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Chapter 1: Problem 75

Sketch by hand the graph of the line passing through the given point and having the given slope. Label two points on the line. $$(-1,3), m=\frac{3}{2}$$

Short Answer

Expert verified
Plot the points \((-1, 3)\) and \((1, 6)\) and connect with a straight line having a slope of \(\frac{3}{2}\).

Step by step solution

01

Understand the Problem

We need to sketch the graph of a line that passes through the point \((-1, 3)\) and has a slope of \(m = \frac{3}{2}\).
02

Use the Point-Slope Formula

The point-slope form of a line is given by the equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. In this case, \((x_1, y_1) = (-1, 3)\) and \(m = \frac{3}{2}\). Substituting these values into the formula gives us \(y - 3 = \frac{3}{2}(x + 1)\).
03

Solve for y

Rearrange the equation to solve for \(y\):\[y - 3 = \frac{3}{2}x + \frac{3}{2}\] Add 3 to both sides to get\[y = \frac{3}{2}x + \frac{9}{2}\]. This is the equation of the line in slope-intercept form \(y = mx + b\), where \(m = \frac{3}{2}\) and \(b = \frac{9}{2}\).
04

Choose Another Point on the Line

Select another point that satisfies the line equation. For example, use \(x = 1\) to find a second point. Substitute \(x = 1\) into the equation \(y = \frac{3}{2}x + \frac{9}{2}\): \[y = \frac{3}{2}(1) + \frac{9}{2} = \frac{3}{2} + \frac{9}{2} = 6\] So, another point is \((1, 6)\).
05

Sketch the Line

On a coordinate plane, plot the points \((-1, 3)\) and \((1, 6)\). Draw a straight line through these points. Ensure the slope of the line is \(\frac{3}{2}\), meaning it rises 3 units for every 2 units it runs.
06

Label Two Points

Clearly label the points \((-1, 3)\) and \((1, 6)\) on the graph since they are specifically determined and often useful for further calculations or analyses involving the line.

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Key Concepts

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

Point-Slope Form
The point-slope form is a helpful way to write the equation of a line when you know a point on the line and its slope. This form is written as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) represents the coordinates of a specific point on the line, and \(m\) is the slope of the line.
To use this formula, follow these simple steps:
  • Identify a point on the line, denoted as \((x_1, y_1)\). In our example, this point is \((-1, 3)\).
  • Determine the slope \(m\) of the line. We have \(m = \frac{3}{2}\) for our example.
  • Substitute the known values into the point-slope formula. For instance, the equation becomes \( y - 3 = \frac{3}{2}(x + 1) \).
Thus, by using the point-slope form, you can easily find the equation of a line when given key information about a point and the slope.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graph lines and other geometric shapes. It comprises two perpendicular number lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically.
Using a coordinate plane allows us to visually represent points and lines in a clear way.
  • Points are represented using ordered pairs \((x, y)\). For example, \((-1, 3)\) tells us to move 1 unit left on the x-axis and 3 units up on the y-axis.
  • Lines are graphed by plotting points and connecting them with a straight path.
  • The origin \((0, 0)\) is where the x-axis and y-axis intersect.
For the problem at hand, plotting points such as \((-1, 3)\) and \((1, 6)\) on the coordinate plane helps us visually establish the necessary line that possesses a slope of \(\frac{3}{2}\). By understanding how to navigate and use the coordinate plane, graphing linear equations becomes an easier task.
Slope-Intercept Form
The slope-intercept form of a line's equation is another popular way to express linear equations. It is written as \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept, representing the point where the line crosses the y-axis.
This form is advantageous because:
  • It quickly reveals the slope \(m\) and the y-intercept \(b\).
  • We can easily derive the line's behavior and where it shifts from.
  • It's straightforward to plot directly onto a coordinate plane.
To convert from the point-slope form to slope-intercept form, the equation \( y - 3 = \frac{3}{2}(x + 1) \) can be rearranged to \( y = \frac{3}{2}x + \frac{9}{2} \), allowing us to instantly ascertain that the slope is \(\frac{3}{2}\) and the y-intercept is \(\frac{9}{2}\). This simple visualization offered by the slope-intercept form gives us a handy guide to sketch the entire line efficiently.

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Most popular questions from this chapter

Find \(f(a), f(b+1),\) and \(f(3 x)\) for the given \(f(x)\) $$f(x)=1-x^{2}$$

90\. Air Temperature When the relative humidity is \(100 \%\) air cools \(5.8^{\circ} \mathrm{F}\) for every 1 -mile increase in altitude. If the temperature is \(80^{\circ} \mathrm{F}\) on the ground, then \(f(x)=80-5.8 x\) calculates the air temperature \(x\) miles above the ground. Find \(f(3)\) and interpret the result. (Source: Battan, L., Weather in Your Life, W.H. Freeman.)

A linear function \(f\) has the ordered pairs listed in the table. Find the slope \(m\) of e table to find the \(y\) -intercept of the line, and give an equation that defines \(f\). begin{array}{rr|r} \text { 70. } & x & f(x) \\ \hline-100 & -4 \\ -50 & -4 \\ 0 & -4 \\ 50 & -4 \\ 100 & -4 \end{array}

Find the slope (if defined) of the line that passes through the given points. $$44 .(-8,2) \text { and }(-8,1)$$

Sketch by hand the graph of the line passing through the given point and having the given slope. Label Through \((-2,-3), m=-\frac{3}{4}\)
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