Build function

We offer to your attention a service for constructing graphs of functions online, all rights to which belong to the company Desmos. Use the left column to enter functions. You can enter manually or using the virtual keyboard at the bottom of the window. To enlarge the window with the graph, you can hide both the left column and the virtual keyboard.

Benefits of online charting

• Visual display of entered functions
• Building very complex graphs
• Construction of graphs specified implicitly (for example, ellipse x^2/9+y^2/16=1)
• The ability to save charts and receive a link to them, which becomes available to everyone on the Internet
• Control of scale, line color
• Possibility of plotting graphs by points, using constants
• Plotting several function graphs simultaneously
• Plotting in polar coordinates (use r and θ(\theta))

With us it’s easy to build charts of varying complexity online. Construction is done instantly. The service is in demand for finding intersection points of functions, for depicting graphs for further moving them into a Word document as illustrations when solving problems, and for analyzing the behavioral features of function graphs. The optimal browser for working with charts on this website page is Google Chrome. Correct operation is not guaranteed when using other browsers.

Graphing an equation is much easier than many people think. You don't have to be a math genius or first in your math class to understand the basic principles of this process. This article describes how to graph linear and quadratic equations and inequalities, as well as equations with moduli.

Steps

Linear equation graph

Use the formula y=mx+b. To graph a linear equation, you simply plug the values ​​into the formula.

• This formula establishes the relationship between variables x And y.
• Parameter m corresponds to the slope of the line. In other words, m indicates the rate of increase (or decrease) y with change x.
• Parameter b indicates where the line corresponding to the equation intersects the axis y.

1. Make a graph. The linear equation is depicted most simply, since there is no need to calculate anything before plotting the graph. First, construct a rectangular coordinate system.

   Find the point where the line intersects the axis y(This b). For example, in the case of the equation y=2x-1 parameter b is equal to -1, that is, the line intersects the axis y at point -1.

   • At the point of intersection of the axis y coordinate x always takes the value 0. Thus, in our example, the intersection point has coordinates (0,-1).
   • Mark on the graph the point where the line intersects the axis y.

2. Find the slope of the line. For a straight line, the slope corresponds to the parameter m. In the case of the equation y=2x-1 this parameter is equal to 2. However, note that the slope indicates a change y with growth x, that is, it should be represented as a fraction. Since at the coordinate x costs the integer 2, you can write the slope as 2/1.

   • To graph slope, start at the axis intercept y. In this case, the change in coordinates y corresponds to the numerator, and the change in coordinate x- the denominator of the fraction.
   • In our example, you can start from point -1 and move from it up by 2 and to the right by 1.
   • A positive slope means that with growth x you're climbing up y, while with a negative slope y decreases. Variable x increases to the right along the horizontal axis and decreases to the left.
   • When determining the slope, you can use as many points as you like, although one point is sufficient.

3. Draw a straight line. After you determine the slope of the line and plot at least one point, you can connect it to the intersection point of the axis y and draw a straight line. Extend the line to the edges of the graph and draw arrows at the ends to indicate that it continues further.

   Inequality graph with one variable

   1. Draw a number line. Since a single axis is sufficient to depict an inequality with one variable, there is no need to draw a rectangular coordinate system. Instead, just draw a straight line.

      Draw the inequality. This is quite simple since there is only one coordinate. Suppose we need to represent the inequality x<1. Для начала следует найти на оси число 1.

      Draw a line. Draw a line from the point you just marked on the number line. If the variable is greater than this number, move the line to the right. If the variable is smaller, draw a line to the left. Place an arrow at the end of the line to show that it is not a final segment and continues further.

      Check the answer. Substitute for the variable x any number and mark its position on the number line. If this number lies on the line you drew, the graph is correct.

   Linear inequality graph

   Use the straight line formula. A similar formula was used above for ordinary linear equations, but in this case, instead of the ‘=’ sign, you should put an inequality sign. This may be one of the following signs:<, >, ≤ (\displaystyle \leq ) or ≥ (\displaystyle \geq ).

   • The equation of a straight line is y=mx+b, Where m corresponds to the slope, and b- intersection with the axis y.
   • The inequality sign means that this expression has multiple solutions.

   1. Draw the inequality. Find the point of intersection of the line with the axis y and its slope, then mark the corresponding coordinates. As an example, consider the inequality y>1/2x+1. In this case, the straight line will intersect the axis y at x=1, and its slope will be ½, that is, when moving to the right by 2 units, we will rise up by 1 unit.

