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Chapter 12: Problem 4

Give the equation and graph for a line with y-intercept equal to -3 and slope equal to 1 .

Short Answer

Expert verified
Answer: The equation of a line with a slope of 1 and a y-intercept of -3 is y = x - 3. To graph it, start at the y-intercept point (0, -3) and plot it. Since the slope is 1, move 1 unit to the right and 1 unit up from the y-intercept point and plot another point. Connect the two points with a straight line, extending infinitely in both directions.

Step by step solution

01

Write the slope-intercept form of a linear equation

The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept. In this problem, we are given m = 1 and b = -3.
02

Plug the slope and y-intercept into the equation

We will now substitute the given slope and y-intercept into the slope-intercept form of the linear equation: y = mx + b. In this case, y = 1x - 3 or simply y = x - 3.
03

Describe the graph of the line

To graph the line y = x - 3, we will use the given slope and y-intercept. The y-intercept (b = -3) tells us that the line passes through the point (0, -3) on the y-axis. The slope (m = 1) indicates that for every step we take to the right on the x-axis, we are going to step up by the same amount on the y-axis.
04

Graph the line

Start by plotting the y-intercept point (0, -3) on the graph. Since the slope is 1, for every unit you move to the right on the x-axis, move up 1 unit on the y-axis. Plot another point by moving 1 unit to the right and 1 unit up from the y-intercept point. Now connect the two points with a straight line, extending infinitely in both directions. This is the graph of the line y = x - 3.

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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 one of the simplest ways to express the equation of a line. It is expressed as \( y = mx + b \), where \( m \) represents the slope, and \( b \) signifies the y-intercept. Understanding this form is crucial when you want to quickly graph a line or comprehend how changes in the equation affect its graph.

- **Slope \( (m) \)**: This is a measure of the line's steepness. It tells us how much the y-value changes for every change in the x-value. For example, a slope of 1 indicates that as x increases by 1 unit, y increases by 1 unit as well.- **Y-intercept \( (b) \)**: This is the point where the line crosses the y-axis. It tells us the value of \( y \) when \( x = 0 \).

This form is particularly convenient because with the values of \( m \) and \( b \), you can instantly determine how to shape and plot the line on a graph.
Graphing Lines
Graphing lines using the slope-intercept form is simple and requires minimal calculations. With the formula \( y = mx + b \), you already have the essential information to start plotting.

Here's how to graph a line step-by-step:
  • Begin with the y-intercept \( (b) \). This is the starting point of the line on the graph, located on the y-axis. For instance, if \( b = -3 \), like in the problem, start at the coordinate \( (0, -3) \).
  • Use the slope \( (m) \) to find other points on the line. The slope indicates the rate of change along the line; that is, how much y changes for a given change in x. If \( m = 1 \), for every 1 unit you move to the right (along the x-axis), you also move 1 unit up (along the y-axis).
  • Connect the points with a straight line. The line should extend infinitely in both directions unless specified otherwise, representing all possible solutions (x, y) to the equation.
Following these simple steps will help you visualize the equation as a graph on a coordinate plane.
Y-Intercept
The y-intercept is a fundamental concept in understanding linear equations and their graphs. It's the point where the line crosses the y-axis, giving you immediate visual information about the line’s position on a graph.

In an equation of the form \( y = mx + b \), the y-intercept is \( b \). This tells us that when \( x = 0 \), the value of \( y \) is exactly \( b \). So, in our problem with the line equation \( y = x - 3 \), the y-intercept is -3, placing it at the point \( (0, -3) \) on the graph.

Understanding the y-intercept helps in:
  • Identifying where to place the starting point of the line on the graph.
  • Recognizing the initial value of y before any changes in x occur.
  • Quickly sketching the line with precision by providing a concrete point to plot first.
The y-intercept offers a clear anchor point, making graphing lines quicker and easier.

