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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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Section 3.3 Graphs of Linear Equations (LF3)

Subsection 3.3.1 Activities

Activity 3.3.1.

(a)
Draw a line that goes through the point \((1,4)\text{.}\)
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Answer.
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(b)
Was this the only possible line that goes through the point \((1,4)\text{?}\)
  1. Yes. The line is unique.
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  2. No. There is exactly one more line possible.
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  3. No. There are a lot of lines that go through \((1,4)\text{.}\)
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  4. No. There are an infinite number of lines that go through \((1,4)\text{.}\)
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Answer.
(Though C is also true. D is just more descriptive.)
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(c)
Now draw a line that goes through the points \((1,4)\) and \((-3,-2)\text{.}\)
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Answer.
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(d)
Was this the only possible line that goes through the points \((1,4)\) and \((-3,-2)\text{?}\)
  1. Yes. The line is unique.
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  2. No. There is exactly one more line possible.
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  3. No. There are a lot of lines that go through \((1,4)\) and \((-3,-2)\text{.}\)
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  4. No. There are an infinite number of lines that go through \((1,4)\) and \((-3,-2)\text{.}\)
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Answer.
(Discussion could include that we would have to make the line curve to connect the points in more than one way. But, a line has to have the same slope everywhere.)
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Observation 3.3.2.

If you are given two points, then you can always graph the line containing them by plotting them and connecting them with a line.
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Activity 3.3.3.

(a)
Graph the line containing the points \((-7,1)\) and \((6,-2)\text{.}\)
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Answer.
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(b)
Graph the line containing the points \((-3,0)\) and \((0,8)\text{.}\)
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Answer.
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(c)
Graph the line given by the table below.
\(x\) \(y\)
\(-3\) \(-12\)
\(-2\) \(-9\)
\(-1\) \(-6\)
\(0\) \(-3\)
\(1\) \(0\)
\(2\) \(3\)
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Answer.
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(d)
Let’s say you are given a table that listed six points that are on the same line. How many of those points are necessary to use to graph the line?
  1. One point is enough.
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  2. Two points are enough.
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  3. Three points are enough.
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  4. You need to plot all six points.
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  5. You can use however many you want.
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Answer.
Discussion could include pointing out that only two are necessary, but we can include more if we want. Also, including more is a good way to catch an error in plotting. One will stick out!
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Remark 3.3.4.

In Activity 3.3.3, we were given at least two points in each question. However, sometimes we are not directly given two points to graph a line. Instead we are given some combination of characteristics about the line that will help us find two points. These characteristics could include a point, the intercepts, the slope, or an equation.
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Activity 3.3.5.

A line has a slope of \(-\dfrac{1}{3}\) and its \(y\)-intercept is \(4\text{.}\)
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(a)

We were given the \(y\)-intercept. What point does that correspond to?
  1. \(\displaystyle (4,0)\)
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  2. \(\displaystyle (0,4)\)
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  3. \(\displaystyle \left(4,-\dfrac{1}{3}\right)\)
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  4. \(\displaystyle \left(-\dfrac{1}{3},4\right)\)
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Answer.
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(b)

After we plot the \(y\)-intercept, how can we use the slope to find another point?
  1. Start at the \(y\)-intercept, then move up one space and to the left three spaces to find another point.
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  2. Start at the \(y\)-intercept, then move up one space and to the right three spaces to find another point.
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  3. Start at the \(y\)-intercept, then move down one space and to the left three spaces to find another point.
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  4. Start at the \(y\)-intercept, then move down one space and to the right three spaces to find another point.
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Answer.
A and D
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(c)

Graph the line that has a slope of \(-\dfrac{1}{3}\) and its \(y\)-intercept is \(4\text{.}\)
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Answer.
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Activity 3.3.6.

A line is given by the equation \(y=-2x+5\text{.}\)
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(a)

What form is the equation given in?
  1. Standard form
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  2. Point-slope form
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  3. Slope-intercept form
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  4. The form it is in doesn’t have a name.
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Answer.
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(b)

The form gives us one point right away: the \(y\)-intercept. Which of the following is the \(y\)-intercept?
  1. \(\displaystyle (-2,0)\)
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  2. \(\displaystyle (0,-2)\)
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  3. \(\displaystyle (5,0)\)
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  4. \(\displaystyle (0,5)\)
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Answer.
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(c)

After we plot the \(y\)-intercept, we can use the slope to find another point. Find another point and graph the resulting line.
Answer.
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Activity 3.3.7.

A line contains the point \((-3,-2)\) and has slope \(\dfrac{1}{5} \text{.}\) Which of the following is the graph of that line?
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Answer.
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Activity 3.3.8.

A line is given by the equation \(y-6=-4(x+2)\text{.}\)
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(a)

What form is the equation given in?
  1. Standard form
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  2. Point-slope form
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  3. Slope-intercept form
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  4. The form it is in doesn’t have a name.
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Answer.
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(b)

The form gives us one point right away. Which of the following is a point on the line?
  1. \(\displaystyle (-2,-6)\)
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  2. \(\displaystyle (-2,6)\)
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  3. \(\displaystyle (2,-6)\)
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  4. \(\displaystyle (2,6)\)
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Answer.
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(c)

After we plot this point, we can use the slope to find another point. Find another point and graph the resulting line.
Answer.
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Activity 3.3.9.

Recall from Definition 3.2.14 that the equation of a horizontal line has the form \(y=k\) where \(k\) is a constant and a vertical line has the form \(x=h\) where \(h\) is a constant.
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(c)

Graph the vertical line that goes through the point \((4,-2)\text{.}\)
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Answer.
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(d)

What is the equation of the vertical line through the point \((4,-2)\text{?}\)
  1. \(\displaystyle x=4\)
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  2. \(\displaystyle y=4\)
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  3. \(\displaystyle x=-2\)
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  4. \(\displaystyle y=-2\)
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Answer.
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(e)

Graph the horizontal line that goes through the point \((4,-2)\text{.}\)
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Answer.
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(f)

What is the equation of the horizontal line through the point \((4,-2)\text{?}\)
  1. \(\displaystyle x=4\)
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  2. \(\displaystyle y=4\)
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  3. \(\displaystyle x=-2\)
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  4. \(\displaystyle y=-2\)
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Answer.
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Activity 3.3.10.

Graph each line described below.
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(a)

The line containing the points \((-3,4)\) and \((5,-2)\text{.}\)
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Answer.
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(b)

The line whose \(x\)-intercept is \(-2\) and whose \(y\)-intercept is \(7\text{.}\)
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Answer.
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(c)

The line whose slope is \(\dfrac{2}{5}\) that goes through the point \((4,6)\text{.}\)
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Answer.
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(d)

The line whose slope is \(-\dfrac{1}{3}\) and whose \(y\)-intercept is \(-4\text{.}\)
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Answer.
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(e)

The vertical line through the point \((-2,-7)\text{.}\)
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Answer.
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(f)

The horizontal line through the point \((-6,3)\text{.}\)
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Answer.
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(g)

The line with equation \(y=-\dfrac{5}{3}x-6\text{.}\)
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Answer.
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(h)

The line with equation \(y-5=\dfrac{7}{2}(x-2)\text{.}\)
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Answer.
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Subsection 3.3.2 Exercises

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