Linear/non-linear functions, equations, inequalities for Grade VII

Linear functions: 

Functions which have x as the input variable, where x has an exponent of only 1.

For example:
y = mx + c
Here we can see that x has an exponent of 1 in each equation.

The above functions such as this yields graphs that is straight line, and, thus, the name linear is alloted to it.

 

Linear functions are of three categories:

1) Slope-Intercept Form

2) Point-Slope Form

3) General Form

 

1) Slope-Intercept Form

A simple way to define a linear function is through the slope-intercept form of a linear function. When drawn on a common (x, y) graph it is given as:

y = mx + b

Or, in a formal function definition:

f(x) = mx + b

 

Basically, this function describes a set, or locus, of (x, y) points, and each of these points are located in a straight line. 'm' represents the slope of the line. 'b' represents the y-coordinate for the spot where the line crosses the y-axis and this point is called the 'y-intercept'.

 

2) Point-Slope Form

In the 'point-slope form' for the equation of a line, the definition does not start with 'y =' or 'f(x)=', so it's not written in a common function definition form.

But it can be very well written in algebra:

y = m(x - x1) + y1

and as a functional definition it can be written as:
f(x) = m(x - x1) + y1

To conclude the equation with the given slopes and points, we can say that the variable m is the slope of the line and the point x1, y1 are the points on the line. If anyone knows the slope of and the coordinates for one point on the line, then the he or she can include or can enter those values into this equation, and the equation would then define a set, or locus, of all the points on that line.

 

3) General Form

Let's say A, B, and C are three numbers forming an equation where the same equation can be assumed as:

0 = 2x + 3y + 4

 

All the points with (x, y) coordinates that can make the above statement true forms a line.

 

Using algebra, this general form can be changed into a slope-intercept form, and then you would know the slope and y-intercept for the line.

 

Non-linear Functions:

 

A non-linear function is defined as a polynomial function of two or even higher degree. The linear function is a polynomial function of degree 1. A quadratic function is a polynomial function of degree 2, defined by an equation of the form

y = ax² + bx + c

The degree of a polynomial function is the degree of the polynomial itself.

For example:

Let us find the degree of the following function and also the points where x and y intercepts.

y= x²-2x-3

The degree of the given equation is 2. To find the point where its graph cuts the x axis, we can make y = 0. now we get,

x²-2x-3 = 0

Solving the equation we get x = 3 and x = -1. These are points where the function graph intercepts the x-axis.
To find the point where the graph intercepts the y-axis, assume x = 0. Now we get y = -3.

 

 

Linear Equations:

 

A linear equation appears like any other form of equation. A linear equation is made up of two expressions sets that are equal to each other. A linear equation is different as, it can have one or even two variables.

No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction. Linear equations graph as straight lines.

A linear equation in two variables shows an experiment between two variables in which the value of one of the variables depends on the value of the other variable. In a linear equation in x and y, x is called an independent variable and y depends on it. We call y the dependent variable. If the variables have other names, yet do have a dependent relationship. The independent variable is plotted along the horizontal axis. Most linear equations are functions (that is, for every value of x, there is only one corresponding value of y). When you assign a value to the independent variable, x, you can compute the value of the dependent variable, y. You can then plot the points named by each (x,y) pair on a coordinate grid.

 

Lines with the Same Slope

Lines with the same slope are either the same line, or parallel lines.

 

Consider 2x + y – 6 = 0. This equation is not in slope-intercept form. There are two ways to put it in slope-intercept form.

1. The actual equation is 2x + y – 6= 0

Subtract y from each side. 2x + y – y – 6 = 0 – y

2x – 6 = 0 – y

Multiply each side by 1.

 

1(2x – 6)= 1(y)

2x + 6 = y

 

2. Show the original equation. 2x + y – 6 = 0

Add 6 to each side. 2x + y – 6 + 6= 0 + 6

2x + y = 6

Subtract 2x from each side. 2x – 2x + y = 6 – 2x

y = 6 – 2x

The given two equations, 2x + 6 = y and y = 6 – 2x are equal because you can change one equation into the other by using the symmetric property of equality, which states that if a = b, then b = a and the commutative property, which states that a + b = b + a.

commutative property 2x + 6 = y 6 – 2x = y

symmetric property 6 – 2x = y y = 6 – 2x

 

Nonlinear Equations:

Equation whose graph does not form a straight line (linear) is called a Nonlinear Equation. In a nonlinear equation, the variables are either of degree greater than 1 or less than 1, but never 1.

For example:

7x⁵+ y = 0 and x³+ 13x²-4xy² = 0 are the examples of nonlinear equations. We can also consider that the form x^1/3 + y^1/3 = 0 is a nonlinear equation.

Lets take an example of an equation:
(x+ 2)²= 6

This is a non-linear equation because, as per the definition, an algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Considering the equation (x+ 2)² = 6

We can see that x can be concluded as x raised to the power 2 or x².
Therefore, (x+ 2)²= 6 is not a linear equation.

Linear inequalities:

Phenomena of linear inequalities is a set of linear inequalities that anyone comes across all in the same time. Normally, a person starts with two or three linear inequalities.

 

For example:

• Solve the following system:

  2x– 3y < 12
  x + 5y < 20
  x > 0

  It is better to solve a large number of linear inequalities for "y" on one side. Solving the first two inequalities, get the rearranged system as:

  y > ( ²/₃ )x – 4
  y < ( – ¹/₅ )x + 4
  x > 0

Nonlinear Inequalities:

To solve non linear inequalities, We can use the following steps the following steps:

1.Place every component on the left hand side so that we have for example equation is greater than (>) 0.

2.Factorize and set equal to zero.

3.Solve and put the answers on the number line. Doing so it will divide the number line into two or three parts.

4.Select the value of each part and analyze them for each region and put that acquired value into each of the factors. Put '+' or '-' over the region looking into the values decide positive or negative values.

5.If the region has two '+' or two '-' then the region is positive. If the region has one of each then the region is negative.

6.If the inequality is "<" then include the negative regions.

7.If the inequality is ">" then include the positive regions.

8.If the inequality is a less than or greater than or equal to then include the endpoints with solid dot and the interval []

9.If the inequality is a less (greater) than then do not include the endpoints by showing an open dot and the interval () Lets take an example, Solve x2 + 3x > -2 1.x2 + 3x + 2 > 0 2.(x + 2)(x + 1) = 0 3.x = -2 or x = -

1 This divides the number line into three regions.

4.For the left region we choose-5 and have, -5 + 2 < 0, -5 + 1 < 0 5.For the middle region we choose -1.5, and get, -1.5 + 2 > 0, -1.5 + 1 < 0 For the left region we choose 0 and get, 0 + 2 >0, 0+ 1 >0 6.

The left and right regions are positive and the middle region is negative. 7.The inequality acquired is a greater in quantity we have the solution: (-,-2) U (-1,)