# Functions & Linear Graphs 1

In this chapter, we will be discussing the below-mentioned topics in detail:

• Cartesian coordinates in two dimensions
• Linear functions
• Graph of a set of ordered pairs as a representation of a relationship between two variables
• Graphs of linear functions

## Cartesian Plane

A Cartesian plane consists of two number lines intersecting at right angles at the point O, known as the origin.

The horizontal number line is known as the x-axis.

The vertical number line is known as the y-axis.

### Cartesian Coordinates

The position of any point on the plane can be determined by its distance from each of the axes.

Point A is 4 units above the x-axis and 3 units to the right of the y-axis.

Hence, A has coordinates (3, 4)

In other words, A(3, 4).

3 is the x-coordinate of A.

4 is the y-coordinate of A.

### Functions

A function is a relationship between two variables, $\displaystyle{x}$ and $y$, such that every input $x$ gives exactly one output $y$.

$\displaystyle{y = x+2}$ \begin{align*} y=x^2+2x+1 \end{align*} \begin{align*} y=x^3+x^2+x+1 \end{align*}
Equation of a Linear Function. Equation of a Quadratic Function. Equation of a Cubic Function.

Let's understand with the help of an example:

If we input a value of $x$ into the function 'machine', each value of $x$ will be added by 2.

Hence, from the other side of the machine, we would get the output “$y$”.

So let's put some values in the above table.

$x$ $0$ $1$ $2$ $3$
$y$ \text{When } x= 0\\ \begin{align} \\ y &=0+2\\ y &=2 \end{align} \text{When } x= 1\\ \begin{align} \\ y &=1+2\\ y &=3 \end{align} \text{When } x= 2\\ \begin{align} \\ y &=2+2\\ y &=4 \end{align} \text{When } x= 3\\ \begin{align} \\ y &=3+2\\ y &=5 \end{align}

$x$ $0$ $1$ $2$ $3$
$y$ \begin{align} &\text{When}\; x=0 \\ \\ &y=(0)^2+(0)+1\\ &y=1 \end{align} \begin{align*} &\text{When} \; x =1,\\\\ &y =(1)^2+(1)+1\\ &y =3 \end{align*} \begin{align*} &\text{When} \; x =2,\\\\ &y =(2)^2+(2)+1\\ &y=7\end{align*} \begin{align*} &\text{When} \; x =3,\\\\ &y =(3)^2+(3)+1\\ &y =13 \end{align*}

Question 1:

The equation of a function is $y=3x+2$. Find

1. The value of $y$ when $x=1.5$,
2. The value of $x$ when $y=-1$.

Solution:

1. Substituting the value of $x=1.5$

​​\begin{align*} y&=3x+2\\ &=3(1.5)+2\\y&=6.5 \end{align*}

1. Substituting the value of $y=-1$

​​\begin{align*} y&=3x+2\\ -1&=3x+2\\-1-2&=3x\\-3&=3x\\x&=-1 \end{align*}

## Graphs of Linear Functions

Question 2:

1. Draw the graph of a linear function $y=2x-1$, for values of $x$ from −2 to 2 inclusive.
Use a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis.
1. the value of $x$ when $y=2$
2. the value of $y$ when $x=0.5$

Solution:

1.

Step 1:

Draw the table of values for $x$ and $y$. These ordered pairs of values for $x$ and $y$ (also known as coordinates) satisfy the equation of the function $y=2x-1$.

 $x$ $-2$ $-1$ $0$ $1$ $y$ $-5$ $-3$ $-1$ $1$

Step 2:

Draw and label the x-axis (horizontal) and y-axis (vertical) using the given scales.

The scale on the graph paper is such that 1 small box is 1 cm.

Step 3:

Plot each pair of coordinates from the table of values as points on the Cartesian plane.

Step 4:

Draw and label the line of best fit.

1. (i)

Using the graph, when $y = 2$, $x = 1.5$.

(ii).

Using the graph, when $x = 0.5$, $y = 0$.

Continue Learning
Basic Geometry Linear Equations
Number Patterns Percentage
Prime Numbers Ratio, Rate And Speed
Functions & Linear Graphs 1 Integers, Rational Numbers And Real Numbers
Basic Algebra And Algebraic Manipulation 1 Approximation And Estimation

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