Graphing with Intercepts

Identify the Intercepts on a Graph

Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.

At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an intercept of the line.

Let’s look at the graph of the lines

First, notice where each of these lines crosses the x- axis:

Figure:	The line crosses the x-axis at:	Ordered pair of this point
42	3	(3,0)
43	4	(4,0)
44	5	(5,0)
45	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 the line. The x- intercept occurs when y is zero.

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

Figure:	The line crosses the y-axis at:	Ordered pair for this point
42	6	(0,6)
43	-3	(0,-3)
44	-5	(0,-5)
45	0	(0,0)

Example

Try it.

Find the \(x\text{- and}\ y\text{-intercepts}\) of each line:

ⓐ \(\ x+2y=4\)
ⓑ \(\ 3x-y=6\)
ⓒ \(\ x+y=-5\)

Solution

ⓐ
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).

ⓑ
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).

ⓒ
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).

Find the Intercepts from an Equation of a Line

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

Example

Try it.

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

Solution

We'll fill in .

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

Substitute 0 for y.
Add.
Divide by 2.
The x-intercept is (3, 0).

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

Substitute 0 for x.
Multiply.
Add.
The y-intercept is (0, 6).

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

Example

Try it.

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

Solution

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

	\(4x-3y=12\)
Substitute 0 for \(y.\)	\(4x-3\cdot 0=12\)
Multiply.	\(4x-0=12\)
Subtract.	\(4x=12\)
Divide by 4.	\(x=3\)

The \(x\text{-intercept}\) is \((3,0).\)

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

	\(4x-3y=12\)
Substitute 0 for \(x.\)	\(4\cdot 0-3y=12\)
Multiply.	\(0-3y=12\)
Simplify.	\(-3y=12\)
Divide by −3.	\(y=-4\)

The \(y\text{-intercept}\) is \((0,-4).\)

The intercepts are the points \((-3,0)\) and \((0,-4).\)

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

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

Example

Try it.

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

Solution

First, find the \(x\text{-intercept}.\) Let \(y=0,\)

\(\begin{array}{l} \\ -x+2y=6 \\ -x+2(0)=6 \\ -x=6 \\ x=-6\end{array}\)

The \(x\text{-intercept}\) is \((-6,0).\)

Now find the \(y\text{-intercept}.\) Let \(x=0.\)

\(\begin{array}{l} \\ -x+2y=6 \\ -0+2y=6 \\ \\ \\ 2y=6 \\ y=3\end{array}\)

The \(y\text{-intercept}\) is \((0,3).\)

Find a third point. We’ll use \(x=2,\)

\(\begin{array}{l} \\ -x+2y=6 \\ -2+2y=6 \\ \\ \\ 2y=8 \\ y=4\end{array}\)

A third solution to the equation is \((2,4).\)

Summarize the three points in a table and then plot them on a graph.

\(-x+2y=6\)
x	y	(x,y)
\(-6\)	\(0\)	\((-6,0)\)
\(0\)	\(3\)	\((0,3)\)
\(2\)	\(4\)	\((2,4)\)

Do the points line up? Yes, so draw line through the points.

Example

Try it.

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

Solution

Find the intercepts and a third point.

We list the points and show the graph.

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

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 intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for \(x,\) so we’ll let \(x\) be \(1\) and \(-1.\)

Organize the points in a table.

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

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

Choose the Most Convenient Method to Graph a Line

While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.

	Equation	Method
#1	\(y=2x+1\)	Plotting points
#2	\(y=\frac{1}{2}x+3\)	Plotting points
#3	\(x=-7\)	Vertical line
#4	\(y=4\)	Horizontal line
#5	\(2x+y=6\)	Intercepts
#6	\(4x-3y=12\)	Intercepts

What is it about the form of equation that can help us choose the most convenient method to graph its line?

Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values.

Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form \(Ax+By=C\). We substituted \(y=0\) and \(x=0\) to find the x- and y- intercepts, and then found a third point by choosing a value for x or y.

This leads to the following strategy for choosing the most convenient method to graph a line.

Example

Try it.

Identify the most convenient method to graph each line:

1. ⓐ \(\ y=-3\)
2. ⓑ \(\ 4x-6y=12\)
3. ⓒ \(\ x=2\)
4. ⓓ \(\ y=\frac{2}{5}x-1\)

Solution

ⓐ \(\ y=-3\)

This equation has only one variable, \(y.\) Its graph is a horizontal line crossing the \(y\text{-axis}\) at \(-3.\)

ⓑ \(\ 4x-6y=12\)

This equation is of the form \(Ax+By=C.\) Find the intercepts and one more point.

