Intercept
| Site: | Saylor Academy |
| Course: | GKT101: General Knowledge for Teachers – Math |
| Book: | Intercept |
| Printed by: | Guest user |
| Date: | Tuesday, July 1, 2025, 4:16 AM |
Description
Another important property of a line (or any curve on a coordinate plane) are its x- and y-intercepts: the points where the line intersects coordinate axes. Watch this lecture series and complete the interactive exercises.
Intro to intercepts
Source: Khan Academy, https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:linear-equations-graphs#x2f8bb11595b61c86:x-intercepts-and-y-intercepts
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
x-intercept of a line
Intercepts from an equation
Intercepts from a table
Intercepts from a graph - Questions
1. Determine the intercepts of the line.
\(y\)-intercept: ( _____, ______ )
\(x\)-intercept: ( _____, ______ )
2. Determine the intercepts of the line.
\(y\)-intercept: ( _____, ______ )
\(x\)-intercept: ( _____, ______ )
3. Determine the intercepts of the line.
\(y\)-intercept: ( _____, ______ )
\(x\)-intercept: ( _____, ______ )
4. Determine the intercepts of the line.
\(y\)-intercept: ( _____, ______ )
\(x\)-intercept: ( _____, ______ )
Answers
1.
The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. Since the \(y\)-axis is also the line \(x = 0\), the \(x\)-value of this point will always be \(0\).
The \(x\)-intercept is the point where the graph intersects the \(x\)-axis. Since the \(x\)-axis is also the line \(y = 0\), the \(y\)-value of this point will always be \(0\).
By looking at the graph, we can see that:
- The \(y\)-intercept is \( (0, 275) \).
- The \(x\)-intercept is \( (125, 0) \).
2.
The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. Since the \(y\)-axis is also the line \(x = 0\), the \(x\)-value of this point will always be \(0\).
The \(x\)-intercept is the point where the graph intersects the \(x\)-axis. Since the \(x\)-axis is also the line \(y = 0\), the \(y\)-value of this point will always be \(0\).
By looking at the graph, we can see that:
- The \(y\)-intercept is \( (0, 0.4) \).
- The \(x\)-intercept is \( (0.3, 0) \).
3.
The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. Since the \(y\)-axis is also the line \(x = 0\), the \(x\)-value of this point will always be \(0\).
The \(x\)-intercept is the point where the graph intersects the \(x\)-axis. Since the \(x\)-axis is also the line \(y = 0\), the \(y\)-value of this point will always be \(0\).
By looking at the graph, we can see that:
- The \(y\)-intercept is \( (0, -45) \).
- The \(x\)-intercept is \( (-10, 0) \).
4.
The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. Since the \(y\)-axis is also the line \(x = 0\), the \(x\)-value of this point will always be \(0\).
The \(x\)-intercept is the point where the graph intersects the \(x\)-axis. Since the \(x\)-axis is also the line \(y = 0\), the \(y\)-value of this point will always be \(0\).
By looking at the graph, we can see that:
- The \(x\)-intercept is \( (-7.5, 0) \).
- The \(y\)-intercept is \( (0, 5.5) \).
Intercepts from an equation - Questions
1. Determine the intercepts of the line.
Do not round your answers.
\(4 x-3 y=17\)
\(y\)-intercept: ( _____, ______ )
\(x\)-intercept: ( _____, ______ )
2. Determine the intercepts of the line.
Do not round your answers.
\(y-3=5(x-2)\)
\(y\)-intercept: ( _____, ______ )
\(x\)-intercept: ( _____, ______ )
3. Determine the intercepts of the line.
Do not round your answers.
\(y=11 x+6\)
\(x\)-intercept: ( _____, ______ )
\(y\)-intercept: ( _____, ______ )
4. Determine the intercepts of the line.
Do not round your answers.
\(-7 x-6 y=-15\)
\(y\)-intercept: ( _____, ______ )
\(x\)-intercept: ( _____, ______ )
Answers
1.
The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. Since the \(y\)-axis is also the line \(x = 0\), the \(x\)-value of this point will always be \(0\).
The \(x\)-intercept is the point where the graph intersects the \(x\)-axis. Since the \(x\)-axis is also the line \(y = 0\), the \(y\)-value of this point will always be \(0\).
To find the \(y\)-intercept, let's substitute \(x = 0\) into the equation and solve for \(y\):
\(\begin{aligned}
4 \cdot 0-3 y &=17 \\
-3 y &=17 \\
y &=-\frac{17}{3}
\end{aligned}\)
So the \(y\)-intercept is \(\left(0,-\frac{17}{3}\right)\).
To find the \(x\) intercept, let's substitute \(y = 0 \) into the equation and solve for \(x\):
\(\begin{array}{r}
4 x-3 \cdot 0=17 \\
4 x=17 \\
x=\frac{17}{4}
\end{array}\)
So the \(x\)-intercept is \(\left(\frac{17}{4}, 0\right)\).
In conclusion,
- The \(y\)-intercept is \(\left(0,-\frac{17}{3}\right)\).
