PROJECT

  

functions

CONSTANT FUNCTION: A constant function is that function that always takes the same image for any value of the independent variable (x), that is, a constant function is of the form f(x)=k, where k is any real number.

Characteristics

• It is a continuous and even function, because the function always takes the same value.

• The constant function is neither increasing nor decreasing, it is a type of function that always has zero slope.

• The limit of the constant function as x approaches plus infinity or minus infinity is equal to the value of the constant.

Domain and range: For a constant function, the domain can be any set of real numbers. The range is simply the constant value of the function. In the example above, the domain is all real numbers, and the range is {3}.

Name of Graph: The graph of a constant function is a horizontal line that passes through the constant value on the y-axis.

Cuts in axes: The constant function does not have cuts in the x or y axes, since the line is parallel to the x axis and does not move vertically.

Increasing or decreasing: Does not apply to constant functions, since the function has the same value for all values ​​of x.

Restrictions: There are no restrictions on the constant function, since it is valid for all values ​​of x.

Steps to graph: Simply draw a horizontal line at the constant value on the y-axis. There is no need for further calculations.

In short, the constant function is a horizontal line in the Cartesian plane, where the value of the function is constant for all values ​​of x.

EXAMPLE OF CONSTANT FUNCTION GRAPH

𝑓(𝑥)=3

 

1. Draw two perpendicular lines on your paper or graph. The horizontal line is the x-axis and the vertical line is the y-axis. Mark the units on both axes to have a reference scale.

2. Find the value 3 on the 𝑦-axis. This is the point at which the function will remain constant. Mark a point at 𝑦=3 on the y-axis.

3. From the point marked at 𝑦=3, draw a horizontal line extending both to the left and to the right, parallel to the 𝑥-axis. This line represents all points (𝑥,3) for any value of x.

LINEAR FUNCTION: A linear function is defined by the equation 𝑓(𝑥)=𝑚𝑥+𝑏, where 𝑚 is the slope and 𝑏 is the independent term. The graph of a linear function is a straight line in the Cartesian plane.

 

Characteristics

• The graph of a linear function is a straight line.

• It is called “graphing the linear equation”.

• The slope (𝑚) indicates the inclination of the line, and the independent term (𝑏) indicates the point where the line crosses the 𝑦-axis.

 

Domain: The domain of a linear function is the set of all real numbers, since it is defined for all values ​​of 𝑥.

Range: The range of a linear function is also the set of all real numbers. It can be determined by noting that the function is a straight line that extends indefinitely in both directions.

Cuts in the axles:The cut on the 𝑥-axis is called the 𝑥-axis intersection or root, and it occurs when

𝑦=0.

The cut on the 𝑦-axis is called the 𝑦-axis intersection and occurs when 𝑥=0, and its value is 𝑏.

Increasing or decreasing: A linear function is increasing if its slope (𝑚) is positive, which means that the line rises to the right.

It is decreasing if its slope (𝑚) is negative, which means that the line falls to the right.

Steps to graph:

1.   Determine the slope (𝑚) and the independent term (𝑏). Mark the point (0,𝑏) on the 𝑦-axis.

2.   Use the slope to find another point. For example, if 𝑚=2 goes up 2 units and goes one unit to the right from the point (0,𝑏).

3.   Join the points with a straight line.

              EXAMPLE OF LINEAR FUNCTION GRAPH

𝑓(𝑥)=2𝑥+3

 

1. Identify the slope and the independent termThe slope (m) is 2

the independent term (b) is 3.

2. Find the point of intersection with the 𝑦-axis. 

The point of intersection with the 𝑦-axis is (0,3).

3. Find another point using the slope

Starting at the point (0,3), use the slope to find another point. Since the slope is 2, go up 2 units and go one unit to the right from the point (0.3). This gives us the point (1,5). 

4. Connect the points with a straight line

Join the points (0,3) and (1,5) with a straight line.

 

QUADRATIC FUNCTION: A quadratic function is a mathematical function that can be expressed in the general form 𝑓(𝑥)=𝑎x2+𝑏𝑥+𝑐, where 𝑎, 𝑏, and 𝑐 are constant coefficients and 𝑥 is the independent variable. The graph of a quadratic function is a parabola.

 

Characteristics

• The graph of a quadratic function is a parabola.

• It is called “graphing the quadratic equation”.

 

Domain: The domain of a quadratic function is the set of all real numbers, since it is defined for all values ​​of x.

 

Range: The range of a quadratic function depends on the opening direction of the parabola.

 

Cuts in the axles:The 𝑥-axis cut is the intersection with the 𝑥-axis and occurs where the parabola crosses the 𝑥-axis.

The cut on the 𝑦-axis is the intersection with the 𝑦-axis and occurs at the point (0,𝑐).

Increasing or decreasing:A quadratic function is increasing if the coefficient a is positive, which means that the parabola opens upward.

It is decreasing if the coefficient a is negative, which means that the parabola opens downwards.

