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Graph with Intercepts

Identify the x - and y - intercepts on a graph

Identify the

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x- axis and the y- axis. These points are called the intercepts of the line.

Let’s look at the graphs of the lines in .

First, notice where each of these lines crosses the \(x\)-axis. See .

FigureThe line crosses the x- axis at:Ordered pair of this point
Figure (a)3\((3,0)\)
Figure (b)4\((4,0)\)
Figure (c)5\((5,0)\)
Figure (d)0\((0,0)\)

Do you see a pattern?

For each row, the y- coordinate of the point where the line crosses the x- axis is zero. The point where the line crosses the x- axis has the form \((a,0)\) and is called the x- intercept of a line. The x- intercept occurs when \(y\) is zero.

Now, let’s look at the points where these lines cross the y- axis. See .

FigureThe line crosses the y-axis at:Ordered pair for this point
Figure (a)6\((0,6)\)
Figure (b)\(-3\)\((0,-3)\)
Figure (c)\(-5\)\((0,-5)\)
Figure (d)0\((0,0)\)
Example

Try it.

Find the x- and y- intercepts on each graph.

Solution
  1. ⓐ The graph crosses the x- axis at the point \((4,0)\). The x- intercept is \((4,0)\).
    The graph crosses the y- axis at the point \((0,2)\). The y- intercept is \((0,2)\).

  2. ⓑ The graph crosses the x- axis at the point \((2,0)\). The x- intercept is \((2,0)\)
    The graph crosses the y- axis at the point \((0,-6)\). The y- intercept is \((0,-6)\).

  3. ⓒ The graph crosses the x- axis at the point \((-5,0)\). The x- intercept is \((-5,0)\).
    The graph crosses the y- axis at the point \((0,-5)\). The y- intercept is \((0,-5)\).

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Find the

Recognizing that the x- intercept occurs when y is zero and that the y- intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x- intercept, let \(y=0\) and solve for x. To find the y- intercept, let \(x=0\) and solve for y.

 

Example

Try it.

Find the intercepts of \(2x+y=6\).

Solution

We will let \(y=0\) to find the x- intercept, and let \(x=0\) to find the y- intercept. We will fill in the table, which reminds us of what we need to find.

To find the x- intercept, let \(y=0\).

Let y = 0.
Simplify.
The x-intercept is(3, 0)
To find the y-intercept, let x = 0.
Let x = 0.
Simplify.
The y-intercept is(0, 6)

The intercepts are the points \((3,0)\) and \((0,6)\) as shown in .

\(2x+y=6\)
\(x\)\(y\)
30
06
Example

Try it.

Find the intercepts of \(4x-3y=12\).

Solution

To find the x-intercept, let y = 0.
Let y = 0.
Simplify.
The x-intercept is(3, 0)
To find the y-intercept, let x = 0.
Let x = 0.
Simplify.
The y-intercept is(0, −4)

The intercepts are the points (3, 0) and (0, −4) as shown in the following table.

\(4x-3y=12\)
\(x\)\(y\)
30
0\(-4\)

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

How to Graph a Line Using Intercepts

Try it.

Graph \(-x+2y=6\) using the intercepts.

Solution

The steps to graph a linear equation using the intercepts are summarized below.

Example

Try it.

Graph \(4x-3y=12\) using the intercepts.

Solution

Find the intercepts and a third point.

We list the points in and show the graph below.

\(4x-3y=12\)
\(x\)\(y\)\((x,y)\)
30\((3,0)\)
0\(-4\)\((0,-4)\)
64\((6,4)\)
Example

Try it.

Graph \(y=5x\) using the intercepts.

Solution

This line has only one intercept. It is the point \((0,0)\).

To ensure accuracy we need to plot three points. Since the x- and y- intercepts are the same point, we need two more points to graph the line.

See .

\(y=5x\)
\(x\)\(y\)\((x,y)\)
00\((0,0)\)
15\((1,5)\)
\(-1\)\(-5\)\((-1,-5)\)

Plot the three points, check that they line up, and draw the line.

Key Concepts

  • Find the x- and y- Intercepts from the Equation of a Line
    • Use the equation of the line to find the x- intercept of the line, let \(y=0\) and solve for x.
    • Use the equation of the line to find the y- intercept of the line, let \(x=0\) and solve for y.
  • Graph a Linear Equation using the Intercepts
    1. Find the x- and y- intercepts of the line.
      Let \(y=0\) and solve for x.
      Let \(x=0\) and solve for y.
    2. Find a third solution to the equation.
    3. Plot the three points and then check that they line up.
    4. Draw the line.