      Draw a line. Before doing this, look at the inequality sign. If this< или >, you should draw a dotted line. If the inequality contains the sign ≤ (\displaystyle \leq ) or ≥ (\displaystyle \geq ), the line must be solid.

      Shade the graph. Since the inequality has many solutions, the graph should show all possible solutions. This means shade the area above or below the line.

   Quadratic Equation Graph

   Look at the formula. In a quadratic equation, at least one variable is squared. Typically, a quadratic equation is written as follows: y=ax 2 +bx+c.

   • When you graph a quadratic equation, you will get a parabola, that is, a curve in the shape of the Latin letter ‘U’.
   • To construct a parabola, you need to know the coordinates of at least three points, including the vertex of the parabola (its central point).

   1. Define a, b And c. For example, in Eq. y=x 2 +2x+1 a=1, b=2 and c=1. Each parameter is a number that precedes the variable to the corresponding power. For example, if before x doesn't cost any number, it means b=1, since the corresponding term can be written in the form 1 x.

      Find the vertex of the parabola. To find the midpoint of a parabola, use the expression -b/2a. For our example, we get -2/2(1), that is -1.

      Make a table. So we know that the coordinate x vertices is equal to -1. However, this is only one coordinate. To find the corresponding coordinate y, as well as the other two points of the parabola, you need to make a table.

      Construct a table of three rows and two columns.

      • Write down the coordinate x the vertices of the parabola in the central cell of the left column.
      • Select two more coordinates x at the same distance to the left and right (in the negative and positive directions along the horizontal axis). For example, you can move 2 units to the left and right from the vertex, that is, write -3 and 1 in the corresponding cells.
      • You can choose any integers that are equidistant from the vertex.
      • If you want to build a more accurate graph, you can use five points instead of three. In this case, you should do the same, only the table will consist not of three, but of five rows.

   2. Use an equation and table to find unknown coordinates y. Take one x coordinate at a time from the table, substitute it into the given equation and find the corresponding y coordinate.

      • In our case, we substitute into the equation y=x 2 +2x+1 instead x-3. As a result we find y= -3 2 +2(-3)+1, that is y=4.
      • We write down the found coordinate y in a cell near its corresponding coordinate x.
      • Find all three (or five if you use more points) coordinates this way y.

   3. Plot the points on the graph. So, you now have at least three points with known coordinates that can be marked on the graph. Connect them with a parabola-shaped curve. Ready!

A rectangular coordinate system is a pair of perpendicular coordinate lines, called coordinate axes, that are placed so that they intersect at their origin.

The designation of coordinate axes by the letters x and y is generally accepted, but the letters can be any. If the letters x and y are used, then the plane is called xy-plane. Different applications may use letters other than x and y, and as shown in the figures below, there are uv plane And ts-plane.

Ordered pair

By ordered pair of real numbers, we mean two real numbers in a certain order. Each point P in the coordinate plane can be associated with a unique ordered pair of real numbers by drawing two lines through P: one perpendicular to the x-axis and the other perpendicular to the y-axis.

For example, if we take (a,b)=(4,3), then on the coordinate strip

To construct a point P(a,b) means to determine a point with coordinates (a,b) on the coordinate plane. For example, various points are plotted in the figure below.

In a rectangular coordinate system, the coordinate axes divide the plane into four regions called quadrants. They are numbered counterclockwise with Roman numerals, as shown in the figure.

Definition of a graph

Schedule equation with two variables x and y, is the set of points on the xy-plane whose coordinates are members of the set of solutions to this equation

Example: draw a graph of y = x 2

Because 1/x is undefined when x=0, we can only plot points for which x ≠0

Example: Find all intersections with axes
(a) 3x + 2y = 6
(b) x = y 2 -2y
(c) y = 1/x

Let y = 0, then 3x = 6 or x = 2

is the desired x-intercept.

Having established that x=0, we find that the point of intersection of the y-axis is the point y=3.

This way you can solve equation (b) and the solution for (c) is given below

x-intercept

Let y = 0

1/x = 0 => x cannot be determined, i.e. there is no intersection with the y-axis

Let x = 0

y = 1/0 => y is also undefined, => no intersection with the y axis

In the figure below, the points (x,y), (-x,y), (x,-y) and (-x,-y) represent the corners of the rectangle.

A graph is symmetrical about the x-axis if for every point (x,y) on the graph, point (x,-y) is also a point on the graph.