ⓒ \(\ x=2\)

There is only one variable, \(x.\) The graph is a vertical line crossing the \(x\text{-axis}\) at \(2.\)

ⓓ \(\ y=\frac{2}{5}x-1\)

Since \(y\) is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.

Condensed: the full section is in OpenStax Prealgebra 2e.

Key Concepts

• Intercepts
  • The x-intercept is the point, \((a,0)\), where the graph crosses the x-axis. The x-intercept occurs when y is zero.
  • The y-intercept is the point, \((0,b)\), where the graph crosses the y-axis. The y-intercept occurs when x is zero.
  • The x-intercept occurs when y is zero.
  • The y-intercept occurs when x is zero.
• Find the x and y intercepts from the equation of a line
  • To find the x-intercept of the line, let \(y=0\) and solve for x.
  • To find the y-intercept of the line, let \(x=0\) and solve for y.

    x	y
    	0
    0

• Graph a line 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.
• Choose the most convenient method to graph a line
  1. Determine if the equation has only one variable. Then 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.
  2. Determine if y is isolated on one side of the equation. The graph by plotting points.
     Choose any three values for x and then solve for the corresponding y- values.
  3. Determine if the equation is of the form \(Ax+By=C\), find the intercepts.
     Find the x- and y- intercepts and then a third point.

Graphing with Intercepts

Identify the Intercepts on a Graph

In the following exercises, find the \(x\text{-}\) and \(y\text{-}\) intercepts.

Find the \(x\) and \(y\) Intercepts from an Equation of a Line

In the following exercises, find the intercepts.

Try it.

\(2x+5y=10\)

Solution

(5,0),(0,2)

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.

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

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

Try it.

\(-x+5y=10\)

Solution

Try it.

\(2x+4y=12\)

Solution

Try it.

\(3x+2y=12\)

Try it.

\(5x-2y=10\)

Try it.

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

Solution

Try it.

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

Choose the Most Convenient Method to Graph a Line

In the following exercises, identify the most convenient method to graph each line.

Try it.

\(y=\frac{2}{3}x-1\)

Solution

plotting points

Try it.

\(y=\frac{4}{5}x-3\)

Try it.

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

Solution

intercepts

Try it.

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

Try it.

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

Solution

plotting points

Try it.

\(y=-\frac{1}{3}x+5\)

Try it.

Road trip Damien is driving from Chicago to Denver, a distance of \(1,000\) miles. The \(x\text{-axis}\) on the graph below shows the time in hours since Damien left Chicago. The \(y\text{-axis}\) represents the distance he has left to drive.

ⓐ Find the \(x\text{-}\) and \(y\text{-}\) intercepts

ⓑ Explain what the \(x\text{-}\) and \(y\text{-}\) intercepts mean for Damien.

Solution

ⓐ (0,1,000),(15,0). ⓑ At (0,1,000) he left Chicago 0 hours ago and has 1,000 miles left to drive. At (15,0) he left Chicago 15 hours ago and has 0 miles left to drive.

Try it.

Road trip Ozzie filled up the gas tank of his truck and went on a road trip. The \(x\text{-axis}\) on the graph shows the number of miles Ozzie drove since filling up. The \(y\text{-axis}\) represents the number of gallons of gas in the truck’s gas tank.

ⓐ Find the \(x\text{-}\) and \(y\text{-}\) intercepts.

ⓑ Explain what the \(x\text{-}\) and \(y\text{-}\) intercepts mean for Ozzie.

Try it.

How do you find the \(x\text{-intercept}\) of the graph of \(3x-2y=6?\)

Solution

Answers will vary.

Try it.

How do you find the \(y\text{-intercept}\) of the graph of \(5x-y=10?\)

Try it.

Do you prefer to graph the equation \(4x+y=-4\) by plotting points or intercepts? Why?

Solution

Answers will vary.

Try it.

Do you prefer to graph the equation \(y=\frac{2}{3}x-2\) by plotting points or intercepts? Why?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

이제 너 계산기는 이것을 해결하지 않지만, 그 조각은 계산 가능합니다. 아래의 하나를 시도하거나 자신의 것을 입력하십시오.

⌨ 종류 ✎ 그려

수학 문제

π 해결

연습 (40)

먼저 종이에 각각을 시도해 보세요. 확인하기 위해 답을 드러내세요. 확인된 답은 각 단계의 솔버에서 열 수 있습니다.