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Most popular questions from this chapter

In Exercise 12.15 (data set EX1215), we measured the armspan and height of eight people with the following results: $$ \begin{array}{l|clll} \text { Person } & 1 & 2 & 3 & 4 \\ \hline \begin{array}{l} \text { Armspan (inches) } \\ \text { Height (inches) } \end{array} & 68 & 62.25 & 65 & 69.5 \\ & 69 & 62 & 65 & 70 \\ \text { Person } & 5 & 6 & 7 & 8 \\ \hline \text { Armspan (inches) } & 68 & 69 & 62 & 60.25 \\ \text { Height (inches) } & 67 & 67 & 63 & 62 \end{array} $$ a. Does the data provide sufficient evidence to indicate that there is a linear relationship between armspan and height? Test at the \(5 \%\) level of significance. b. Construct a \(95 \%\) confidence interval for the slope of the line of means, \(\beta\). c. If Leonardo da Vinci is correct, and a person's armspan is roughly the same as the person's height, the slope of the regression line is approximately equal to \(1 .\) Is this supposition confirmed by the confidence interval constructed in part b? Explain.

You can refresh your memory about regression lines and the correlation coefficient by doing the MyApplet Exercises at the end of Chapter \(3 .\) a. Graph the line corresponding to the equation \(y=0.5 x+3\) by graphing the points corresponding to \(x=0,1,\) and 2 . Give the \(y\) -intercept and slope for the line. b. Check your graph using the How a Line Works applet.

The following data (Exercises 12.16 and 12.24 ) were obtained in an experiment relating the dependent variable, \(y\) (texture of strawberries), with \(x\) (coded storage temperature). $$ \begin{array}{l|rrrrr} x & -2 & -2 & 0 & 2 & 2 \\ \hline y & 4.0 & 3.5 & 2.0 & 0.5 & 0.0 \end{array} $$ a. Estimate the expected strawberry texture for a coded storage temperature of \(x=-1 .\) Use a \(99 \%\) confidence interval. b. Predict the particular value of \(y\) when \(x=1\) with a \(99 \%\) prediction interval. c. At what value of \(x\) will the width of the prediction interval for a particular value of \(y\) be a minimum, assuming \(n\) remains fixed?

You are given five points with these coordinates: $$ \begin{array}{c|rrrrrrr} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 1 & 1 & 3 & 5 & 5 \end{array} $$ a. Use the data entry method on your scientific or graphing calculator to enter the \(n=5\) observations. Find the sums of squares and cross-products, \(S_{x x} S_{x y},\) and \(S_{y y}\) b. Find the least-squares line for the data. c. Plot the five points and graph the line in part b. Does the line appear to provide a good fit to the data points? d. Construct the ANOVA table for the linear regression.

A marketing research experiment was conducted to study the relationship between the length of time necessary for a buyer to reach a decision and the number of alternative package designs of a product presented. Brand names were eliminated from the packages to reduce the effects of brand preferences. The buyers made their selections using the manufacturer's product descriptions on the packages as the only buying guide. The length of time necessary to reach a decision was recorded for 15 participants in the marketing research study. $$ \begin{array}{l|l|l|l} \begin{array}{l} \text { Length of Decision } \\ \text { Time, } y(\mathrm{sec}) \end{array} & 5,8,8,7,9 & 7,9,8,9,10 & 10,11,10,12,9 \\ \hline \text { Number of } & & & \\ \text { Alternatives, } x & 2 & 3 & 4 \end{array} $$ a. Find the least-squares line appropriate for these data. b. Plot the points and graph the line as a check on your calculations. c. Calculate \(s^{2}\). d. Do the data present sufficient evidence to indicate that the length of decision time is linearly related to the number of alternative package designs? (Test at the \(\alpha=.05\) level of significance.) e. Find the approximate \(p\) -value for the test and interpret its value. f. If they are available, examine the diagnostic plots to check the validity of the regression assumptions. g. Estimate the average length of time necessary to reach a decision when three alternatives are presented, using a \(95 \%\) confidence interval.
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