Skip to main content

Precalculus for Team-Based Inquiry Learning: 2024 Development Edition — Instructor Version

TBIL Fellows

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 \((1,4)\text{.}\)
Answer.
Many answers possible.
(b)
Was this the only possible line that goes through the point \((1,4)\text{?}\)
  1. Yes. The line is unique.
  2. No. There is exactly one more line possible.
  3. No. There are a lot of lines that go through \((1,4)\text{.}\)
  4. No. There are an infinite number of lines that go through \((1,4)\text{.}\)
Answer.
D
(Though C is also true. D is just more descriptive.)
(c)
Now draw a line that goes through the points \((1,4)\) and \((-3,-2)\text{.}\)
Answer.
Just the one possible line.
(d)
Was this the only possible line that goes through the points \((1,4)\) and \((-3,-2)\text{?}\)
  1. Yes. The line is unique.
  2. No. There is exactly one more line possible.
  3. No. There are a lot of lines that go through \((1,4)\) and \((-3,-2)\text{.}\)
  4. No. There are an infinite number of lines that go through \((1,4)\) and \((-3,-2)\text{.}\)
Answer.
A
(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.)

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.

Activity 3.3.3.

(a)
Graph the line containing the points \((-7,1)\) and \((6,-2)\text{.}\)
Answer.
(b)
Graph the line containing the points \((-3,0)\) and \((0,8)\text{.}\)
Answer.
(c)
Graph the line given by the table below.
\(x\) \(y\)
\(-3\) \(-12\)
\(-2\) \(-9\)
\(-1\) \(-6\)
\(0\) \(-3\)
\(1\) \(0\)
\(2\) \(3\)
Answer.
(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.
  2. Two points are enough.
  3. Three points are enough.
  4. You need to plot all six points.
  5. You can use however many you want.
Answer.
B
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!

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.

Activity 3.3.5.

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

(a)

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

(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.
  2. Start at the \(y\)-intercept, then move up one space and to the right three spaces to find another point.
  3. Start at the \(y\)-intercept, then move down one space and to the left three spaces to find another point.
  4. Start at the \(y\)-intercept, then move down one space and to the right three spaces to find another point.
Answer.
A and D

(c)

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

Activity 3.3.6.

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

(a)

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

(b)

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

(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.

Activity 3.3.7.

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

Activity 3.3.8.

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

(a)

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

(b)

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

(c)

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

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.

(a)

Which type of line has a slope of zero?
  1. Horizontal
  2. Vertical
Answer.
A

(b)

Which type of line has an undefined slope?
  1. Horizontal
  2. Vertical
Answer.
B

(c)

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

(d)

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

(e)

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

(f)

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

Activity 3.3.10.

Graph each line described below.

(a)

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

(b)

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

(c)

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

(d)

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

(e)

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

(f)

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

(g)

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

(h)

The line with equation \(y-5=\frac{7}{2}(x-2)\text{.}\)
Answer.

(i)

The line with equation \(3x-6y=8\text{.}\)
Answer.

Subsection 3.3.2 Videos

It would be great to include videos down here, like in the Calculus book!
PrevTopNext