- The \(x\)-intercept is \(\left(\frac{17}{4}, 0\right)\).
2.
The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. Since the \(y\)-axis is also the line \(x = 0\), the \(x\)-value of this point will always be \(0\).
The \(x\)-intercept is the point where the graph intersects the \(x\)-axis. Since the \(x\)-axis is also the line \(y = 0\), the \(y\)-value of this point will always be \(0\).
To find the \(y\)-intercept, let's substitute \(x = 0\) into the equation and solve for \(y\):
\(\begin{aligned}
y-3 &=5(0-2) \\
y-3 &=-10 \\
y &=-7
\end{aligned}\)
So the \(y\)-intercept is \( (0, -7)\).
To find the \(x\) intercept, let's substitute \(y = 0 \) into the equation and solve for \(x\):
\(\begin{aligned}
0-3 &=5(x-2) \\
-3 &=5 x-10 \\
7 &=5 x \\
1.4 &=x
\end{aligned}\)
So the \(x\)-intercept is \( (1.4, 0)\).
In conclusion,
- The \(y\)-intercept is \( (0, -7)\).
- The \(x\)-intercept is \( (1.4, 0)\).
3.
The \(x\)-intercept is the point where the graph intersects the \(x\)-axis. Since the \(x\)-axis is also the line \(y = 0\), the \(y\)-value of this point will always be \(0\).
The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. Since the \(y\)-axis is also the line \(x = 0\), the \(x\)-value of this point will always be \(0\).
To find the \(x\) intercept, let's substitute \(y = 0 \) into the equation and solve for \(x\):
\(\begin{aligned}
0 &=11 x+6 \\
-6 &=11 x \\
-\frac{6}{11} &=x
\end{aligned}\)
So the \(x\)-intercept is \(\left(-\frac{6}{11}, 0\right)\).
To find the \(y\)-intercept, let's substitute \(x = 0\) into the equation and solve for \(y\):
\(\begin{aligned}
&y=11 \cdot 0+6 \\
&y=6
\end{aligned}\)
So the \(y\)-intercept is \( (0, 6) \). Generally, in linear equations of the form \(y=m x+b\) (which is called slope-intercept form), the \(y\)-intercept is \(( 0, b) \).
In conclusion,
- The \(x\)-intercept is \(\left(-\frac{6}{11}, 0\right)\).
- The \(y\)-intercept is \( (0, 6) \).
4.
The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. Since the \(y\)-axis is also the line \(x = 0\), the \(x\)-value of this point will always be \(0\).
The \(x\)-intercept is the point where the graph intersects the \(x\)-axis. Since the \(x\)-axis is also the line \(y = 0\), the \(y\)-value of this point will always be \(0\).
To find the \(y\)-intercept, let's substitute \(x = 0\) into the equation and solve for \(y\):
\(\begin{aligned}
-7 \cdot 0-6 y &=-15 \\
-6 y &=-15 \\
y &=\frac{15}{6} \\
y &=\frac{5}{2}
\end{aligned}\)
So the \(y\)-intercept is \(\left(0, \frac{5}{2}\right)\).
To find the \(x\) intercept, let's substitute \(y = 0 \) into the equation and solve for \(x\):
\(\begin{aligned}
-7 x-6 \cdot 0 &=-15 \\
-7 x &=-15 \\
x &=\frac{15}{7}
\end{aligned}\)
So the \(x\)-intercept is \(\left(\frac{15}{7}, 0\right)\).
In conclusion,
- The \(y\)-intercept is \(\left(0, \frac{5}{2}\right)\).
- The \(x\)-intercept is \(\left(\frac{15}{7}, 0\right)\).
Intercepts from a table
1. This table gives a few \((x,y)\) pairs of a line in the coordinate plane.
| \(x\) | \(y\) |
|---|---|
| 33 | -22 |
| 52 | -33 |
| 71 | -44 |
What is the \(x\)-intercept of the line?
2. This table gives a few \((x,y)\) pairs of a line in the coordinate plane.
| \(x\) | \(y\) |
|---|---|
| -28 | -54 |
| -21 | -40 |
| -14 | -26 |
What is the \(y\)-intercept of the line?
3. This table gives a few \((x,y)\) pairs of a line in the coordinate plane.
| \(x\) | \(y\) |
|---|---|
| -38 | 40 |
| -23 | 30 |
| -8 | 20 |
What is the \(x\)-intercept of the line?
4. This table gives a few \((x,y)\) pairs of a line in the coordinate plane.
| \(x\) | \(y\) |
|---|---|
| 32 | 22 |
| 48 | 17 |
| 64 |
12 |
What is the \(y\)-intercept of the line?
Answers
1.
An \(x\)-intercept is a point on the line that is on the \(x\)-axis, which is a point where the \(y\)-value is \(0\).
For points on a line, a constant change in the \(x\)-value brings a constant change in the \(y\)-value. Let's use this fact to find the point where the \(y\)-value is \(0\).
