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Chapter 2: Problem 7

Find the linear function \(f\) passing through the given points. (-12,22) and (6,-20)

Short Answer

Expert verified
The linear function is \( f(x) = -\frac{7}{3}x - 6 \).

Step by step solution

01

Calculate the Slope

To find the slope \( m \) of the line passing through the points \((-12, 22)\) and \((6, -20)\), use the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the values: \( m = \frac{-20 - 22}{6 - (-12)} = \frac{-42}{18} = -\frac{7}{3} \).
02

Use Point-Slope Form

Now, use the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \). Let's use the point \((6, -20)\) and the slope \( m = -\frac{7}{3} \): \( y - (-20) = -\frac{7}{3}(x - 6) \).
03

Simplify the Equation

Simplify the equation from step 2: \( y + 20 = -\frac{7}{3}x + 14 \). Subtract 20 from both sides to obtain the linear equation: \( y = -\frac{7}{3}x - 6 \).

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Key Concepts

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

Understanding the Slope
The slope of a linear function is a measure of how steep a line is. It is calculated using two points on the line. The formula for the slope, denoted as \( m \), is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula involves subtracting the y-values of the points and dividing by the difference in the x-values.

Consider the points \( (-12, 22) \) and \( (6, -20) \). To find the slope, substitute these points into the formula:
  • Subtract the y-values: \( -20 - 22 = -42 \).
  • Subtract the x-values: \( 6 - (-12) = 18 \).
  • Divide: \( m = \frac{-42}{18} = -\frac{7}{3} \).
This slope tells us that for every 3 units the line moves horizontally, it moves 7 units down.
It is a negative slope, indicating the line falls from left to right.
Introducing Point-Slope Form
The point-slope form of a linear equation is a powerful tool that helps you write the equation of a line when you know the slope and a point on the line. The point-slope form is expressed as:

\[ y - y_1 = m(x - x_1) \]
Here, \( (x_1, y_1) \) is a known point on the line, and \( m \) is the slope.
In our case, we use the point \( (6, -20) \) and the slope \( -\frac{7}{3} \). Plug these into the formula:
  • Start with \( y - (-20) = -\frac{7}{3}(x - 6) \).
  • Transform it into: \[ y + 20 = -\frac{7}{3}(x - 6) \]
The point-slope form sets the ground for an easy transition to the slope-intercept form, which is the typical presentation of a linear equation.
Simplifying to Linear Equations
Linear equations in their simplest form often appear as \( y = mx + b \), which is called the slope-intercept form. Here, \( m \) is the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

Let's simplify our point-slope equation step-by-step:
  • From point-slope: \( y + 20 = -\frac{7}{3}x + 14 \).
  • We isolate \( y \) by subtracting 20 from both sides, getting: \( y = -\frac{7}{3}x + 14 - 20 \).
  • This simplifies to: \( y = -\frac{7}{3}x - 6 \).
In this equation, \( -\frac{7}{3} \) is the slope, showing the line's inclination, while \( -6 \) is the y-intercept, indicating that the line crosses the y-axis at \( (0, -6) \).
This form allows for quick graphing of the line and understanding how it moves relative to the axes.

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