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Chapter 12: Problem 5

Give the equation and graph for a line with y-intercept and slope given in Exercises. $$y \text { -intercept }=-3 ; \text { slope }=1$$

Short Answer

Expert verified
Answer: The equation of the line with a y-intercept of -3 and a slope of 1 is y = x - 3. Two points on this line are (0, -3) and (1, -2).

Step by step solution

01

1. Understand the concept of y-intercept and slope

The y-intercept is the point where the line crosses the y-axis. In other words, it is the value of y when the value of x is 0. The slope, on the other hand, is a measure of how steep the line is. In general, the equation of a straight line is given by $$ y = mx + b $$, where 'm' is the slope and 'b' is the y-intercept.
02

2. Formulate the equation

Given that the y-intercept (b) is -3 and the slope (m) is 1, we can now plug in these values into our general equation for a straight line: $$y = mx + b$$ Replacing 'm' and 'b' with their given values, we have: $$ y = 1x - 3 $$, or simplifying further: $$ y = x - 3 $$
03

3. Plot the graph of the given line

To plot the graph of the given line ($$y = x - 3$$), follow these steps: a. Identify the y-intercept (b = -3). This is where the line crosses the y-axis, which is the point (0, -3). b. Choose another point on the x-axis (other than x = 0) to determine the corresponding value of y given the slope. In our case, since the slope is 1, that means that for every unit increase in x, the y value also increases by 1. Hence, if we choose x = 1, the y value is $$1 * 1 -3 = -2$$. So, the second point is (1, -2). c. Now we have two points, (0, -3) and (1, -2). Plot these points on a cartesian plane and draw a straight line passing through them. The graph of the given line ($$y = x - 3$$) is now visualized. To summarize, the equation for the line with y-intercept -3 and slope 1 is $$ y = x - 3 $$, and its graph is a straight line passing through the points (0, -3) and (1, -2).

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Key Concepts

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

Y-Intercept
The y-intercept is a fundamental element when it comes to understanding linear equations. It specifically refers to the point where the line crosses the y-axis of a graph. In simpler terms, it's the 'y' value when 'x' is zero. For example, if we have an equation with a y-intercept of -3, this indicates that the point (0, -3) is where the line will intersect the y-axis.

When plotting a graph or interpreting an equation, the y-intercept serves as a starting point. It tells us where to place our first point on the graph. In real-world scenarios, this could represent a base level from which changes begin. For instance, the starting balance in a bank account before any deposits or withdrawals have occurred. Understanding this concept allows students to draw accurate graphs and to better comprehend how linear equations model various phenomena.
Slope
Slope is the term that describes the steepness or incline of a line. Mathematically, it is often denoted as 'm' and represents the ratio of the rise (vertical change) over the run (horizontal change) between any two points on the line. If the slope is positive, the line ascends from left to right; if it's negative, the line descends. A slope of 1, as given in our example, means that for every step you take to the right on the x-axis, you also move up one step on the y-axis.

The concept of slope is vital, as it shows the rate at which one variable changes in relation to another. Applied to real-life situations, this could exemplify gradients, such as the slope of a hill, or rates of change, such as speed where distance changes over time. Knowing the slope allows students not only to graph a line but also to understand the behavior of the line - whether it's increasing or decreasing and at what rate.
Graph of a Line
The graph of a line represents a visual interpretation of all the solutions to a linear equation. It is a straight path that extends infinitely in both directions on a cartesian plane. The two key pieces of information needed to graph a line are the y-intercept and the slope. Once we plot the y-intercept on the graph, we can use the slope to find another point on the line and draw the entirety of the line through these points.

For the graphing process, consistency is important. Maintaining consistent intervals on the axes ensures that the slope is correctly represented. Graphs are not just about plotting points; they are powerful tools that help visualize relationships between variables. They can provide a clearer understanding of concepts like speed, growth rates, and trends in economics or science by depicting how one quantity varies with another.
Equation of a Straight Line
The equation of a straight line, typically expressed in the form \( y = mx + b \), encapsulates the entire relationship of the variables 'x' and 'y' for a line on a graph. In this equation, 'm' represents the slope, and 'b' is the y-intercept. This format is known as the slope-intercept form of a linear equation and is one of the simplest ways to write the equation of a line.

Knowing how to create and manipulate this equation is a key skill in solving many mathematical problems. It allows for quick determination of the y-value for any given x-value and thus can be used in various applied maths, from economics to engineering. Understanding the equation of a straight line empowers students to predict, analyze, and convey information regarding linear relationships in a precise and efficient manner.