The table shows that for each increase of \(19\) in \(x\), there's a decrease of \(11\) in \(y\).
| \(x\) | \(y\) |
|---|---|
| \(33\) | \(-22\) |
| \(\stackrel{+19}{\longrightarrow} 52\) | \(-33 \stackrel{\longleftarrow}{-11}\) |
| \(\stackrel{+19}{\longrightarrow} 71\) | \(-44 \stackrel{\longleftarrow}{-11}\) |
Let's start at \((33,-22)\) and extend the table backwards to get to a \(y\)-value of \(0\):
| \(x\) | \(y\) |
|---|---|
| \(33\) | \(-22\) |
| \(\stackrel{-19}{\longrightarrow} 14\) | \(-11 \stackrel{-(-11)}{\longleftarrow}\) |
| \(\stackrel{-19}{\longrightarrow} -5\) | \(0 \stackrel{-(-11)}{\longleftarrow}\) |
In conclusion, the line's \(x\)-intercept is \((-5,0)\).
To verify, here is the graph of the line. You can see it passes through all the points we've seen, including the \(x\)-intercept at \((-5, 0)\).
2.
A \(y\)-intercept is a point on the line that is on the \(y\)-axis, which is a point where the \(x\)-value is \(0\).
For points on a line, a constant change in the \(x\)-value brings a constant change in the \(y\)-value. Let's use this fact to find the point where the \(x\)-value is \(0\).
The table shows that for each increase of \(7\) in \(x\), there's an increase of \(14\) in \(y\).
| \(x\) | \(y\) |
|---|---|
| \(-28\) | \(-54\) |
| \(\stackrel{+7}{\longrightarrow} -21\) | \(-40 \stackrel{\longleftarrow}{+14}\) |
| \(\stackrel{+7}{\longrightarrow} -14\) | \(-26 \stackrel{\longleftarrow}{+14}\) |
Let's start at \((-14,-26)\) and extend the table to get to an \(x\)-value of \(0\):
| \(x\) | \(y\) |
|---|---|
| \(-14\) | \(-26\) |
| \(\stackrel{+7}{\longrightarrow} -7\) | \(-12 \stackrel{\longleftarrow}{+14}\) |
| \(\stackrel{+7}{\longrightarrow} 0\) | \(2 \stackrel{\longleftarrow}{+14}\) |
In conclusion, the line's \(y\)-intercept is \((0,2)\).
To verify, here is the graph of the line. You can see it passes through all the points we've seen, including the \(y\)-intercept at \((0,2)\).
3.
An \(x\)-intercept is a point on the line that is on the \(x\)-axis, which is a point where the \(y\)-value is \(0\).
For points on a line, a constant change in the \(x\)-value brings a constant change in the \(y\)-value. Let's use this fact to find the point where the \(y\)-value is \(0\).
The table shows that for each increase of \(15\) in \(x\), there's a decrease of \(10\) in \(y\).
| \(x\) | \(y\) |
|---|---|
| \(-38\) | \(40\) |
| \(\stackrel{+15}{\longrightarrow} -23\) | \(30 \stackrel{\longleftarrow}{-10}\) |
| \(\stackrel{+15}{\longrightarrow} -8\) | \(20 \stackrel{\longleftarrow}{-10}\) |
Let's start at \((-8,20)\)and extend the table to get to a \(y\)-value of \(0\):
| \(x\) | \(y\) |
|---|---|
| \(-8\) | \(20\) |
| \(\stackrel{+15}{\longrightarrow} -7\) | \(10 \stackrel{\longleftarrow}{-10}\) |
| \(\stackrel{+15}{\longrightarrow} -22\) | \(0 \stackrel{\longleftarrow}{-10}\) |
In conclusion, the line's \(x\)-intercept is \((22,0)\).
To verify, here is the graph of the line. You can see it passes through all the points we've seen, including the \(x\)-intercept at \((22, 0) \).
4.
A \(y\)-intercept is a point on the line that is on the \(y\)-axis, which is a point where the \(x\)-value is \(0\).
For points on a line, a constant change in the \(x\)-value brings a constant change in the \(y\)-value. Let's use this fact to find the point where the \(x\)-value is \(0\).
The table shows that for each increase of \(16\) in \(x\), there's an increase of \(5\) in \(y\).
| \(x\) | \(y\) |
|---|---|
| \(32\) | \(40\) |
| \(\stackrel{+16}{\longrightarrow} 48\) | \(17 \stackrel{\longleftarrow}{-5}\) |
| \(\stackrel{+16}{\longrightarrow} 64\) | \(12 \stackrel{\longleftarrow}{-5}\) |
Let's start at \((32,22)\)and extend the table backwards to get to an \(x\)-value of \(0\):
| \(x\) | \(y\) |
|---|---|
| \(32\) | \(22\) |
| \(\stackrel{-16}{\longrightarrow} 16\) | \(27 \stackrel{-(-5)}{\longleftarrow}\) |
| \(\stackrel{-16}{\longrightarrow} 0\) | \(32 \stackrel{-(-5)}{\longleftarrow}\) |
In conclusion, the line's \(y\)-intercept is \((0,32)\).
