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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 the line is y = x - 3. To graph this line, create a table of values using the equation, plot the points on a coordinate plane, and draw a straight line passing through these points and the y-intercept at -3 with a slope of 1.

Step by step solution

01

1. Identify the slope and the y-intercept

The problem states that the slope (m) is equal to 1, and the y-intercept (b) is equal to -3.
02

2. Use the slope-intercept form to write the equation

Now that we have the slope and the y-intercept, substitute these values into the slope-intercept form \(y = mx + b\): \(y = 1x - 3\). Simplifying it, we have \(y = x - 3\).
03

3. Create a table of values to help graph the line

To help graph the line, create a table of values using the equation \(y = x - 3\). Here are some example values: x | y ------ -2 | -5 -1 | -4 0 | -3 1 | -2 2 | -1
04

4. Plot the points and draw the line

Using the table of values, plot the points on a coordinate plane. The points plotted will be: (-2, -5), (-1, -4), (0, -3), (1, -2), and (2, -1). Draw a straight line through these points, making sure it passes through the point (0, -3), which is our y-intercept. You should now have the following: 1. The equation of the line: \(y = x - 3\). 2. The graph of the line, passing through the y-intercept at -3 with a slope of 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
Being one of the most common ways to express the equation of a straight line, the slope-intercept form is incredibly useful.
This form is written as \( y = mx + b \).
Here, \( m \) represents the slope of the line and \( b \) is the y-intercept.
  • The slope \( m \) tells us how steep the line is.
  • The y-intercept \( b \) is where the line crosses the y-axis.
Using this form allows us to easily graph a line and understand its behavior based on these two components. If you know the slope and the y-intercept, you can write the equation of any straight line.
Graphing Lines
Graphing a line involves plotting points on the coordinate plane and then connecting them to form a straight path.
Once you have the equation in the slope-intercept form, graphing becomes straightforward.
  • Start by plotting the y-intercept \( b \) on the y-axis.
  • Use the slope \( m \) to find your next point.
The slope \( m \) is a ratio, usually given as a fraction \( \frac{rise}{run} \).
For example, a slope of 1 means you move up 1 unit and 1 unit to the right for each step.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can graph lines and other points.
It consists of two axes: the horizontal x-axis and the vertical y-axis.
These axes divide the plane into four quadrants, labeled as:
  • Quadrant I: where both x and y values are positive.
  • Quadrant II: where x is negative and y is positive.
  • Quadrant III: where both x and y are negative.
  • Quadrant IV: where x is positive and y is negative.
Graphing a line involves finding the points that satisfy the line's equation and plotting them within these quadrants.
Slope and Intercept
Understanding the slope and intercept is crucial to mastering linear equations.

Slope

The slope represents the line's angle or direction.
  • A positive slope means the line ascends from left to right.
  • A negative slope means it descends.
  • A slope of zero indicates a horizontal line.

Intercept

The y-intercept is where the line meets the y-axis.
  • This point is essential for setting off your graphing process.
  • It provides a starting point and is vital for calculating other points along the line.
Both slope and intercept together help in understanding and drawing a line effectively on the coordinate plane.

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Most popular questions from this chapter

The table gives the numbers of Octolasmis tridens and \(O\). lowei barnacles on each of 10 lobsters. \({ }^{8}\) Does it appear that the barnacles compete for space on the surface of a lobster? $$ \begin{array}{ccc} \text { Lobster } & & \\ \text { Field Number } & \text { 0. tridens } & \text { 0. lowei } \\ \hline \text { A061 } & 645 & 6 \\\ \text { A062 } & 320 & 23 \\ \text { A066 } & 401 & 40 \\ \text { A070 } & 364 & 9 \\ \text { A067 } & 327 & 24 \\ \text { A069 } & 73 & 5 \\ \text { A064 } & 20 & 86 \\ \text { A068 } & 221 & 0 \\ \text { A065 } & 3 & 109 \\\ \text { A063 } & 5 & 350 \end{array} $$ a. If they do compete, do you expect the number \(x\) of O. tridens and the number \(y\) of \(O .\) lowei barnacles to be positively or negatively correlated? Explain. b. If you want to test the theory that the two types of barnacles compete for space by conducting a test of the null hypothesis "the population correlation coefficient \(\rho\) equals 0 ," what is your alternative hypothesis? c. Conduct the test in part \(b\) and state your conclusions.

You are given these data: $$ \begin{array}{l|llllll} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 7 & 5 & 5 & 3 & 2 & 0 \end{array} $$ a. Plot the six points on graph paper. b. Calculate the sample coefficient of correlation \(r\) and interpret. c. By what percentage was the sum of squares of deviations reduced by using the least-squares predictor \(\hat{y}=a+b x\) rather than \(\bar{y}\) as a predictor of \(y ?\)

Refer to Exercise \(12.8 .\) The data, along with the \(M S\) Excel analysis of variance table are reproduced below: $$ \begin{array}{l|lllllll} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 9.7 & 6.5 & 6.4 & 4.1 & 2.1 & 1.0 \end{array} $$ a. Do the data provide sufficient evidence to indicate that \(y\) and \(x\) are linearly related? Use the information in the printout to answer this question at the \(5 \%\) level of significance. b. Calculate the coefficient of determination \(r^{2}\). What information does this value give about the usefulness of the linear model?

The table below, a subset of the data given in Exercise 3.33 , shows the gestation time in days and the average longevity in years for a variety of mammals in captivity. $$ \begin{array}{lrr} & \text { Gestation } & \text { Avg Longevity } \\\ \text { Animal } & \text { (days) } & \text { (yrs) } \\ \hline \text { Baboon } & 187 & 20 \\ \text { Bear (black) } & 219 & 18 \\ \text { Bison } & 285 & 15 \\ \text { Cat (domestic) } & 63 & 12 \\ \text { Elk } & 250 & 15 \\\ \text { Fox (red) } & 52 & 7 \\ \text { Goat (domestic) } & 151 & 8 \\\ \text { Gorilla } & 258 & 20 \\ \text { Horse } & 330 & 20 \\ \text { Monkey (rhesus) } & 166 & 15 \\ \text { Mouse (meadow) } & 21 & 3 \\\ \text { Pig (domestic) } & 112 & 10 \\ \text { Puma } & 90 & 12 \\ \text { Sheep (domestic) } & 154 & 12 \\ \text { Wolf (maned) } & 63 & 5 \end{array} $$ a. If you want to estimate the average longevity of an animal based on its gestation time, which variable is the response variable and which is the independent predictor variable? b. Assume that there is a linear relationship between gestation time and longevity. Calculate the leastsquares regression line describing longevity as a linear function of gestation time. c. Plot the data points and the regression line. Does it appear that the line fits the data? d. Use the appropriate statistical tests and measures to explain the usefulness of the regression model for predicting longevity.

What diagnostic plot can you use to determine whether the assumption of equal variance has been violated? What should the plot look like when the variances are equal for all values of \(x ?\)
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