1. Solve: \(3x+4y=-12\) for \(x\) when \(y=0.\)

   끝내주는 걸 힌트

   답을 드러내세요

   \(-4\)

2. Is the point \((0,-5)\) on the \(x\text{-axis}\) or \(y\text{-axis?}\)

   끝내주는 걸 힌트

   답을 드러내세요

   \(y\text{-axis}\)

3. Which ordered pairs are solutions to the equation \(2x-y=6?\)
   ⓐ \(\ (6,0)\)ⓑ \(\ (0,-6)\)ⓒ \(\ (4,-2).\)

   끝내주는 걸 힌트

   답을 드러내세요

   b

4. Find the \(x\text{- and}\ y\text{-intercepts}\) of each line:

   ⓐ \(\ x+2y=4\)
   ⓑ \(\ 3x-y=6\)
   ⓒ \(\ x+y=-5\)

   끝내주는 걸 힌트

   답을 드러내세요

   ⓐ
   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).

   ⓑ
   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).

   ⓒ
   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).

5. Find the \(x\text{-}\) and \(y\text{-intercepts}\) of the graph: \(x-y=2.\)

   끝내주는 걸 힌트

   답을 드러내세요

   x-intercept (2,0): y-intercept (0,−2)

6. Find the \(x\text{-}\) and \(y\text{-intercepts}\) of the graph: \(2x+3y=6.\)

   끝내주는 걸 힌트

   답을 드러내세요

   x-intercept (3,0); y-intercept (0,2)

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

   끝내주는 걸 힌트

   답을 드러내세요

   We'll fill in .

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

   Substitute 0 for y.
   Add.
   Divide by 2.
   The x-intercept is (3, 0).

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

   Substitute 0 for x.
   Multiply.
   Add.
   The y-intercept is (0, 6).

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

8. Find the intercepts: \(3x+y=12\)

   끝내주는 걸 힌트

   답을 드러내세요

   \((4,0)\) and \((0,12)\)

9. Find the intercepts: \(x+4y=8\)

   끝내주는 걸 힌트

   답을 드러내세요

   \((8,0)\) and \((0,2)\)

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

    끝내주는 걸 힌트

    답을 드러내세요

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

    	\(4x-3y=12\)
    Substitute 0 for \(y.\)	\(4x-3\cdot 0=12\)
    Multiply.	\(4x-0=12\)
    Subtract.	\(4x=12\)
    Divide by 4.	\(x=3\)

    The \(x\text{-intercept}\) is \((3,0).\)

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

    	\(4x-3y=12\)
    Substitute 0 for \(x.\)	\(4\cdot 0-3y=12\)
    Multiply.	\(0-3y=12\)
    Simplify.	\(-3y=12\)
    Divide by −3.	\(y=-4\)

    The \(y\text{-intercept}\) is \((0,-4).\)

    The intercepts are the points \((-3,0)\) and \((0,-4).\)

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

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

    끝내주는 걸 힌트

    답을 드러내세요

    x-intercept (4,0); y-intercept: (0,−3)

12. Find the intercepts of the line: \(2x-4y=8.\)

    끝내주는 걸 힌트

    답을 드러내세요

    x-intercept (4,0); y-intercept: (0,−2)

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

    끝내주는 걸 힌트

    답을 드러내세요

    First, find the \(x\text{-intercept}.\) Let \(y=0,\)

    \(\begin{array}{l} \\ -x+2y=6 \\ -x+2(0)=6 \\ -x=6 \\ x=-6\end{array}\)

    The \(x\text{-intercept}\) is \((-6,0).\)

    Now find the \(y\text{-intercept}.\) Let \(x=0.\)

    \(\begin{array}{l} \\ -x+2y=6 \\ -0+2y=6 \\ \\ \\ 2y=6 \\ y=3\end{array}\)

    The \(y\text{-intercept}\) is \((0,3).\)

    Find a third point. We’ll use \(x=2,\)

    \(\begin{array}{l} \\ -x+2y=6 \\ -2+2y=6 \\ \\ \\ 2y=8 \\ y=4\end{array}\)

    A third solution to the equation is \((2,4).\)

    Summarize the three points in a table and then plot them on a graph.

    \(-x+2y=6\)
    x	y	(x,y)
    \(-6\)	\(0\)	\((-6,0)\)
    \(0\)	\(3\)	\((0,3)\)
    \(2\)	\(4\)	\((2,4)\)

    Do the points line up? Yes, so draw line through the points.

14. Graph the line using the intercepts: \(x-2y=4.\)

    끝내주는 걸 힌트

15. Graph the line using the intercepts: \(-x+3y=6.\)

    끝내주는 걸 힌트

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

    끝내주는 걸 힌트

    답을 드러내세요

    Find the intercepts and a third point.

