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Magazine Chapter 2: Problem 75
Find a function whose graph is the given curve. The line segment joining the points \((-2,1)\) and \((4,-6)\)
Short Answer
Expert verified
The function is \(y = \frac{-7}{6}x - \frac{4}{3}\).
Step by step solution
01
Identify the Type of Function
Since the problem involves a straight line joining two points, we need to find the equation of a line. The standard form of a line is given by the linear function \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02
Calculate the Slope
The slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points \((-2, 1)\) and \((4, -6)\), we have \(m = \frac{-6 - 1}{4 + 2} = \frac{-7}{6}\).
03
Use the Point-Slope Form
With the slope \(m = \frac{-7}{6}\) and one of the points, we can use the point-slope form of a line, which is \(y - y_1 = m(x - x_1)\). Using the point \((-2, 1)\), it becomes \(y - 1 = \frac{-7}{6}(x + 2)\).
04
Simplify to Find the Equation
Simplify the equation \(y - 1 = \frac{-7}{6}(x + 2)\) to find the equation of the line. Distribute \(\frac{-7}{6}\) on the right side: \(y - 1 = \frac{-7}{6}x - \frac{14}{6}\). Then solve for \(y\): \[y = \frac{-7}{6}x - \frac{14}{6} + 1 = \frac{-7}{6}x - \frac{14}{6} + \frac{6}{6}.\] This simplifies to \[y = \frac{-7}{6}x - \frac{8}{6} = \frac{-7}{6}x - \frac{4}{3}.\] Thus, the function is \(y = \frac{-7}{6}x - \frac{4}{3}\).
05
Verify the Result
Check if both points lie on the line. For \((-2,1)\): \(y = \frac{-7}{6}(-2) - \frac{4}{3} = \frac{14}{6} - \frac{4}{3} = 1\). For \((4,-6)\): \(y = \frac{-7}{6}(4) - \frac{4}{3} = -\frac{28}{6} - \frac{4}{3} = -6\), confirming the points lie on the line.
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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. It indicates how much the line rises or falls as you move from left to right across a graph. To find the slope, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula helps us calculate the vertical change (rise) over the horizontal change (run) between two points
- The slope is "rise" over "run".
- A positive slope means the line ascends as you move to the right.
- A negative slope means the line descends as you move to the right.
Point-Slope Form
The point-slope form is a way to easily write the equation of a line when you know its slope and one point it passes through. This form is expressed as \( y - y_1 = m(x - x_1) \), where
- \(m\) is the slope.
- \((x_1, y_1)\) is a known point on the line.
Linear Function
A linear function is a function that graphs a straight line. It is of the form \( y = mx + b \), where
- \(m\) is the slope of the line.
- \(b\) is the y-intercept, or where the line crosses the y-axis.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. In a function expressed as \( y = mx + b \), "b" is the y-intercept. This is an important part of graphing linear functions since it helps you define where the line starts on the y-axis. To find the y-intercept from an equation, you can set \( x = 0 \) in the equation and solve for \( y \). From the simplified form of our equation, \( y = \frac{-7}{6}x - \frac{4}{3} \), the y-intercept is \(-\frac{4}{3} \). Visualizing this, it means that the line will cross the vertical axis at the point \( (0, -\frac{4}{3}) \). Knowing the y-intercept, alongside the slope, can help you plot the entire line.
