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Chapter 4: Problem 10

Each equation below represents a family of lines. Describe what the lines in eac form have in common. a. \(y=a+3 x\) (a) b. \(y=5+b x\) c. \(y=a\) d. \(x=c\)

Short Answer

Expert verified
Lines have constant slopes or intercepts influenced by parameters, with vertical and horizontal uniqueness.

Step by step solution

01

Analyze Equation a

The equation provided is \(y = a + 3x\). This is in the form of a linear equation, \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, \(m = 3\), which means all lines in this family have a slope of 3. The parameter \(a\) represents the y-intercept, which means each line will have a different y-intercept but the same slope.
02

Analyze Equation b

The equation given is \(y = 5 + bx\). Again, this is in a linear formula form, \(y = mx + c\). The y-intercept here is 5. The parameter \(b\) represents the slope, so lines in this family have a constant y-intercept of 5 but different slopes depending on \(b\).
03

Analyze Equation c

The equation \(y = a\) represents horizontal lines. All such lines have a slope of 0. The parameter \(a\) determines the vertical position of these lines, making each line parallel to the x-axis at different y-values.
04

Analyze Equation d

The equation \(x = c\) represents vertical lines. These lines have an undefined slope and are parallel to the y-axis. The constant \(c\) indicates the x-intercept, hence each line in this family will be located at different positions along the x-axis.

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Key Concepts

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

Slope
The slope is a fundamental aspect of linear equations that defines how a line tilts in the coordinate plane. It essentially measures the steepness or the incline of the line. Perfectly expressed as the ratio of the change in the output values to the change in the input values, this concept is mathematical but simple.
  • In mathematical terms, slope \(m\) is expressed as \(m = \frac{\Delta y}{\Delta x}\), where \(\Delta y\) is the change in \(y\)-value and \(\Delta x\) is the change in \(x\)-value.
  • For a line with a positive slope, as in equation \(y = a + 3x\), the line rises from left to right, indicating that as \(x\) increases, \(y\) increases.
  • Conversely, a negative slope would mean the line falls as it moves from left to right.
  • Zero slope represents a completely horizontal line, while undefined slope corresponds to a vertical line.
Understanding the slope is crucial, whether you're dealing with simple linear equations or analyzing more complex linear models. It helps you determine trends, predict outcomes, and understand relationships between variables.
Y-Intercept
The y-intercept of a line is where the line crosses the y-axis in a coordinate plane. Denoted in the equation format as \(c\) in \(y = mx + c\), it's the point where the value of \(x\) is zero.
  • For instance, in the equation \(y = 5 + bx\), the y-intercept is \(5\). This means regardless of the slope, every line in this family will intersect the y-axis at the point \((0, 5)\).
  • If the y-intercept is positive, the line will cross above the origin on the y-axis. If it's negative, it will cross below.
  • The y-intercept provides a starting point for graphing a line, allowing you to plot the initial point before applying the slope to find other points on the line.
Recognizing the y-intercept helps in quickly sketching graphs and analyzing lines without needing additional calculations or points.
Horizontal Lines
Horizontal lines are unique in that they have a slope of zero. This means no matter the changes in \(x\), the \(y\) value remains consistent, creating a flat line parallel to the x-axis.
  • For example, in the equation \(y = a\), all lines are horizontal.
  • This implies that for any line, as determined by the parameter \(a\), the \(y\)-value will remain constant.
  • A real-world example might involve visualizing a line representing a fixed price over time, with zero economic inflation or deflation.
These lines demonstrate situations where variability does not influence the outcome, and are simple to visualize once you understand the concept of a zero slope.
Vertical Lines
Vertical lines show a distinctive behavior on the graph; they are parallel to the y-axis and have an undefined slope. This means that for any point on the line, the \(x\) value remains consistent, while the \(y\) values can change freely.
  • Represented by the equation \(x = c\), vertical lines have a consistent x-value, \(c\).
  • They do not touch or cross the y-axis, since they extend infinitely up and down parallel to the y-axis.
  • Vertical lines can illustrate situations where a variable, like a day of the week, does not affect the spatial position on a graph, such as tracking consistent temperature readings.
Due to their undefined slope, vertical lines present unique challenges in mathematical contexts, particularly in slope-intercept form and graphing.

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Most popular questions from this chapter

Imagine that a classmate has been out of school for the past few days with the flu. Write him or her an e-mail describing how to convert an equation such as \(y=4+2(x-3)\) from point-slope form to slope-intercept form. Be sure to include examples and explanations. End your note by telling your classmate how to find out if the two equations are equivalent.

Scoop has a rolling ice cream cart. He recorded his daily sales for the last seven days and the mean daytime temperature for each day. $$ \begin{aligned} &\text { Ice Cream Sales }\\\ &\begin{array}{|l|r|r|r|r|r|r|r|} \hline \text { Day } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Temperature ('F) } & 83 & 79 & 75 & 70 & 71 & 67 & 62 \\ \hline \text { Sales (cones) } & 66 & 47 & 51 & 23 & 33 & 30 & 21 \\ \hline \end{array} \end{aligned} $$ a. Find the equation of the line that passes through the points \((79,47)\) and \((67,30)\). (Use the second point as the point in the point-slope form.) (a) b. Graph the data and your line from 8 a on your calculator. Sketch the result. You should have noticed in \(8 \mathrm{~b}\) that the line does not fit the data well. In fact, no two points from this data set make a good model. In \(8 \mathrm{c}-\) e you'll adjust the values of \(y_{1}\) c. Copy the table shown, and begin by changing the value of \(y_{1}\). Write two new equations, one with a larger value for \(y_{1}\) and one Value with a smaller value for \(y_{1}\). Graph each equation, and describe how the graphs compare to your original equation. (a) d. Now write two new equations that have the same values of \(x_{1}\) and \(y_{1}\) as the original, but larger and smaller values of \(b\). Graph each equation, and describe how the graphs compare to your original equation. e. Continue to adjust your values for \(y_{1}\) and \(b\) until you find a line that fits the data well. Record your final equation, Graph your equation with the data and sketch the result.

At 2:00 P.M., elevator B passes the 94th floor of the same building going down. The table shows the floors and the times in seconds after 2:00. \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline Floor \(x\) & 94 & 92 & 90 & 88 & 86 & 84 & 80 \\ \hline Time after 2:00 (s) \(y\) & 0 & \(1.3\) & \(2.5\) & \(3.8\) & 5 & \(6.3\) & \(8.6\) \\\ \hline \end{tabular} a. What is the line of fit based on Q-points for the data? b. Give a real-world meaning of the slope. c. About what time will this elevator pass the 10 th floor if it makes no stops? d. Where will this elevator be at \(2: 00: 34\) if it makes no stops?

A line passes through the points \((-2,-1)\) and \((5,13)\). a. Find the slope of this line. (a) b. Write an equation in point-slope form using the slope you found in \(3 a\) and the point \((-2,-1)\). (a) c. Write an equation in point-slope form using the slope you found in \(3 \mathrm{a}\) and the point \((5,13)\). d. Verify that the equations in \(3 \mathrm{~b}\) and \(\mathrm{c}\) represent the same line. Enter the equations into \(\mathrm{Y}_{1}\) and \(\mathrm{Y}_{2}\) on your calculator, and compare their graphs and tables.

A 4 oz bottle of mustard costs \(\$ 0.88\), a \(7.5\) oz bottle costs \(\$ 1.65\), and an 18 oz bottle costs \(\$ 3.99\). Is the size of the mustard bottle directly proportional to the price? If so, show how you know. If not, suggest the change of one or two prices so that they will be directly proportional. (Ti)
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