/*! 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 6 In Exercises \(5-36,\) graph the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 6

In Exercises \(5-36,\) graph the given functions. $$y=-2 x$$

Short Answer

Expert verified
Graph the line passing through (0, 0) and (1, -2).

Step by step solution

01

Identify the Function Type

The function given is \( y = -2x \). This is a linear function of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = -2 \) and \( b = 0 \).
02

Determine the Slope and Y-Intercept

For the equation \( y = -2x \), the slope \( m \) is \(-2\) and the y-intercept \( b \) is \(0\). This means the line crosses the y-axis at (0, 0).
03

Plot the Y-Intercept

Begin plotting the graph by marking the y-intercept point at (0, 0) on the coordinate plane. This is where the line will cross the y-axis.
04

Use the Slope to Find Another Point

From the y-intercept (0, 0), use the slope to find another point. The slope of \(-2\) indicates a change of -2 in \( y \) for every +1 change in \( x \). From (0, 0), move 1 unit right (to x = 1) and 2 units down (to y = -2) to get the point (1, -2).
05

Draw the Line

With the two points, (0, 0) and (1, -2), plotted on the graph, draw a straight line through these points, extending the line across the grid.

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 is a standardized way to write the equation of a straight line, given by the formula \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) denotes the y-intercept, which is where the line crosses the y-axis. This form is especially helpful when graphing a line because it directly provides you with two important pieces of information about the line's geometry.
  • **Slope (\( m \))**: The slope tells you the angle or steepness of the line. A positive slope means the line ascends from left to right, while a negative slope means it descends. In the equation \( y = -2x \), the slope \( m \) is -2, indicating that for every 1 unit you move right along the x-axis, you move 2 units down.
  • **Y-Intercept (\( b \))**: The y-intercept is the point where the line crosses the y-axis (where \( x = 0 \)). In the equation \( y = -2x \), \( b \) is 0, hence the y-intercept is at the origin (0, 0).
By knowing the slope and y-intercept, you can quickly sketch the graph of a line without needing a table of values. Simply start at the y-intercept and use the slope to find another point on the line.
Linear Equations
Linear equations, like \( y = -2x \), describe straight lines on the coordinate plane. These equations are called "linear" because their graphs form straight lines. Linear equations are typically solved to find the value of \( y \) for given values of \( x \) and follow a consistent pattern of addition and multiplication.
  • **Form**: Most linear equations can be written in the form \( Ax + By = C \). However, the slope-intercept form \( y = mx + b \) is often easier for graphing.
  • **Properties**: Every linear equation has a constant rate of change. This means that the difference in \( y \) divided by the difference in \( x \) (i.e., the slope) is always the same. For \( y = -2x \), the slope is constant at -2.
  • **Applications**: Linear equations model a variety of real-world situations, such as predicting expenses or income based on a flat rate plus a variable amount that increases steadily over time.
Understanding how to manipulate and graph linear equations is foundational in algebra and helps in solving practical problems efficiently.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional number line created by the intersection of a horizontal line, the x-axis, and a vertical line, the y-axis. Every point on this plane is identified by a pair of numerical coordinates (x, y), which corresponds to its position relative to the x and y axes.
  • **Axes**: The x-axis runs horizontally, and the y-axis runs vertically. These axes divide the plane into four quadrants.
  • **Plotting Points**: A point is plotted on the coordinate plane by identifying its x-coordinate along the x-axis and its y-coordinate parallel to the y-axis. In our example, the point (0, 0) is where the line \( y = -2x \) crosses the y-axis, and the point (1, -2) helps in defining the slope.
  • **Graphing Lines**: After plotting two or more points of a linear equation, you can draw a line through these points to extend it across the plane. This line represents all solutions \( (x, y) \) that satisfy the equation.
The coordinate plane is instrumental in graphing functions and understanding spatial relationships between different algebraic quantities.

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

Solve the given problems. A person considering the purchase of a car has two options: (1) purchase it for \(\$ 35,000,\) or (2) lease it for three years for payments of \(\$ 450 /\) month plus \(\$ 2000\) down, with the option of buying the car at the end of the lease for \(\$ 18,000\). (a) For the lease option, express the amount \(P\) paid as a function of the numbers of months \(m\). (b) What is the difference in the total cost if the person keeps the car for the three years, and then decides to buy it?

Graph the indicated functions. A measure of the light beam that can be passed through an optic fiber is its numerical aperture \(N\). For a particular optic fiber, \(N\) is a function of the index of refraction \(n\) of the glass in the fiber, given by \(N=\sqrt{n^{2}-1.69} .\) Plot the graph for \(n \leq 2.00\).

Solve the indicated equations graphically. Assume all data are accurate to two significant digits unless greater accuracy is given. In finding the illumination at a point \(x\) feet from one of two light sources that are \(100 \mathrm{ft}\) apart, it is necessary to solve the equation \(9 x^{3}-2400 x^{2}+240,000 x-8,000,000=0 .\) Find \(x\)

Use the following table, which gives the fraction (as a decimal) of the total heating load of a certain system that will be supplied by a solar collector of area \(A\) (in \(\mathrm{m}^{2}\) ). Find the indicated values by linear interpolation. $$\begin{array}{l|c|c|c|c|c|c|c} f & 0.22 & 0.30 & 0.37 & 0.44 & 0.50 & 0.56 & 0.61 \\ \hline A\left(\mathrm{m}^{2}\right) & 20 & 30 & 40 & 50 & 60 & 70 & 80 \end{array}$$ For \(f=0.27,\) find \(A.\)

Solve the given problems. A demolition ball is used to tear down a building. Its distance \(s\) (in \(\mathrm{m}\) ) above the ground as a function of time \(t\) (in s) after it is dropped is \(s=17.5-4.9 t^{2} .\) Because \(s=f(t),\) find \(f(1.2)\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.