  • Strategy for Choosing the Most Convenient Method to Graph a Line:
    • Consider the form of the equation.
    • If it only has one variable, it is a vertical or horizontal line.
      \(x=a\) is a vertical line passing through the x- axis at \(a\)
      \(y=b\) is a horizontal line passing through the y- axis at \(b\).
    • If y is isolated on one side of the equation, graph by plotting points.
    • Choose any three values for x and then solve for the corresponding y- values.
    • If the equation is of the form \(ax+by=c\), find the intercepts. Find the x- and y- intercepts and then a third point.

Graph with Intercepts

Identify the x- and y- Intercepts on a Graph

In the following exercises, find the x- and y- intercepts on each graph.

Try it.

Solution

\((3,0),(0,3)\)

Try it.

Try it.

Solution

\((5,0),(0,-5)\)

Try it.

Try it.

Solution

\((-2,0),(0,-2)\)

Try it.

Try it.

Solution

\((-1,0),(0,1)\)

Try it.

Try it.

Solution

\((6,0),(0,3)\)

Try it.

Try it.

Solution

\((0,0)\)

Try it.

Find the x- and y- Intercepts from an Equation of a Line

In the following exercises, find the intercepts for each equation.

Try it.

\(x+y=4\)

Solution

\((4,0),(0,4)\)

Try it.

\(x+y=3\)

Try it.

\(x+y=-2\)

Solution

\((-2,0),(0,-2)\)

Try it.

\(x+y=-5\)

Try it.

\(x-y=5\)

Solution

\((5,0),(0,-5)\)

Try it.

\(x-y=1\)

Try it.

\(x-y=-3\)

Solution

\((-3,0),(0,3)\)

Try it.

\(x-y=-4\)

Try it.

\(x+2y=8\)

Solution

\((8,0),(0,4)\)

Try it.

\(x+2y=10\)

Try it.

\(3x+y=6\)

Solution

\((2,0),(0,6)\)

Try it.

\(3x+y=9\)

Try it.

\(x-3y=12\)

Solution

\((12,0),(0,-4)\)

Try it.

\(x-2y=8\)

Try it.

\(4x-y=8\)

Solution

\((2,0),(0,-8)\)

Try it.

\(5x-y=5\)

Try it.

\(2x+5y=10\)

Solution

\((5,0),(0,2)\)

Try it.

\(2x+3y=6\)

Try it.

\(3x-2y=12\)

Solution

\((4,0),(0,-6)\)

Try it.

\(3x-5y=30\)

Try it.

\(y=\frac{1}{3}x+1\)

Solution

\((-3,0),(0,1)\)

Try it.

\(y=\frac{1}{4}x-1\)

Try it.

\(y=\frac{1}{5}x+2\)

Solution

\((-10,0),(0,2)\)

Try it.

\(y=\frac{1}{3}x+4\)

Try it.

\(y=3x\)

Solution

\((0,0)\)

Try it.

\(y=-2x\)

Try it.

\(y=-4x\)

Solution

\((0,0)\)

Try it.

\(y=5x\)

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

Try it.

\(-x+5y=10\)

Solution

Try it.

\(-x+4y=8\)

Try it.

\(x+2y=4\)

Solution

Try it.

\(x+2y=6\)

Try it.

\(x+y=2\)

Solution

Try it.

\(x+y=5\)

Try it.

\(x+y=-3\)

Solution

Try it.

\(x+y=-1\)

Try it.

\(x-y=1\)

Solution

Try it.

\(x-y=2\)

Try it.

\(x-y=-4\)

Solution

Try it.

\(x-y=-3\)

Try it.

\(4x+y=4\)

Solution

Try it.

\(3x+y=3\)

Try it.

\(2x+4y=12\)

Solution

Try it.

\(3x+2y=12\)

Try it.

\(3x-2y=6\)

Solution

Try it.

\(5x-2y=10\)

Try it.

\(2x-5y=-20\)

Solution

Try it.

\(3x-4y=-12\)

Try it.

\(3x-y=-6\)

Solution

Try it.

\(2x-y=-8\)

Try it.

\(y=-2x\)

Solution

Try it.

\(y=-4x\)

Try it.

\(y=x\)

Solution

Try it.

\(y=3x\)

Try it.

