linear equation is always a straight line. To graph a linear equation like \[ y = -x + 1 \],we follow a series of systematic steps:
  • Identify the slope (\( m \)) and the y-intercept (\( b \)) in the equation's slope-intercept form: \( y = mx + b \).
  • The slope \( m \) dictates the steepness and direction of the line; in this case, \( m = -1 \), indicating the line decreases by 1 unit in y for each increase of 1 unit in x.
  • The y-intercept \( b \) is the point where the line crosses the y-axis, providing an easy starting point for graphing.
To graph this equation, you start by plotting the intercepts and then draw a line through them, ultimately illustrating the equation visually. This graphical representation allows you to quickly understand the relationship between x and y on the Cartesian coordinate system.
x-intercept
The x-intercept of a graph is the point where the line crosses the x-axis. It鈥檚 a key feature in understanding the behavior of a linear equation. Finding the x-intercept involves setting the y value to zero and solving for x.

Take the equation \( y = -x + 1 \). To find the x-intercept, set \( y = 0 \):
  • Plug \( y = 0 \) into the equation: \[ 0 = -x + 1 \].
  • Solve for \( x \) to find \( x = 1 \).
The x-intercept of the equation is the point (1, 0). This means the line crosses the x-axis at 1. By identifying this intercept, you gain insight into where the line touches the x-axis, which is important for graphing the entire equation. It helps verify that your graph covers all aspects of the linear relationship.
y-intercept
The y-intercept is an essential concept in graphing linear equations, representing the point where the line intersects the y-axis. For equations in slope-intercept form \( y = mx + b \), the y-intercept is directly visible as \( b \).For the equation \( y = -x + 1 \):
  • The slope-intercept form highlights \( b = 1 \), so the line crosses the y-axis at (0, 1).
  • To explicitly determine the y-intercept: set \( x = 0 \) in the equation:\[ y = -0 + 1 \].
  • Thus, \( y = 1 \), confirming (0, 1)as the y-intercept.
Plotting this intercept point is one of the first steps in graphing the equation. It provides a fixed reference on the graph to connect with other points, such as the x-intercept, ultimately forming the straight line representing the equation. Understanding the y-intercept helps in visualizing the start of the graph in relation to the axes.

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

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. Vertex \((0,5),\) passing through \((2,-3)\)

Exer. 73-74: Graph the lines on the same coordinate plane, and estimate the coordinates of the points of Intersection. Identify the polygon determined by the lines. $$2 x-y=-1 ; \quad x+2 y=-2 ; \quad 3 x+y=11$$

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. Vertex \((0,-2),\) passing through \((3,25)\)

(a) Sketch the graph of \(f\) on the given interval \([a, b] .\) (b) Estimate the range of \(f\) on \([a, b] .\) (c) Estimate the intervals on which \(f\) is increasing or is decreasing. $$f(x)=\frac{x^{1 / 3}}{1+x^{4}}$$

If \(S(r)=4 \pi r^{2}\) and \(D(t)=2 t+5,\) find \((S \circ D)(t)\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.