A graph is symmetrical about the y-axis if for every point on the graph (x,y), point (-x,y) also belongs to the graph.

A graph is symmetrical about the center of coordinates if for each point (x,y) on the graph, point (-x,-y) also belongs to this graph.

Definition:

Schedule functions on the coordinate plane is defined as the graph of the equation y = f(x)

Plot f(x) = x + 2

Example 2. Plot a graph of f(x) = |x|

The graph coincides with the line y = x for x > 0 and with line y = -x

for x< 0 .

graph of f(x) = -x

Combining these two graphs we get

graph f(x) = |x|

Example 3: Plot a graph

t(x) = (x 2 - 4)/(x - 2) =

= ((x - 2)(x + 2)/(x - 2)) =

= (x + 2) x ≠ 2

Therefore, this function can be written as

y = x + 2 x ≠ 2

Graph h(x)= x 2 - 4 Or x - 2

graph y = x + 2 x ≠ 2

Example 4: Plot a graph

Graphs of functions with displacement

Suppose that the graph of the function f(x) is known

Then we can find the graphs

y = f(x) + c - graph of function f(x), moved

UP c values

y = f(x) - c - graph of function f(x), moved

DOWN by c values

y = f(x + c) - graph of function f(x), moved

LEFT by c values

y = f(x - c) - graph of the function f(x), moved

Right by c values

Example 5: Build

graph y = f(x) = |x - 3| + 2

Let's move the graph y = |x| 3 values ​​to the RIGHT to get the graph

Let's move the graph y = |x - 3| UP 2 values ​​to get the graph y = |x - 3| + 2

Plot a graph

y = x 2 - 4x + 5

Let's transform the given equation as follows, adding 4 to both sides:

y + 4 = (x 2 - 4x + 5) + 4 y = (x 2 - 4x + 4) + 5 - 4

y = (x - 2) 2 + 1

Here we see that this graph can be obtained by moving the graph of y = x 2 to the right by 2 values, because x - 2, and up by 1 value, because +1.

y = x 2 - 4x + 5

Reflections

(-x, y) is a reflection of (x, y) about the y-axis

(x, -y) is a reflection of (x, y) about the x axis

The graphs y = f(x) and y = f(-x) are reflections of each other relative to the y axis

The graphs y = f(x) and y = -f(x) are reflections of each other relative to the x-axis

The graph can be obtained by reflecting and moving:

Draw a graph

Let's find its reflection relative to the y-axis and get a graph

Let's move this graph right by 2 values ​​and we get a graph

Here is the graph you are looking for

If f(x) is multiplied by a positive constant c, then

the graph f(x) is compressed vertically if 0< c < 1

the graph f(x) is stretched vertically if c > 1

The curve is not a graph of y = f(x) for any function f

A linear equation in two variables is any equation that has the following form: a*x + b*y =с. Here x and y are two variables, a,b,c are some numbers.

The solution to the linear equation a*x + b*y = c is any pair of numbers (x,y) that satisfies this equation, that is, turns the equation with variables x and y into a correct numerical equality. A linear equation has an infinite number of solutions.

If each pair of numbers that are solutions to a linear equation in two variables is depicted on the coordinate plane as points, then all these points form the graph of a linear equation in two variables. The coordinates of the points will be our x and y values. In this case, the x value will be the abscissa, and the y value will be the ordinate.

Graph of a Linear Equation in Two Variables

The graph of a linear equation with two variables is the set of all possible points on the coordinate plane, the coordinates of which will be solutions to this linear equation. It is easy to guess that the graph will be a straight line. That is why such equations are called linear.

Construction algorithm

Algorithm for plotting a linear equation in two variables.

1. Draw coordinate axes, label them and mark the unit scale.

2. In a linear equation, put x = 0, and solve the resulting equation for y. Mark the resulting point on the graph.

3. In a linear equation, take the number 0 as y, and solve the resulting equation for x. Mark the resulting point on the graph

4. If necessary, take an arbitrary value of x and solve the resulting equation for y. Mark the resulting point on the graph.

5. Connect the resulting points and continue the graph beyond them. Sign the resulting straight line.

Example: Graph the equation 3*x - 2*y =6;

Let's put x=0, then - 2*y =6; y= -3;

Let's put y=0, then 3*x = 6; x=2;

We mark the obtained points on the graph, draw a straight line through them and label it. Look at the figure below, the graph should look exactly like this.