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Most popular questions from this chapter

An informal experiment was conducted at McNair Academic High School in Jersey City, New Jersey. Twenty freshman algebra students were given a survey at the beginning of the semester, measuring his or her skill level. They were then allowed to use laptop computers both at school and at home. At the end of the semester, their scores on the same survey were recorded \((x)\) along with their score on the final examination \((y) .^{9}\) The data and the MINITAB printout are shown here. $$ \begin{array}{ccc} \hline \text { Student } & \text { End-of-Semester Survey } & \text { Final Exam } \\ \hline 1 & 100 & 98 \\ 2 & 96 & 97 \\ 3 & 88 & 88 \\ 4 & 100 & 100 \\ 5 & 100 & 100 \\ 6 & 96 & 78 \\ 7 & 80 & 68 \\ 8 & 68 & 47 \\ 9 & 92 & 90 \\ 10 & 96 & 94 \\ 11 & 88 & 84 \\ 12 & 92 & 93 \\ 13 & 68 & 57 \\ 14 & 84 & 84 \\ 15 & 84 & 81 \\ 16 & 88 & 83 \\ 17 & 72 & 84 \\ 18 & 88 & 93 \\ 19 & 72 & 57 \\ 20 & 88 & 83 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { Analysis of Variance }\\\ &\begin{array}{lrrrrr} \text { Source } & \text { DF } & \text { Adj SS } & \text { AdjMS } & \text { F-Value } & \text { P-Value } \\ \hline \text { Regression } & 1 & 3254.03 & 3254.03 & 56.05 & 0.000 \\ \text { Error } & 18 & 1044.92 & 58.05 & & \\ \text { Total } & 19 & 4298.95 & & & \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Model Summary }\\\ &\begin{array}{ccc} \mathrm{S} & \mathrm{R}-\mathrm{sq} & \mathrm{R}-\mathrm{sq}(\mathrm{adj}) \\ \hline 7.61912 & 75.69 \% & 74.34 \% \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { Coefficients }\\\ &\begin{array}{lrrrr} \text { Term } & \text { Coef } & \text { SE Coef } & \text { T-Value } & \text { P-Value } \\ \hline \text { Constant } & -26.8 & 14.8 & -1.82 & 0.086 \\ \mathrm{x} & 1.262 & 0.169 & 7.49 & 0.000 \end{array} \end{aligned} $$ Regression Equation $$ y=-26.8+1.262 x $$ a. Construct a scatterplot for the data. Does the assumption of linearity appear to be reasonable? b. What is the equation of the regression line used for predicting final exam score as a function of the endof-semester survey score? c. Do the data present sufficient evidence to indicate that final exam score is linearly related to the end-ofsemester survey score? Use \(\alpha=.01\). d. Find a \(99 \%\) confidence interval for the slope of the regression line. e. Use the MINITAB printout to find the value of the coefficient of determination, \(r^{2}\). Show that \(r^{2}=\) SSR/Total SS. f. What percentage reduction in the total variation is achieved by using the linear regression model?

Sleep Deprivation A study was conducted to determine the effects of sleep deprivation on DS1206 people's ability to solve problems. Ten subjects participated in the study, two at each of five sleep deprivation levels \(-8,12,16,20,\) and 24 hours. After his or her specified sleep deprivation period, each subject answered a set of simple addition problems, and the number of errors was recorded. These results were obtained: d. What is the least-squares prediction equation? e. Use the prediction equation to predict the number of errors for a person who has not slept for 10 hours. $$\begin{aligned}&\begin{array}{l|l|l|l}\text { Number of Errors, } y & 8,6 & 6,10 & 8,14 \\\\\hline \begin{array}{l}\text { Number of Hours without } \\\\\text { Sleep, } x\end{array} & 8 & 12 & 16\end{array}\\\&\begin{array}{l|l|l}\text { Number of Errors, } y & 14,12 & 16,12 \\\\\hline \begin{array}{l}\text { Number of Hours without } \\\\\text { Sleep. } x\end{array} & 20 & 24 \\\\\hline\end{array}\end{aligned}$$ a. How many pairs of observations are in the experiment? b. What are the total number of degrees of freedom? c. Complete the MS Exce/ printout.

What are the assumptions made about the random error \(\epsilon\) in the probabilistic model \(y=\alpha+\beta x+\epsilon ?\)

Use the data given in Exercises 5-6 (Exercises 17-18, Section 12.1). Do the data provide sufficient evidence to indicate that \(y\) and \(x\) are linearly related? Test using the \(t\) statistic at the 1\% level of significance. Construct a \(99 \%\) confidence interval for the slope of the line. What does the phrase "99\% confident" mean? $$ \begin{array}{r|rrrrr} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 1 & 1 & 3 & 5 & 5 \end{array} $$

Basics Use the information given in Exercises \(1-2\) (Exercises 1 and 3 , Section 12.2 ) to construct an ANOVA table for a simple linear regression analysis. Use the ANOVA \(F\) -test to test \(H_{0}: \beta=0\) with \(\alpha=.05 .\) Then calculate \(b\) and its standard error: Use a t statistic to test \(H_{0}: \beta=0\) with \(\alpha=.05 .\) Verify that within rounding \(t^{2}=F\). $$ n=8 \text { pairs }(x, y), S_{x x}=4, S_{y y}=20, S_{x y}=8 $$
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