/*! 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 26 Sketch a graph of the equation. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 26

Sketch a graph of the equation. $$ y-1=3(x+4) $$

Short Answer

Expert verified
The equation \(y-1 = 3(x+4)\) represents a straight line with a slope of 3 and y-intercept at \(y =13\). The graph starts at point (0,13) and rises by 3 units and runs 1 unit to the right to find other points.

Step by step solution

01

Identify the slope and y-intercept

The equation given is \(y-1 = 3(x+4)\). If it's rearranged to the form \(y = mx + c\), it becomes \(y = 3x + 13\). The coefficient of x is the slope, which is 3 in this case, and the constant term is the y-intercept, which is 13.
02

Plot the y-intercept

The y-intercept is a point on the graph where the line crosses the y-axis. Since the y-intercept is 13, plot a point at \(y = 13\) on the y-axis.
03

Use the slope to find other points

The slope of the line, 3, represents the ratio of vertical change (rise) to horizontal change (run), as you move along the line. Here, rise is 3 and run is 1 (since 3 can be written as 3/1). Starting from the y-intercept, count 3 units up (since rise is positive) and 1 unit to the right (since run is positive). This is a new point that the line passes through.
04

Draw the line

Draw a straight line through the plotted points. This line represents the graph of the given equation \(y-1=3(x+4)\).

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.

Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is crucial for graphing and analyzing lines efficiently. This form is expressed as
\[ y = mx + b \]
where \(m\) represents the slope and \(b\) indicates the y-intercept. The slope is a number that describes both the direction and the steepness of the line. Meanwhile, the y-intercept is the point where the line crosses the y-axis.

Transforming to Slope-Intercept Form

To sketch the graph of a linear equation, first, ensure it is in slope-intercept form. Take the equation
\( y - 1 = 3(x + 4) \)
and rearrange it to get
\( y = 3x + 13 \).
In this format, you can easily identify the slope, \(m = 3\), and the y-intercept, \(b = 13\), guiding you to graph the line quickly. It's like having a map where 'm' tells you the direction to go (up if positive, down if negative) and 'b' shows you where to start on the y-axis.
Y-Intercept
The y-intercept of a line is one of the first pieces of information you should find when graphing. It is the point where the line crosses the y-axis and is exclusively defined by its y-coordinate, as the x-coordinate is always zero at this point. In the equation \( y = 3x + 13 \), the y-intercept is \( (0, 13) \).

Plotting the Y-Intercept

To plot the y-intercept on a graph, locate the number represented by 'b' along the y-axis, and mark that point. This marks the starting point for your line. Even without knowing the slope at first, the y-intercept gives you the exact location where your line will intersect the y-axis, anchoring the line in place. For the given equation, start at 13 on the y-axis, ensuring your graph begins accurately.
Slope of a Line
The slope is a measure of the line's tilt, often referred to as 'rise over run.' It tells us how much the line rises vertically (the rise) for a given horizontal movement (the run). The slope is represented by the variable \(m\) in slope-intercept form and can be either positive, negative, zero, or undefined, depending on the direction of the line.

Calculating Slope

Given the slope-intercept form \( y = mx + b \), our slope \(m\) is the coefficient in front of \(x\). For \( y = 3x + 13 \), the slope \(m\) is 3, which means for every one unit you move to the right (the run), you move three units up (the rise).

Using Slope to Graph

When graphing, start at the y-intercept and use the slope to find other points. From \( (0, 13) \) move 3 units up and 1 unit to the right to find a new point. Then draw a straight line through these points to represent the graph of the equation. In this way, the slope directly determines the angle and direction of the line on a graph.

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

True or False? In Exercises \(50-53\), determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Use the \(\varepsilon-\delta\) definition of infinite limits to prove that \(\lim _{x \rightarrow 3^{+}} \frac{1}{x-3}=\infty\)

Boyle's Law For a quantity of gas at a constant temperature, the pressure \(P\) is inversely proportional to the volume \(V\). Find the limit of \(P\) as \(V \rightarrow 0^{+}\).

$$ \lim _{x \rightarrow 2} f(x)=3, \text { where } f(x)=\left\\{\begin{array}{ll} 3, & x \leq 2 \\ 0, & x>2 \end{array}\right. $$

Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power on \(x\) in the denominator is greater than \(3 ?\) $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 0.5 & 0.2 & 0.1 & 0.01 & 0.001 & 0.0001 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array} $$ (a) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x}\) (b) \(\lim _{x \rightarrow 0^{-}} \frac{x-\sin x}{x^{2}}\) (c) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{3}}\) (d) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}\)

Use the position function \(s(t)=-4.9 t^{2}+150\), which gives the height (in meters) of an object that has fallen from a height of 150 meters. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). Find the velocity of the object when \(t=3\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.