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Chapter 2: Problem 3

Calculate the slope and write an equation for the linear function represented by each of the given tables. a. $$ \begin{array}{cc} \hline x & y \\ \hline 2 & 7.6 \\ 4 & 5.1 \\ \hline \end{array} $$ b. $$ \begin{array}{cc} \hline A & W \\ \hline 5 & 12 \\ 7 & 16 \\ \hline \end{array} $$

Short Answer

Expert verified
a) Slope: -1.25, Equation: y = -1.25x + 10.1. b) Slope: 2, Equation: y = 2x + 2.

Step by step solution

01

- Identify the given points

First, list the points given in each table as coordinate pairs. For table (a), the points are (2, 7.6) and (4, 5.1). For table (b), the points are (5, 12) and (7, 16).
02

- Calculate the slope for (a)

To find the slope for table (a), use the formula for slope, \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, \[ y_2 = 5.1, y_1 = 7.6, x_2 = 4, x_1 = 2 \]. Calculate \[ m = \frac{5.1 - 7.6}{4 - 2} = \frac{-2.5}{2} = -1.25 \].
03

- Write the equation for (a)

With the slope (m = -1.25) and using the point-slope form \[ y - y_1 = m(x - x_1) \], choose one of the points (2, 7.6). The equation becomes: \[ y - 7.6 = -1.25(x - 2) \]. Simplify to get the slope-intercept form: \[ y = -1.25x + 10.1 \].
04

- Calculate the slope for (b)

For table (b), use the slope formula with points (5, 12) and (7, 16). Thus, \[ y_2 = 16, y_1 = 12, x_2 = 7, x_1 = 5 \]. Calculate \[ m = \frac{16 - 12}{7 - 5} = \frac{4}{2} = 2 \].
05

- Write the equation for (b)

With the slope (m = 2) and using the point-slope form \[ y - y_1 = m(x - x_1) \], choose one of the points (5, 12). The equation becomes: \[ y - 12 = 2(x - 5) \]. Simplify to get the slope-intercept form: \[ y = 2x + 2 \].

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Key Concepts

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

slope calculation
Understanding how to calculate the slope is essential in working with linear functions. The slope is a measure of how steep a line is. It can be calculated using the formula: ewline \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. ewline This formula essentially takes the difference in the y-values of two points and divides it by the difference in the x-values. Make sure to: ewline
  • Identify your two points, let's call them (x1, y1) and (x2, y2).
  • Substitute the values into the formula correctly.

We apply this formula in the exercise as follows: for point (2, 7.6) and (4, 5.1), ewline we get \[ m = \frac{5.1 - 7.6}{4 - 2} = \frac{-2.5}{2} = -1.25 \]. And for the points (5, 12) and (7, 16), \[ m = \frac{16 - 12}{7 - 5} = \frac{4}{2} = 2 \] Following these steps will always give you the slope of the line that passes through the two points.
point-slope form
Using the point-slope form of a linear equation is a straightforward way to write the equation of a line when you know the slope and one point on the line. The point-slope form is given by the formula: \[ y - y_1 = m(x - x_1) \] ewline This format is useful because it directly incorporates the slope m and any point (x1, y1) on the line. Let's break it down:
  • Identify the slope, m.
  • Select a point (x1, y1) which lies on the line.
  • Plug in these values into the formula.
ewline For example, in the exercise when slope m = -1.25 and the point (2, 7.6), the equation becomes: \[ y - 7.6 = -1.25(x - 2) \]. And for m = 2 and the point (5, 12), we get: \[ y - 12 = 2(x - 5) \].
slope-intercept form
The slope-intercept form of a linear equation is one of the most common formats used, because it easily shows both the slope and the y-intercept of a line. This form is written as: \[ y = mx + b \] Where
  • m is the slope,

  • and b is the y-intercept (the value of y when x is 0).
To convert from point-slope to slope-intercept form, you need to solve for y. Let's illustrate this with examples from our exercise: For equation \[ y - 7.6 = -1.25(x - 2) \], solving for y gives: \[ y - 7.6 = -1.25(x - 2) \Rightarrow y = -1.25x + 10.1 \] ewlineSimilarly, for \[ y - 12 = 2(x - 5) \], it simplifies to: \[ y = 2x + 2 \]. ewline Understanding these steps helps you quickly get to a form that tells you a lot about the line, like how steep it is and where it crosses the y-axis.

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Most popular questions from this chapter

In 1977 a math professor bought her condominium in Cambridge, Massachusetts, for \(\$ 70,000 .\) The value of the condo has risen steadily so that in 2007 real estate agents tell her the condo is now worth \(\$ 850,000\). a. Find a formula to represent these facts about the value of the condo \(V(t),\) as a function of time, \(t\). b. If she retires in 2010 , what does your formula predict her condo will be worth then?

a. Write an equation that describes the total cost to produce \(x\) items if the startup cost is \(\$ 200,000\) and the production cost per item is \(\$ 15\). b. Why is the total average cost per item less if the item is produced in large quantities?

Identify the slopes and the vertical intercepts of the lines with the given equations. a. \(y=3+5 x\) b. \(f(t)=-t\) c. \(y=4\) d. \(Q=35 t-10\) e. \(f(E)=10,000+3000 E\)

Construct an equation and sketch the graph of its line with the given slope, \(m,\) and vertical intercept, \(b .\) (Hint: Find two points on the line.) a. \(m=2, b=-3\) b. \(m=-\frac{3}{4}, b=1\) c. \(m=0, b=50\)

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.
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