Road trip. Damien is driving from Chicago to Denver, a distance of 1000 miles. The x- axis on the graph below shows the time in hours since Damien left Chicago. The y- axis represents the distance he has left to drive.

  1. ⓐ Find the x- and y- intercepts.
  2. ⓑ Explain what the x- and y- intercepts mean for Damien.
Solution

ⓐ \((0,1000),(15,0)\)
ⓑ At \((0,1000)\), he has been gone 0 hours and has 1000 miles left. At \((15,0)\), he has been gone 15 hours and has 0 miles left to go.

Try it.

Road trip. Ozzie filled up the gas tank of his truck and headed out on a road trip. The x- axis on the graph below shows the number of miles Ozzie drove since filling up. The y- axis represents the number of gallons of gas in the truck’s gas tank.

  1. ⓐ Find the x- and y- intercepts.
  2. ⓑ Explain what the x- and y- intercepts mean for Ozzie.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (38)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Solve: \(3\cdot 0+4y=-2.\)
    If you missed this problem, review .

    ഉത്തരം വെളിപ്പെടുത്തുക

    \(-\frac{1}{2}\)

  2. Find the x- and y- intercepts on each graph.

    ഉത്തരം വെളിപ്പെടുത്തുക
    1. ⓐ The graph crosses the x- axis at the point \((4,0)\). The x- intercept is \((4,0)\).
      The graph crosses the y- axis at the point \((0,2)\). The y- intercept is \((0,2)\).

    2. ⓑ The graph crosses the x- axis at the point \((2,0)\). The x- intercept is \((2,0)\)
      The graph crosses the y- axis at the point \((0,-6)\). The y- intercept is \((0,-6)\).

    3. ⓒ The graph crosses the x- axis at the point \((-5,0)\). The x- intercept is \((-5,0)\).
      The graph crosses the y- axis at the point \((0,-5)\). The y- intercept is \((0,-5)\).

  3. Find the x- and y- intercepts on the graph.

    ഉത്തരം വെളിപ്പെടുത്തുക

    x- intercept: \((2,0)\); y- intercept: \((0,-2)\)

  4. Find the x- and y- intercepts on the graph.

    ഉത്തരം വെളിപ്പെടുത്തുക

    x- intercept: \((3,0)\), y- intercept: \((0,2)\)

  5. Find the intercepts of \(2x+y=6\).

    ഉത്തരം വെളിപ്പെടുത്തുക

    We will let \(y=0\) to find the x- intercept, and let \(x=0\) to find the y- intercept. We will fill in the table, which reminds us of what we need to find.

    To find the x- intercept, let \(y=0\).

    Let y = 0.
    Simplify.
    The x-intercept is(3, 0)
    To find the y-intercept, let x = 0.
    Let x = 0.
    Simplify.
    The y-intercept is(0, 6)

    The intercepts are the points \((3,0)\) and \((0,6)\) as shown in .

    \(2x+y=6\)
    \(x\)\(y\)
    30
    06

  6. Find the intercepts of \(3x+y=12.\)

    ഉത്തരം വെളിപ്പെടുത്തുക

    x- intercept: \((4,0)\), y- intercept: \((0,12)\)

  7. Find the intercepts of \(x+4y=8.\)

    ഉത്തരം വെളിപ്പെടുത്തുക

    x- intercept: \((8,0)\), y- intercept: \((0,2)\)

  8. Find the intercepts of \(4x-3y=12\).

    ഉത്തരം വെളിപ്പെടുത്തുക

    To find the x-intercept, let y = 0.
    Let y = 0.
    Simplify.
    The x-intercept is(3, 0)
    To find the y-intercept, let x = 0.
    Let x = 0.
    Simplify.
    The y-intercept is(0, −4)

    The intercepts are the points (3, 0) and (0, −4) as shown in the following table.

    \(4x-3y=12\)
    \(x\)\(y\)
    30
    0\(-4\)

  9. Find the intercepts of \(3x-4y=12.\)

    ഉത്തരം വെളിപ്പെടുത്തുക

    x- intercept: \((4,0)\), y- intercept: \((0,-3)\)

  10. Find the intercepts of \(2x-4y=8.\)

    ഉത്തരം വെളിപ്പെടുത്തുക

    x- intercept: \((4,0)\), y- intercept: \((0,-2)\)

  11. Graph \(-x+2y=6\) using the intercepts.

  12. Graph \(x-2y=4\) using the intercepts.

    ഉത്തരം വെളിപ്പെടുത്തുക

  13. Graph \(-x+3y=6\) using the intercepts.

    ഉത്തരം വെളിപ്പെടുത്തുക

  14. Graph \(4x-3y=12\) using the intercepts.

    ഉത്തരം വെളിപ്പെടുത്തുക

    Find the intercepts and a third point.

