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Chapter 8: Problem 19

Determine the linear function whose graph is a line that contains the points \((-3,-1)\) and \((2,-6)\). $$ f(x)=-x-4 $$

Short Answer

Expert verified
The linear function is \(f(x) = -x - 4\).

Step by step solution

01

Understand the Problem

We need to find the linear equation of a line that passes through the points \((-3, -1)\) and \((2, -6)\). This equation will be in the form of \(f(x) = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
02

Calculate the Slope

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given points: \((-3, -1)\) and \((2, -6)\). This gives:\[ m = \frac{-6 - (-1)}{2 - (-3)} = \frac{-6 + 1}{2 + 3} = \frac{-5}{5} = -1 \]
03

Find the Y-intercept

With the slope \(m = -1\), use the point-slope form of a line \(y - y_1 = m(x - x_1)\) and one of the given points, say \((-3, -1)\). Substitute the slope and point into the equation:\[ y + 1 = -1(x + 3) \] Expand and simplify: \[ y + 1 = -x - 3 \]\[ y = -x - 4 \] Thus, the y-intercept \(b\) is -4.
04

Write the Linear Function

Now that we know the slope \(m = -1\) and the y-intercept \(b = -4\), the linear function is:\[ f(x) = -x - 4 \]
05

Verify the Solution

Verify by checking if both given points \((-3, -1)\) and \((2, -6)\) satisfy the equation \(f(x) = -x - 4\). For \((-3, -1)\): \[ f(-3) = -(-3) - 4 = 3 - 4 = -1 \] For \((2, -6)\):\[ f(2) = -(2) - 4 = -2 - 4 = -6 \]Both calculations match, confirming our function.

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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 is a measure of its steepness and direction. To find the slope given two points on the line, we use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

The variables \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points.
The numerator \((y_2 - y_1)\) is the change in the y-values, and the denominator \((x_2 - x_1)\) is the change in the x-values.
This essentially measures the vertical change per unit of horizontal change, defined as the "rise over run."
  • If the slope is positive, the line rises as we move from left to right.
  • If the slope is negative, the line falls.
For example, using the points \((-3, -1)\) and \((2, -6)\), the slope calculation would be:
  • \( m = \frac{-6 - (-1)}{2 - (-3)} = \frac{-5}{5} = -1 \)

This tells us the line inclines downward with a slope of \(-1\), meaning for each step right, we step down one unit.
Y-intercept
The y-intercept of a line is the point where the line crosses the y-axis.
In the equation of a line, represented as \( y = mx + b \), the y-intercept is the constant \( b \).
It tells us the value of \( y \) when \( x = 0 \).
  • This can be thought of as the starting point of the line on the graph's vertical axis.
To find the y-intercept, you can rearrange the equation or read it directly once you have used one point and the slope.Using the slope-point equation:
  • If you have the slope \( m = -1 \) and a point \((-3, -1)\), use \( y - y_1 = m(x - x_1) \) to derive the y-intercept.

Solving, we have the equation:
  • \( y + 1 = -1(x + 3) \)
  • Simplifying, \( y = -x - 4 \)
This shows the y-intercept is \(-4\), meaning the line crosses the y-axis at \((0, -4)\).
Point-Slope Form
The point-slope form of a linear equation is a useful way to write the equation of a line when you know one point on the line and the slope.
The general formula is:
  • \( y - y_1 = m(x - x_1) \)
Here, \((x_1, y_1)\) is a known point, and \( m \) is the slope of the line.
  • This form is particularly handy when you want to swiftly find the equation without first calculating the y-intercept.
For the points \((-3, -1)\) and slope \( m = -1 \):
  • \( y + 1 = -1(x + 3) \)
  • By expanding, we arrive at \( y = -x - 4 \).

This gives us the standard form of the line equation, where it's easy to recognize both the slope and the y-intercept right away. It's an efficient process, especially useful in coordinate geometry.

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Most popular questions from this chapter

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