    We list the points and show the graph.

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

17. Graph the line using the intercepts: \(5x-2y=10.\)

    끝내주는 걸 힌트

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

    끝내주는 걸 힌트

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

    끝내주는 걸 힌트

    답을 드러내세요

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

    To ensure accuracy, we need to plot three points. Since the intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for \(x,\) so we’ll let \(x\) be \(1\) and \(-1.\)

    Organize the points in a table.

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

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

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

    끝내주는 걸 힌트

21. Graph using the intercepts: \(y=-x.\)

    끝내주는 걸 힌트

22. Identify the most convenient method to graph each line:

    1. ⓐ \(\ y=-3\)
    2. ⓑ \(\ 4x-6y=12\)
    3. ⓒ \(\ x=2\)
    4. ⓓ \(\ y=\frac{2}{5}x-1\)

    끝내주는 걸 힌트

    답을 드러내세요

    ⓐ \(\ y=-3\)

    This equation has only one variable, \(y.\) Its graph is a horizontal line crossing the \(y\text{-axis}\) at \(-3.\)

    ⓑ \(\ 4x-6y=12\)

    This equation is of the form \(Ax+By=C.\) Find the intercepts and one more point.

    ⓒ \(\ x=2\)

    There is only one variable, \(x.\) The graph is a vertical line crossing the \(x\text{-axis}\) at \(2.\)

    ⓓ \(\ y=\frac{2}{5}x-1\)

    Since \(y\) is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.

23. Identify the most convenient method to graph each line:

    1. ⓐ \(\ 3x+2y=12\)
    2. ⓑ \(\ y=4\)
    3. ⓒ \(\ y=\frac{1}{5}x-4\)
    4. ⓓ \(\ x=-7\)
    끝내주는 걸 힌트

    답을 드러내세요

    1. ⓐ intercepts
    2. ⓑ horizontal line
    3. ⓒ plotting points
    4. ⓓ vertical line

24. Identify the most convenient method to graph each line:

    1. ⓐ \(\ x=6\)
    2. ⓑ \(\ y=-\frac{3}{4}x+1\)
    3. ⓒ \(\ y=-8\)
    4. ⓓ \(\ 4x-3y=-1\)
    끝내주는 걸 힌트

    답을 드러내세요

    1. ⓐ vertical line
    2. ⓑ plotting points
    3. ⓒ horizontal line
    4. ⓓ intercepts

25. \(2x+5y=10\)

    끝내주는 걸 힌트

    답을 드러내세요

    (5,0),(0,2)

26. \(3x-2y=12\)

    끝내주는 걸 힌트

    답을 드러내세요

    (4,0),(0,−6)

27. \(3x-5y=30\)

    끝내주는 걸 힌트

28. \(y=\frac{1}{3}x-1\)

    끝내주는 걸 힌트

    답을 드러내세요

    (3,0),(0,−1)

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

    끝내주는 걸 힌트

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

    끝내주는 걸 힌트

    답을 드러내세요

    (−10,0),(0,2)

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

    끝내주는 걸 힌트

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

    끝내주는 걸 힌트

33. \(2x+4y=12\)

    끝내주는 걸 힌트

34. \(3x+2y=12\)

    끝내주는 걸 힌트

35. \(5x-2y=10\)

    끝내주는 걸 힌트

36. \(2x-5y=-20\)

    끝내주는 걸 힌트

37. \(3x-4y=-12\)

    끝내주는 걸 힌트

38. \(y=\frac{2}{3}x-1\)

    끝내주는 걸 힌트

    답을 드러내세요

    plotting points

39. \(y=\frac{4}{5}x-3\)

    끝내주는 걸 힌트

40. \(3x-2y=-12\)

    끝내주는 걸 힌트

    답을 드러내세요

    intercepts

자신의 작업을 유지

무료 계정은 모든 수업에 노트를 추가, 당신이 완료 한 기록, 당신이 해결 한 문제를 한 곳에, 그리고 당신이이 페이지에 대해 물어 수업 지도자. 수학 자체는 모두에게 열려 있습니다, 로그인 여부.

가입하기 로그인

여기서 사용된 기호

기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.

\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\sqrt{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.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

어떻게: Graphing with Intercepts

1. Identify the intercepts on a graph
2. Find the intercepts from an equation of a line
3. Graph a line using the intercepts
4. Choose the most convenient method to graph a line
5. the
6. the
7. Find the
8. Let

사람들이 묻는 질문

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.

이 페이지의 일부는 다음에서 변경되었습니다. OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). 여기서 압축하고 다시 설명; 오류는 우리의.

에 더 Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value