Study S1 Mathematics Maths - Linear Graphs - Geniebook

# 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, $$x$$ and $$y$$, such that every input $$x$$ gives exactly one output $$y$$.

#### \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 machine, each value of $$x$$ will add 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.

 $$x$$ $$0$$ $$1$$ $$2$$ $$3$$ $$y$$ \begin{align*} &\text{When}\; x= 0 \\ \\ &y =0+2\\ &y =2 \end{align*} \begin{align*} &\text{When} \; x =1,\\ \\ &y =1+2\\ &y=3 \end{align*} \begin{align*} &\text{When} \; x =2,\\\\ &y =2+2\\ &y =4 \end{align*} \begin{align*} &\text{When} \; x =3,\\\\ &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.
2. Using your graph, find
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 $$x = 0.5$$, $$y = 0$$.

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