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Chapter 8: Problem 27

Find an equation of the line passing through the given points. Use function notation to write the equation. $$ (2,0),(4,6) $$

Short Answer

Expert verified
The equation is \(f(x) = 3x - 6\).

Step by step solution

01

Find the Slope (m)

To find the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]For the points \((2,0)\) and \((4,6)\), substitute into the formula:\[ m = \frac{6 - 0}{4 - 2} = \frac{6}{2} = 3 \].Thus, the slope \(m\) is 3.
02

Use the Point-Slope Form

The point-slope form of the equation of a line is:\[ y - y_1 = m(x - x_1) \]We can use either of the given points, \((2,0)\). Thus, substituting \(m = 3\), \(x_1 = 2\), and \(y_1 = 0\) into the formula, we get:\[ y - 0 = 3(x - 2) \].This simplifies to:\[ y = 3(x - 2) \].
03

Simplify to Slope-Intercept Form

Expand and simplify the equation to get it into the slope-intercept form \(y = mx + b\):\[ y = 3x - 6 \].The simplified equation of the line is \(y = 3x - 6\).
04

Write the Equation in Function Notation

Function notation for a line is written as \(f(x)\) instead of \(y\). Thus, rewrite the equation \(y = 3x - 6\) as:\[ f(x) = 3x - 6 \].This conveys the same linear relationship using function notation, indicating \(f(x)\) is the output for input \(x\).

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Key Concepts

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

Slope
The slope of a line is a measure of its steepness and direction. To find the slope between two points, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. In our exercise, to find the slope for the points \((2,0)\) and \((4,6)\):
  • Subtract the first \( y \)-value from the second: \( 6 - 0 = 6 \).
  • Subtract the first \( x \)-value from the second: \( 4 - 2 = 2 \).
Divide the difference in \( y \)-values by the difference in \( x \)-values: \( m = \frac{6}{2} = 3 \). This tells us the line rises 3 units vertically for every 1 unit it moves horizontally. The slope is positive, indicating the line goes upwards from left to right.
Point-Slope Form
The point-slope form is useful for writing the equation of a line when we know the slope and a single point on the line. The formula is: \[ y - y_1 = m(x - x_1) \] In this formula:
  • \( m \) is the slope of the line.
  • \((x_1, y_1)\) are the coordinates of the given point.
We used the point \((2,0)\) from our example with the slope \( m = 3 \): \[ y - 0 = 3(x - 2) \] Simplifying this equation gives: \[ y = 3(x - 2) \]. This form allows us to visualize how each change in \( x \) affects \( y \), starting from the given point on the line.
Function Notation
Function notation is a way of expressing a mathematical relationship that pairs each input with exactly one output. Instead of using \( y \), we use \( f(x) \) to denote that \( y \) is a function of \( x \). This highlights that for every value of \( x \) you plug into the equation, you will get a corresponding \( y \) value. In our solution, we converted \( y = 3x - 6 \) into function notation as \( f(x) = 3x - 6 \). Function notation is particularly helpful because it explicitly indicates that the output depends on the input. Key features of function notation:
  • Clarifies the variable relationships.
  • Allows for easy substitution of different \( x \) values.
  • Makes the function easy to interpret and compare.
By using function notation, you are identifying the function as a tool to calculate new outputs for given inputs.
Slope-Intercept Form
The slope-intercept form of a linear equation is widely used because it clearly shows the slope and the y-intercept. This form is given by: \( y = mx + b \) where:
  • \( m \) is the slope.
  • \( b \) is the y-intercept (the point where the line crosses the y-axis).
In our problem, after simplifying the equation derived from the point-slope form, we obtained \( y = 3x - 6 \). Here, the slope \( m = 3 \) and the y-intercept \( b = -6 \). By glancing at the equation, you can immediately determine how steep the line is (thanks to \( m \)), and where it hits the y-axis (thanks to \( b \)). This form is handy for quickly graphing or understanding the behavior of the line.

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