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Chapter 12: Problem 1

Graph the line corresponding to the equation \(y=2 x+1\) by graphing the points corresponding to \(x=0,1,\) and 2 . Give the \(y\) -intercept and slope for the line.

Short Answer

Expert verified
Answer: The coordinates of the points are (0,1), (1,3), and (2,5). The slope of the line is 2 and the y-intercept is 1.

Step by step solution

01

Find the corresponding y-values for each x-value

Given the equation \(y=2x+1\), we need to find the \(y\)-values for each of the given \(x\)-values: 1. For \(x=0\): \(y=2(0)+1 = 1\) 2. For \(x=1\): \(y=2(1)+1 = 3\) 3. For \(x=2\): \(y=2(2)+1 = 5\) The points corresponding to these \(x\) and \(y\) values are: 1. For \(x=0\): \((0,1)\) 2. For \(x=1\): \((1,3)\) 3. For \(x=2\): \((2,5)\)
02

Plot the points on the graph

Plot the points \((0,1)\), \((1,3)\) and \((2,5)\) on the coordinate plane. Join these points to create a straight line.
03

Determine the slope and y-intercept

The slope of the line is the change in \(y\) divided by the change in \(x\). We can use the two points \((1,3)\) and \((2,5)\) to calculate the slope: Slope, \(m = \frac{(5-3)}{(2-1)} = \frac{2}{1} = 2\) The \(y\)-intercept is the point where the line crosses the \(y\)-axis. Since the coordinates of the y-intercept are \((0,1)\), the \(y\)-intercept is 1. The line represents the equation \(y=2x+1\), with a slope of 2 and a \(y\)-intercept of 1.

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Key Concepts

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

Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where we graph equations. It consists of two perpendicular lines.
  • The horizontal line is the x-axis.
  • The vertical line is the y-axis.
These axes intersect at the origin, (0, 0). On the coordinate plane, we plot points using pairs of numbers that indicate positions. Each point is expressed as (x, y), where 'x' is the coordinate on the x-axis, and 'y' is the coordinate on the y-axis.
The coordinate plane is essential for visualizing mathematical equations like linear equations, providing a graphical representation that makes data easier to understand.
Y-Intercept
The y-intercept is a crucial point on the graph of an equation. It's where the line crosses the y-axis. For a linear equation, the y-intercept can be found by setting x to 0 in the equation.
In the equation \(y = 2x + 1\), when \(x = 0\), the value of \(y = 1\). Therefore, the y-intercept is at the point (0,1). This means the line crosses the y-axis at y = 1.
Identifying the y-intercept is helpful in graphing as it gives a starting point for plotting the line. It also provides insight into how the graph behaves in relation to the axes.
Slope
The slope is a measure of how steep a line is on the graph. It tells us how much the y-value changes as the x-value changes. The slope is represented by the letter \(m\) in the equation of a line.
For an equation like \(y = 2x + 1\), the number before x, which is 2, represents the slope. This means for every increase of 1 in x, y increases by 2. To calculate the slope manually, you can use two points on the line. Take the change in y-values divided by the change in x-values:
\(m = \frac{\Delta y}{\Delta x} = \frac{(5-3)}{(2-1)} = 2\). A positive slope indicates an upward trend as you move from left to right, while a negative slope indicates a downward trend.
Graphing
Graphing is the action of plotting points on the coordinate plane to form shapes, lines, or curves. For linear equations, graphing typically results in straight lines.
Here's how to graph a line like \(y = 2x + 1\):
  • Start by plotting the y-intercept, which is (0,1).
  • Calculate the next points using the slope. For example, starting from (0,1), rise two units and run one unit to reach point (1,3).
  • Repeat to find another point, like (2,5).
  • Draw a line through your plotted points.
This visual representation helps in understanding the relationships in linear equations and provides a clear picture of how variables interact.
Points on a Graph
Points on a graph serve as visual markers that depict the relationship between variables within an equation. Each point is represented by coordinates \((x, y)\), providing a location on the coordinate plane.
For \(y = 2x + 1\), calculate the y-values for given x-values to find these points.
  • When \(x = 0\), \(y = 1\) gives point (0,1).
  • When \(x = 1\), \(y = 3\) gives point (1,3).
  • When \(x = 2\), \(y = 5\) gives point (2,5).
Connecting these points forms the line of the equation. The points not only define the line but also provide essential data for understanding how the variables x and y relate.

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Most popular questions from this chapter

Why is it that one person may tend to gain weight, even if he eats no more and exercises no less than a slim friend? Recent studies suggest that the factors that control metabolism may depend on your genetic makeup. One study involved 11 pairs of identical twins fed about 1000 calories per day more than needed to maintain initial weight. Activities were kept constant, and exercise was minimal. At the end of 100 days, the changes in body weight (in kilograms) were recorded for the 22 twins. \({ }^{16}\) Is there a significant positive correlation between the changes in body weight for the twins? Can you conclude that this similarity is caused by genetic similarities? Explain. $$ \begin{array}{rrr} \text { Pair } & \text { Twin A } & \text { Twin B } \\\ \hline 1 & 4.2 & 7.3 \\ 2 & 5.5 & 6.5 \\ 3 & 7.1 & 5.7 \\ 4 & 7.0 & 7.2 \\\ 5 & 7.8 & 7.9 \\ 6 & 8.2 & 6.4 \\ 7 & 8.2 & 6.5 \\ 8 & 9.1 & 8.2 \\ 9 & 11.5 & 6.0 \\ 10 & 11.2 & 13.7 \\ 11 & 13.0 & 11.0 \end{array} $$

A marketing research \end{tabular}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 and 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. $$ \begin{array}{c|c|c|c} \text { Length of Decision } & & & \\ \text { Time, } y \text { (sec) } & 5,8,8,7,9 & 7,9,8,9,10 & 10,11,10,12,9 \\ \hline \text { Number of } & & & & \\ \text { Alternatives, } x & 2 & & 3 & & 4 \end{array} $$ 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.

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

You are given these data: $$ \begin{array}{l|rrrrrrr} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 2 & 2 & 3 & 4 & 4 \end{array} $$ a. Plot the data points. Based on your graph, what will be the sign of the sample correlation coefficient? b. Calculate \(r\) and \(r^{2}\) and interpret their values.

Refer to Exercise \(12.45 .\) a. Estimate the average number of passing yards for games in which Brees throws 20 completed passes using a \(95 \%\) confidence interval. b. Predict the actual number of passing yards for games in which Brees throws 20 completed passes using a \(95 \%\) confidence interval. c. Would it be advisable to use the least-squares line from Exercise 12.45 to predict Brees' total number of passing yards for a game in which he threw only five completed passes? Explain.
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