Steps to graph:

    1. Find the vertex of the parabola.

    2. Find the intersections with the 𝑥 and 𝑦-axes.

    3. Find other points symmetrical about the vertex.

    4. Join the points to form the parabola.

EXAMPLE OF QUADRATIC FUNCTION GRAPH

1. Find the vertex

         So the vertex is (2, -1).

2. Find the intersections with the 𝑥 and 𝑦 axes:

The 𝑦-intercept is (0,3) (the constant term in the equation).

To find the 𝑥-intercepts, we set the function equal to zero:     

    𝑥2−4𝑥+3=0

Solving this quadratic equation, we find the roots 𝑥=1 and 𝑥=3 

3.Find other points symmetrical with respect to the vertex:

  You can take points on both sides of the vertex to maintain symmetry, such as (1,0) and (3,0).

4.Join the points with a smooth curve to form the parabola:

Using the calculated points, you can plot the curve of the parabola.

CUBIC FUNCTION: A cubic function is a type of polynomial function of the third degree, generally expressed as 𝑓(𝑥)=𝑎𝑥3+𝑏𝑥2+𝑐𝑥+d, where 𝑎, 𝑏, 𝑐, and 𝑑 are constant coefficients and 𝑥 is the independent variable.

 

Characteristics

• The graph of a cubic function is a smooth curve that can have an "S" shape or an inverted "S" shape, depending on the values ​​of the coefficients.

• It is called "graphing the cubic equation".

 

 

Domain: The domain of a cubic function is the set of all real numbers.

 

Range: The range of a cubic function covers all real numbers, since this function can assume any real value depending on the coefficients used.

 

Cuts in the axles: The 𝑥-axis cut is the intersection with the 𝑥-axis and occurs where the curve crosses the 𝑥-axis.

The cut on the 𝑦-axis is the intersection with the 𝑦-axis and occurs at the point (0,𝑑).

 

Increasing or decreasing: A cubic function can be increasing or decreasing depending on the values ​​of the coefficients. If the leading coefficient 𝑎 is positive, the function is increasing. If 𝑎 is negative, the function is decreasing.

 

Steps to graph:

1. Find the critical points such as the vertex and the intersections with the 𝑥 and 𝑦 axes.

2. Use symmetry to find other points on the curve, such as points symmetrical about the vertex.

3. Join the points with a smooth curve to form the cubic function.

EXAMPLE OF CUBIC FUNCTION GRAPH

f(x)=x³−3x²+2x

1. Find the vertex

2. Find the x- and 𝑦-intercepts:

The 𝑦-intercept is (0,0) (the constant term in the equation).

To find the 𝑥-intercepts, we set the function equal to zero:

𝑥³−3𝑥²+2𝑥=0

x(x²−3x+2) =0

 

1. x = 0

2. x² – 3x + 2 = 0

Solving this cubic equation, we find the roots 𝑥=0, 𝑥=1, and 𝑥=2.

3. Find other points symmetrical with respect to the vertex:

You can take points on both sides of the vertex to maintain symmetry, such as (1,0) and (2,0).

4. Join the points with a smooth curve to form the cubic curve:

Using the calculated points, you can plot the curve of the cubic function.

ABSOLUTE VALUE FUNCTION: An absolute value function, commonly denoted as ∣𝑥∣, is a mathematical function that takes the absolute value of its argument. In other words, it returns the positive value of the real number 𝑥 if 𝑥 is positive or zero, and the negative value of 𝑥 if 𝑥 is negative.

 

Characteristics

• The graph of an absolute value function is an inverted "V" or a "V"-shaped line in the Cartesian plane.

• It is called "graphing the absolute value function".

 

Domain: The domain of an absolute value function is the set of all real numbers.

 

Range: The range of an absolute value function is also the set of all real numbers, since absolute value always produces a non-negative real number.

 

Cuts in the axles: The cut in the 𝑥-axis occurs where the function vanishes, that is, when

∣𝑥∣=0. This occurs at 𝑥=0

The cut on the 𝑦-axis also occurs at 𝑦=0, since the absolute value of any number is zero if the number is zero.

 

Increasing or decreasing: The absolute value function is constant or increasing on intervals where the argument 𝑥 is positive or zero.

It is decreasing on intervals where the argument 𝑥 is negative.

 

Steps to graph:

1. Mark key points on the graph, such as the cut on the 𝑥 and 𝑦-axes and other points of interest.

2. Draw the inverted "V" by connecting these points with lines

straight.

EXAMPLE OF ABSOLUTE VALUE FUNCTION GRAPH

F(x) = ∣x∣

1. Mark the key points

- Cut on the X axis: (0.0)

- Cut on the y axis: (0.0)

- Another point: (1,1), (-1,1)

 

2. Draw the inverted “V”:

- Join the points (0,0), (1,1), and (-1,1) with straight lines for the inverted “V”.

NAMES: Lady Camila Díaz Bermúdez - Sol Yulima Velasco Valencia

GRADE: 11-B

DATE: 29/05/2024 

•Mathematics: Paola Andrea Valencia Estrada. 

•English: Leidy Alexandra Pinto Garcés. 

•Technology: Jorge Iván Castaño Cárdenas.