/*! 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 41 Find the intercepts and graph th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 41

Find the intercepts and graph them. $$ y=2 x+4 $$

Short Answer

Expert verified
X-intercept: (-2, 0); Y-intercept: (0, 4).

Step by step solution

01

Identify the Y-Intercept

The y-intercept of a linear equation in the form \(y = mx + c\) is the value of \(y\) when \(x = 0\). In the equation \(y = 2x + 4\), the y-intercept \(c\) is 4. Therefore, the y-intercept is the point \((0, 4)\).
02

Find the X-Intercept

The x-intercept is the value of \(x\) when \(y = 0\). Set \(y = 0\) in the equation: \[0 = 2x + 4\] Solving for \(x\), we get:\[2x = -4\] \[x = -2\] Thus, the x-intercept is the point \((-2, 0)\).
03

Graph the Intercepts

To graph the intercepts, plot the points \((0, 4)\) and \((-2, 0)\) on a coordinate plane. Draw a straight line through these two points, which represents the graph of the equation \(y = 2x + 4\).

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.

Understanding Intercepts
Intercepts are specific points where a line crosses the axes on a coordinate plane. In linear equations like \(y = 2x + 4\), these intercept points help us understand the behavior of the line quickly. There are two types of intercepts:
  • Y-Intercept: This is the point where the line crosses the y-axis. To find it, we set \(x = 0\) and solve for \(y\). In our equation, substituting \(x = 0\) gives \(y = 4\). Hence, the y-intercept is \((0, 4)\).
  • X-Intercept: This is where the line meets the x-axis. Set \(y = 0\) and solve for \(x\). For the given equation, setting \(y = 0\) results in \(x = -2\), so the x-intercept is \((-2, 0)\).
Once you find these intercepts, you can plot them on the coordinate plane, providing insightful starting points for graphing the entire line.
Graphing the Line
Graphing a line from an equation involves plotting its intercepts on the coordinate plane and connecting them. This provides a visual representation of the equation's behavior. To graph the line for our equation \(y = 2x + 4\):
  • First, plot the y-intercept \((0, 4)\) on the plane. This means from the origin \((0,0)\), you move 4 units up along the y-axis.
  • Then, plot the x-intercept \((-2, 0)\). You place this point 2 units to the left of the origin on the x-axis.
  • With both points plotted, draw a straight line through them. This line is a visual representation of the equation, showing the slope and intercepts clearly.
Remember, accurate placements of these points are key to correctly graphing lines on the coordinate plane.
The Coordinate Plane
The coordinate plane is a crucial element in graphing linear equations. It's a two-dimensional plane where we can plot points, lines, and curves. Understanding the coordinate plane involves:
  • Axes: The plane consists of the x-axis (horizontal) and y-axis (vertical), which divide the plane into four quadrants. Each point is defined by a pair of numbers \((x, y)\).
  • Quadrants: They are numbered counterclockwise starting from the upper right. The line equation \(y = 2x + 4\) will extend through different quadrants based on the slope and intercepts.
  • Plotting Points: Each point \((x, y)\) on the plane has distinct horizontal and vertical placements from the origin (0,0), making it easy to plot lines by linking the intercepts.
Understanding the coordinate plane helps in visualizing mathematical concepts and solving equations by graphically locating solutions.

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

Determine the slope, given two points. (0,-5) and (-6,3)

Find the equation of the line given two points on the line. (-6,12) and (3,-3)

The monthly revenue generated by selling subscriptions to a tutoring website is given by the function \(R(x)=29 x\), where \(x\) represents the number of subscription sales per month. How many subscriptions were sold if the revenues for the month totaled \(\$ 1,508 ?\)

Find the equation of a line, given the slope and a point on the line. $$ m=-3 / 4 ;(2,-3) $$

Find the equation of the line given two points on the line. \((5 / 2,-2)\) and \((-5,5 / 2)\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

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.