Q65. - Algebra 1

Question

Observe the table below.

 

Equation	Related Expression	Power of ‘x’	Linear or non linear
$y=x$
$y=x^2$
$y=x^3$

 

a. For each equation, write the related expression and record the power of x.

 

b. Graph each equation using a graphing calculator.

 

c. Classify each graph as linear or nonlinear.

 

d. Explain how to determine whether an equation, or its related expression, is linear or nonlinear without graphing.

Step-by-Step Solution

Answer

a. 

 

Equation	Related expression	Power of ‘x’
$y=x$	x	1
$y=x^2$	$x^2$	2
$y=x^3$	$x^3$	3

 

b.

 

The graph of $y=x$ is shown in figure (1).

(figure 1)

The graph of $y=x^2$ is shown in figure (2).

(figure 2)

The graph of $y=x^3$ is shown in figure (3).

(figure 3)

c.

 

Equation	Linear or nonlinear
$y=x$	Linear
$y=x^2$	Nonlinear
$y=x^3$	Nonlinear

 

d.

 

Without using graph, the equation or the related expression is said to be linear, if the degree of an equation or the related expression is ‘1’.

 

Without using graph, the equation or the related expression is said to be nonlinear, if the degree of an equation or the related expression is ‘2’ or more than ‘2’.

Part a. Step 1. State the concept of related expression.

The related expression are the terms in the right hand side of the equations.

Part a. Step 2. State the concept of power of variables.

The number in the exponent of a given term is called the power of that term.

Part a. Step 3. Find the related expression.

Using the definitions,

The related expression of $y=x$ is ‘x’.

The related expression of $y=x^2$ is ‘$x^2$’.

The related expression of $y=x^3$ is ‘$x^3$’.

Part a. Step 4. Find the power of ‘ x ’.

By definition,

In the equation $y=x$, the power of x is ‘1’. (Since there is no number in the exponent, it is considered as 1).

In the equation $y=x^2$, the power of x is ‘2’.

In the equation $y=x^3$, the power of x is ‘3’.

Part b. Step 1. State the meaning of graphing calculator.

A calculator in which the graph of the given function is graphed directly just by inputting the function is known as graphing calculator.

 

Desmos is one of the online graphing calculator.

Part b. Step 2. Graph the functions using graphing calculator.

Using graphing calculator,

 

The graph of $y=x$ is shown in figure (1).

(figure 1)

The graph of $y=x^2$ is shown in figure (2).

(figure 2)

The graph of $y=x^3$ is shown in figure (3).

(figure 3)

Part b. Step 3. State the conclusion.

Therefore the graphs are as follows:

The graph of $y=x$ is shown in figure (1).

(figure 1)

The graph of $y=x^2$ is shown in figure (2).

(figure 2)

The graph of $y=x^3$ is shown in figure (3).

(figure 3)

Part c. Step 1. State the graphical definition of linearity.

A graph is said to be linear if it is a straight line.

Part c. Step 2. State the graphical definition of nonlinearity.

A graph is said to be non-linear if it is a curve(not a straight line).

Part c. Step 3. State the conclusion.

The graph of $y=x$ is shown in figure (1).

(figure 1)

In the above figure see that the graph is a straight line.

 

Therefore by definition, $y=x$ is a linear graph.

 

The graph of $y=x^2$ is shown in figure (2).

(figure 2)

In the above figure see that the graph is not a straight line. It is a curve.

 

Therefore by definition, $y=x^2$ is a nonlinear graph.

 

The graph of $y=x^3$ is shown in figure (3).

(figure 3)

In the above figure see that the graph is not a straight line. It is a curve.

 

Therefore by definition, $y=x^3$ is a nonlinear graph.

Part d. Step 1. State the definition of linearity.

An equation is said to be linear, if its maximum degree is equal to 1.

Part d. Step 2. State the definition of nonlinearity.

An equation is said to be nonlinear, it its degree is 2 or more than 2.

Part d. Step 3. State the conclusion.

The degree of the equation $y=x$ is ‘1’.

Hence, it is a linear equation.

The degree of the equation $y=x^2$ is ‘2’.

Hence, it is a nonlinear equation.

The degree of the equation $y=x^3$ is ‘3’.

Hence, it is a nonlinear equation.