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Chapter 2: Problem 99

Write an equation for a linear function whose graph has the given characteristics. Passes through \((1,7)\) and \((-2,1)\)

Short Answer

Expert verified
The linear function is given by the equation \( y = 2x + 5 \).

Step by step solution

01

Find the Slope

To find the slope of the line, use the formula for the slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \( (x_1, y_1) = (1, 7) \) and \( (x_2, y_2) = (-2, 1) \).So, \[ m = \frac{1 - 7}{-2 - 1} = \frac{-6}{-3} = 2 \]
02

Write the Point-Slope Form

With the slope \( m = 2 \) and a point \( (1, 7) \), use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \]Plug in the values:\[ y - 7 = 2(x - 1) \]
03

Simplify to Slope-Intercept Form

Expand and simplify the equation from Step 2 to write it in the slope-intercept form \( y = mx + b \):\[ y - 7 = 2(x - 1) \]\[ y - 7 = 2x - 2 \]Add 7 to both sides:\[ y = 2x + 5 \]

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Key Concepts

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

Slope of a Line
The slope of a line represents how steep the line is. It is a crucial component in understanding linear functions. The slope is often denoted by the letter \( m \) and is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line. The difference \( y_2 - y_1 \) represents the change in the vertical direction (often called the "rise"), while \( x_2 - x_1 \) is the change in the horizontal direction (the "run").
  • Positive slope: As you move from left to right, the line rises.
  • Negative slope: As you move from left to right, the line falls.
  • A slope of zero: The line is horizontal.
  • Undefined slope: The line is vertical.
This calculation is a straightforward way to see how two different values (points) change in relation to each other and ensures a consistent description of a line's incline.
Point-Slope Form
The point-slope form of a line is particularly useful when you have one point on the line and the slope. This form is written as: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a known point on the line and \( m \) is the slope of the line. This equation helps us organize our information efficiently and offers a quick way to develop a linear equation.
  • Start with the slope \( m \) you have calculated.
  • Substitute the known point \( (x_1, y_1) \) into the equation.
By substituting the slope and the chosen point into the equation, you create a versatile tool for quickly writing equations of lines.
Slope-Intercept Form
The slope-intercept form of a line is among the most popular ways to express a linear equation. It is expressed as: \[ y = mx + b \] where \( m \) stands for the slope and \( b \) represents the y-intercept, the point at which the line crosses the y-axis.
  • It provides a clear visualization of the slope and the starting point of the line on the y-axis.
  • Easily determine the y-intercept by setting \( x = 0 \).
  • Transforms quickly from point-slope or standard forms through simple algebraic manipulation.
This form is particularly beneficial as it allows you to predict one variable based on another and visually analyze the position and direction of the line's graph much easier.

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Most popular questions from this chapter

For each of the following functions, first sketch the graph of its associated function, \(f(x)=x^{2}, f(x)=x^{3},\) or \(f(x)=|x|\) Then draw the graph of function g using trans/ations and/or a reflection. See Examples 7 and 8 . $$ g(x)=-x^{2} $$

Wood Production. The total world wood production can be modeled by a linear function. In 1960 , approximately \(2,400\) million cubic feet of wood were produced. since then, the amount of increase has been approximately 25.5 million cubic feet per year. (Source: Earth Policy Institute) a. Let \(t\) be the number of years after 1960 and \(W\) be the number of million cubic feet of wood produced. Write a linear function \(W(t)\) to model the production of wood. b. Use your answer to part a to estimate how many million cubic feet of wood the world produced in 2010.

Fire Protection. City growth and the number of fires for a certain city are related by a linear equation. Records show that 113 fires occurred in a year when the local population was \(150,000\) and that the rate of increase in the number of fires was 1 for every \(1,000\) new residents. a. Using the variables \(p\) for population and \(F\) for fires, write an equation (in slope-intercept form) that the fire department can use to predict future fire statistics. b. How many fires can be expected when the population reaches \(200,000 ?\)

Graph each equation. \(y=\frac{x-3}{|x-3|}\)

Explain what \(m, x_{1},\) and \(y_{1}\) represent in the point-slope form of the equation of a line.
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