    We list the points in and show the graph below.

    \(4x-3y=12\)
    \(x\)\(y\)\((x,y)\)
    30\((3,0)\)
    0\(-4\)\((0,-4)\)
    64\((6,4)\)

  15. Graph \(5x-2y=10\) using the intercepts.

    ഉത്തരം വെളിപ്പെടുത്തുക

  16. Graph \(3x-4y=12\) using the intercepts.

    ഉത്തരം വെളിപ്പെടുത്തുക

  17. Graph \(y=5x\) using the intercepts.

    ഉത്തരം വെളിപ്പെടുത്തുക

    This line has only one intercept. It is the point \((0,0)\).

    To ensure accuracy we need to plot three points. Since the x- and y- intercepts are the same point, we need two more points to graph the line.

    See .

    \(y=5x\)
    \(x\)\(y\)\((x,y)\)
    00\((0,0)\)
    15\((1,5)\)
    \(-1\)\(-5\)\((-1,-5)\)

    Plot the three points, check that they line up, and draw the line.

  18. Graph \(y=4x\) using the intercepts.

    ഉത്തരം വെളിപ്പെടുത്തുക

  19. Graph \(y=\text{-}x\) the intercepts.

    ഉത്തരം വെളിപ്പെടുത്തുക

  20. \(2x+5y=10\)

    ഉത്തരം വെളിപ്പെടുത്തുക

    \((5,0),(0,2)\)

  21. \(3x-2y=12\)

    ഉത്തരം വെളിപ്പെടുത്തുക

    \((4,0),(0,-6)\)

  22. \(3x-5y=30\)

  23. \(y=\frac{1}{3}x+1\)

    ഉത്തരം വെളിപ്പെടുത്തുക

    \((-3,0),(0,1)\)

  24. \(y=\frac{1}{4}x-1\)

  25. \(y=\frac{1}{5}x+2\)

    ഉത്തരം വെളിപ്പെടുത്തുക

    \((-10,0),(0,2)\)

  26. \(y=\frac{1}{3}x+4\)

  27. \(-x+5y=10\)

    ഉത്തരം വെളിപ്പെടുത്തുക

  28. \(2x+4y=12\)

    ഉത്തരം വെളിപ്പെടുത്തുക

  29. \(3x+2y=12\)

  30. \(5x-2y=10\)

  31. \(2x-5y=-20\)

    ഉത്തരം വെളിപ്പെടുത്തുക

  32. \(3x-4y=-12\)

  33. Road trip. Damien is driving from Chicago to Denver, a distance of 1000 miles. The x- axis on the graph below shows the time in hours since Damien left Chicago. The y- axis represents the distance he has left to drive.

    1. ⓐ Find the x- and y- intercepts.
    2. ⓑ Explain what the x- and y- intercepts mean for Damien.
    ഉത്തരം വെളിപ്പെടുത്തുക

    ⓐ \((0,1000),(15,0)\)
    ⓑ At \((0,1000)\), he has been gone 0 hours and has 1000 miles left. At \((15,0)\), he has been gone 15 hours and has 0 miles left to go.

  34. Road trip. Ozzie filled up the gas tank of his truck and headed out on a road trip. The x- axis on the graph below shows the number of miles Ozzie drove since filling up. The y- axis represents the number of gallons of gas in the truck’s gas tank.

    1. ⓐ Find the x- and y- intercepts.
    2. ⓑ Explain what the x- and y- intercepts mean for Ozzie.

  35. How do you find the x- intercept of the graph of \(3x-2y=6\)?

    ഉത്തരം വെളിപ്പെടുത്തുക

    Answers will vary.

  36. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation \(4x+y=-4\)? Why?

  37. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation \(y=\frac{2}{3}x-2\)? Why?

    ഉത്തരം വെളിപ്പെടുത്തുക

    Answers will vary.

  38. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation \(y=6\)? Why?

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Graph with Intercepts

  1. Identify the x - and y - intercepts on a graph
  2. Find the x - and y - intercepts from an equation of a line
  3. Graph a line using the intercepts
  4. the
  5. the
  6. Find the
  7. Let
  8. Let

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

നീ സ്വയം ശ്രമിക്ക്.

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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