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Chapter 2: Problem 5

Plot each pair of points, then determine the equation of the line that goes through the points. a. (2,3),(4,0) b. (-2,3),(2,1) c. (2,0),(0,2) d. (4,2),(-5,2)

Short Answer

Expert verified
a) \[ y = -\frac{3}{2}x + 6 \] b) \[ y = -\frac{1}{2}x + 2 \] c) \[ y = -x + 2 \] d) \[ y = 2 \]

Step by step solution

01

Plot the Points

Plot each pair of points on a coordinate plane to visualize the line segment between them.
02

Calculate the Slope

For two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope of the line that passes through them is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
03

Part (a): Calculate the Slope for Points (2,3) and (4,0)

Using the formula for slope: \[ m = \frac{0 - 3}{4 - 2} = \frac{-3}{2} = -\frac{3}{2} \]
04

Part (a): Write the Equation of the Line

Using the point-slope form \( y - y_1 = m(x - x_1) \) and point (2, 3): \[ y - 3 = -\frac{3}{2}(x - 2) \] Simplify to get: \[ y = -\frac{3}{2}x + 6 \]
05

Part (b): Calculate the Slope for Points (-2,3) and (2,1)

Using the formula for slope: \[ m = \frac{1 - 3}{2 - (-2)} = \frac{-2}{4} = -\frac{1}{2} \]
06

Part (b): Write the Equation of the Line

Using the point-slope form and point (-2, 3): \[ y - 3 = -\frac{1}{2}(x + 2) \] Simplify to get: \[ y = -\frac{1}{2}x + 2 \]
07

Part (c): Calculate the Slope for Points (2,0) and (0,2)

Using the formula for slope: \[ m = \frac{2 - 0}{0 - 2} = \frac{2}{-2} = -1 \]
08

Part (c): Write the Equation of the Line

Using the point-slope form and point (2, 0): \[ y - 0 = -1(x - 2) \] Simplify to get: \[ y = -x + 2 \]
09

Part (d): Calculate the Slope for Points (4,2) and (-5,2)

Using the formula for slope: \[ m = \frac{2 - 2}{-5 - 4} = \frac{0}{-9} = 0 \]
10

Part (d): Write the Equation of the Line

Since the slope is 0, this line is horizontal and passes through y = 2: \[ y = 2 \]

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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 is a two-dimensional surface where we can plot points, lines, and curves. It's essential for visually representing equations and geometric shapes. The plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). Each point on the plane is defined by a pair of numbers called coordinates, written as (x, y).

For example, in our exercise, when we plot the points (2, 3) and (4, 0), we place them on the coordinate plane. This helps us to visualize the line segment connecting them.
  • The first number (x-coordinate) tells how far to move left or right from the origin (0, 0).
  • The second number (y-coordinate) tells how far to move up or down.

By plotting these points, we can proceed to find the relationship between them as a straight line.
Slope Calculation
The slope of a line measures its steepness and direction. It is a ratio of the vertical change ('rise') to the horizontal change ('run') between two points on the line. The formula for calculating the slope (m) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In our exercise, for points (2, 3) and (4, 0), the slope is calculated as follows:
\[ m = \frac{0 - 3}{4 - 2} = -\frac{3}{2} \]
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
  • A slope of zero indicates a horizontal line.

Understanding the slope helps in determining the orientation of the line and is an essential step in forming the line's equation.
Point-Slope Form
The point-slope form of a linear equation is useful for writing the equation when you know the slope and a point on the line. The formula is:

\[ y - y_1 = m(x - x_1) \]
Here, \( (x_1, y_1) \) is a known point and \( m \) is the slope.

For example, using the points from our exercises:
For points (2, 3) with slope \( m = -\frac{3}{2} \), the equation becomes:
\[ y - 3 = -\frac{3}{2}(x - 2) \]
This method makes it easy to translate the geometric representation into a mathematical equation that can be analyzed or graphed.
  • The equation can be transformed into slope-intercept form \( y = mx + b \) by simplifying.
  • This is especially useful for predicting values and understanding the line's behavior.
Horizontal Line
A horizontal line is a special type of line where all points have the same y-coordinate. This means it has a slope of zero because there is no vertical change as you move along the line.

In our exercise, we see an example of this with the points (4, 2) and (-5, 2). The slope calculation is:
\[ m = \frac{2 - 2}{-5 - 4} = 0 \]
Since the slope is zero, the equation of the line simplifies to \( y = 2 \).
  • This indicates that no matter what the x-coordinate is, the y-coordinate will always be 2.
  • Horizontal lines are parallel to the x-axis and visually appear flat.

Horizontal lines are straightforward and provide a constant function which is helpful in various applications like finding limits and analyzing graphs.

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Most popular questions from this chapter

A residential customer in the Midwest purchases gas from a utility company that charges according to the formula \(C(g)=11+10.50(g)\), where \(C(g)\) is the cost, in dollars, for \(g\) thousand cubic feet of gas. a. Find \(C(0), C(5),\) and \(C(10)\). b. What is the cost if the customer uses no gas? c. What is the rate per thousand cubic feet charged for using the gas? d. How much would it cost if the customer uses 96 thousand cubic feet of gas (the amount an average Midwest household consumes during the winter months)?

Sketch a graph for each of the following situations. a. The amount in your savings account over a month, where you direct-deposit your paycheck each week and make one withdrawal during the month. b. The amount of money in an ATM machine over one day, where the ATM is stocked with dollars at the beginning of the day, and then ATM withdrawals of various sizes are made.

The greatest integer function \(y=[x]\) is defined as the greatest integer \(\leq x\) (i.e., it rounds \(x\) down to the nearest integer below it). a. What is [2]\(?[2.5] ?[2.9999999] ?\) b. Sketch a graph of the greatest integer function for \(0 \leq x<5 .\) Be sure to indicate whether each endpoint is included or excluded. [Note: A bank employee embezzled hundreds of thousands of dollars by inserting software to round down transactions (such as generating interest on an account) to the nearest cent, and siphoning the round-off differences into his account. He was eventually caught.

The gas gauge on your car is broken, but you know that the car averages 22 miles per gallon. You fill your 15.5 -gallon gas tank and tell your friend, 'I can travel 300 miles before 1 need to fill up the tank again." Explain why this is true,

Calculate the average rate of change between adjacent points for the following functions and place the values in a third column in each table. (The first entry is "n.a.") $$ \begin{array}{rr} \hline x & f(x) \\ \hline 0 & 5 \\ 10 & 25 \\ 20 & 45 \\ 30 & 65 \\ 40 & 85 \\ 50 & 105 \\ \hline \end{array} $$ $$ \begin{array}{rl} \hline x & g(x) \\ \hline 0 & 270 \\ 10 & 240 \\ 20 & 210 \\ 30 & 180 \\ 40 & 150 \\ 50 & 120 \\ \hline \end{array} $$ a. Are the functions \(f(x)\) and \(g(x)\) increasing, decreasing, or constant throughout? b. Is the average rate of change of each function increasing, decreasing, or constant throughout?
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