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Magazine Chapter 1: Problem 45
Use the y-intercept and slope to sketch the graph of each equation. The line through \((-3,-2)\) with slope \(-1 / 2\)
Short Answer
Expert verified
Equation: \(y = -\frac{1}{2}x - \frac{7}{2}\). Y-intercept: \(-3.5\).
Step by step solution
01
Identify the given information
The given point on the line is \((-3, -2)\), and the slope \(m\) is \(-\frac{1}{2}\).
02
Use the point-slope form of the equation
To find the equation of the line, use the point-slope form: \(y - y_1 = m(x - x_1)\). Plug in \((-3, -2)\) for \(x_1, y_1\) and \(-\frac{1}{2}\) for \(m\).
03
Substitute the values into the equation
Using the point-slope form, we get: \(y - (-2) = -\frac{1}{2}(x - (-3))\) which simplifies to \(y + 2 = -\frac{1}{2}(x + 3)\).
04
Simplify the equation to slope-intercept form
Distribute the slope and then isolate \(y\): \(y + 2 = -\frac{1}{2}x - \frac{3}{2}\). Subtract 2 from both sides to get \(y = -\frac{1}{2}x - \frac{3}{2} - 2\). Simplify the constant term to get \(-\frac{3}{2} - \frac{4}{2} = -\frac{7}{2}\). Thus, \(y = -\frac{1}{2}x - \frac{7}{2}\).
05
Identify the y-intercept from the equation
From the slope-intercept form \(y = mx + b\), the constant term \(-\frac{7}{2}\) is the y-intercept. Thus, the y-intercept is \(-\frac{7}{2}\) or \(-3.5\).
06
Plot the y-intercept on the graph
Plot the point \(0, -3.5\) on the graph. This is the point where the line crosses the y-axis.
07
Use the slope to find another point
From the y-intercept, use the slope \(-\frac{1}{2}\) which means go down 1 unit and right 2 units. Starting from \(0, -3.5\), moving to the right 2 units gets \(2, -4.5\). Plot this point as well.
08
Draw the line
Draw a straight line through the points \(0, -3.5\) and \(2, -4.5\). This is the graph of the line through \((-3,-2)\) with slope \(-\frac{1}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope-intercept form
The slope-intercept form is a way to write the equation of a line so that important features are immediately visible. The general form is \(y = mx + b\) where:
- \(m\) is the slope - it measures how steep the line is.
- \(b\) is the y-intercept - the point where the line crosses the y-axis.
- The slope \(-\frac{1}{2}\) means for every 2 units you move horizontally, the line moves down 1 unit.
- The y-intercept \(-3.5\) tells us that the line will cross the y-axis at \(0, -3.5\).
point-slope form
The point-slope form is another useful way to express the equation of a line, especially when you know one point on the line and its slope. The general form is \(y - y_1 = m(x - x_1)\), where:
- \((x_1, y_1)\) is a specific point on the line.
- \(m\) is the slope.
- \(y - (-2) = -\frac{1}{2}(x - (-3))\)
- which simplifies to \(y + 2 = -\frac{1}{2}(x + 3)\).
- This indicates that at the point \((-3, -2)\), if you move 2 units right, you move 1 unit down.
plotting points
Plotting points is an essential part of graphing linear equations. It involves marking specific points on the graph based on their coordinates \((x, y)\). Here is a step-by-step method:
- Start with the y-intercept. For \(y = -\frac{1}{2}x - \frac{7}{2}\), plot \(0, -3.5\).
- Use the slope to find another point. With a slope of \(-\frac{1}{2}\), move right by 2 units and down by 1 unit from \(0, -3.5\). This gives us the point \(2, -4.5\).
- Mark these points on the graph.
- Draw a straight line through these points to represent the equation.
