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Magazine Chapter 8: Problem 33
Find an equation of the line passing through the given points. Use function notation to write the equation. $$ (-3,-8),(-6,-9) $$
Short Answer
Expert verified
The equation of the line is \( f(x) = \frac{1}{3}x - 7 \).
Step by step solution
01
Calculate the Slope
To find the slope (m) of the line passing through two points, we use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the given points \((-3, -8)\) and \((-6, -9)\): \( m = \frac{-9 - (-8)}{-6 - (-3)} = \frac{-9 + 8}{-6 + 3} = \frac{-1}{-3} = \frac{1}{3} \). Thus, the slope of the line is \( \frac{1}{3} \).
02
Write the Point-Slope Form
Use the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \). Here, we can use point \((-3, -8)\) and the slope \( \frac{1}{3} \). Substitute these values into the formula: \( y - (-8) = \frac{1}{3}(x - (-3)) \). This simplifies to \( y + 8 = \frac{1}{3}(x + 3) \).
03
Convert to Slope-Intercept Form
To convert from point-slope form to slope-intercept form (\( y = mx + b \)), you need to solve for \( y \). Start with the equation from the previous step: \( y + 8 = \frac{1}{3}(x + 3) \). Distribute \( \frac{1}{3} \): \( y + 8 = \frac{1}{3}x + 1 \). Then, subtract 8 from both sides to solve for \( y \): \( y = \frac{1}{3}x + 1 - 8 = \frac{1}{3}x - 7 \).
04
Express in Function Notation
In function notation, an equation of a line is written as \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Thus, from the slope-intercept form found previously, we can write \( f(x) = \frac{1}{3}x - 7 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope Calculation
The slope of a line is a crucial concept in mathematics because it measures how steep a line is. To find the slope, we need to use two specific points that the line passes through. The formula to calculate the slope \( m \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. In our exercise, we use the points \((-3, -8)\) and \((-6, -9)\). Plugging these into the formula gives: \[ m = \frac{-9 - (-8)}{-6 - (-3)} = \frac{-1}{-3} = \frac{1}{3} \] Thus, the slope of the line is \( \frac{1}{3} \). Things to remember:
- The slope can be positive, negative, zero, or undefined, each representing a different type of line.
- A positive slope means the line inclines upwards; a negative slope indicates it slopes downwards.
Point-Slope Form
The point-slope form of the equation of a line is extremely useful because it allows us to write the equation quickly using a known point on the line and the slope. The point-slope form is expressed as: \[ y - y_1 = m(x - x_1) \] Where \( m \) is the slope and \( (x_1, y_1) \) is the point the line passes through. Given the slope \( \frac{1}{3} \) and the point \(-3, -8\), the equation becomes: \[ y - (-8) = \frac{1}{3}(x - (-3)) \] Simplifying this, we get: \[ y + 8 = \frac{1}{3}(x + 3) \] Point-slope form is advantageous for quickly describing lines when a point and the slope are known.
Slope-Intercept Form
The slope-intercept form is widely used because it provides a straightforward way to describe any line on a coordinate plane. It looks like this: \[ y = mx + b \] Where \( m \) is the slope and \( b \) is the y-intercept (the point where the line crosses the y-axis). Starting from the point-slope form, let's convert it. Given: \[ y + 8 = \frac{1}{3}(x + 3) \] Distribute \( \frac{1}{3} \) to both \( x \) and 3: \[ y + 8 = \frac{1}{3}x + 1 \] Subtract 8 from both sides to isolate \( y \): \[ y = \frac{1}{3}x - 7 \] Thus, the line's slope-intercept form is \( y = \frac{1}{3}x - 7 \). The ease of this format makes it preferred for graphing lines directly and quickly.
Function Notation
Function notation lets us express our equations in a way that clearly represents how inputs (x-values) are transformed into outputs (y-values). For a line, the equation in function notation is given by: \[ f(x) = mx + b \] This represents the same equation as the slope-intercept form, just with a function name \( f \). Using the previously found slope-intercept form: \[ y = \frac{1}{3}x - 7 \] We translate it to function notation like this: \[ f(x) = \frac{1}{3}x - 7 \] Remember:
- Function notation almost always uses \( f \), but other letters like \( g \) or \( h \) may be used based on context.
- \( f(x) \) tells us that every x-value has a particular y-value associated with it.
