A function in which the maximum degree or power of the dependent variable is one is called a linear function and its graph will always be a line. The most basic linear function is: y=x.

To plot the graph of a function, we first plot a set of ordered pairs that satisfy the given equation and then join them to form a smooth line. The ordered pairs for this equation will be (1, 1), (0, 0), (2, 2) etc. The graph is shown below:

How will the graph change if we have an equation of the type y=k x?

**Consider 2 cases**

- If k >1, then the absolute value of y will always be greater than x except when x =0. This means the value of y will increase faster for positive x compared to the graph y=x for the same change in the value of x. We can understand this better once we plot its graph.

Let y=2x

XY

0

0

1

2

-1

-2

2

4

Graph of y=2x moves closer to the y axis compared to the graph of y=x

- If k < 1, then the value of y will always be less than x except when x =0. This means the value of y will increase slowly compared to the graph of y=x for the same change in the value of x.

Let y=0.5x

XY

0

0

1

0.5

-1

-0.5

2

1

Graph of y=0.5x moves away from the y axis and closer to the x axis compared to the graph of y=x

**Example**

Plot the graph for the same statement as in example 3, “The speed of a car increases by 0.5 km/hr after every 30 minutes.”. How can we predict the speed after 6 hours?

**Answer:**

To plot the graph, we need a set of ordered pairs.

Time (t) – hrsSpeed (s) (km/hr)

0

0

0.5

0.5

1.0

1.0

1.5

1.5

2

2.5

We get the following graph

Graphs can be used to find the value of a dependent variable for a given value of an independent variable. To find the value of speed at t= 6 hours from the graph, we draw a vertical line at t=6 .Suppose it meets the graph line at point A. From there we draw a horizontal line towards the y axis. The point where this line meets the y axis is the value of speed at t=6 hours.

Also it can be seen that as time increases by 0.5 hours, speed increases by 0.5 km/hr and at any time value of speed s=t.

**Example **

The growth rate of the two companies between the year 2000 and 2009 is shown. Answer the questions below

- Which company has better growth rate and why?

**Answer:**

It will be the company whose graph has a steeper line. Since the graph of company P is steeper than that of B, it means for same change in x i.e. number of years, the profit of company P is greater than that of Q. - Approximately how much more profit does Company P make than Q in the year 2005.

**Answer:**

In 2005, P’s profit = 24% (approx) and Q’s profit = 19% (approx)

Hence P’s profit is 5 % more than Q’s.

A linear function is written as y=mx + b where m is the slope of the line and b is its y intercept. This form of the equation of a line is called slope intercept form. To find the equation of a line in this form we need to know the slope and y intercept.

If the change in value of x given by ∆x changes the value of y by ∆y then the ratio of ∆y / ∆x is called the slope of the function. It represents how the value of y changes with a change in the value of x. The slope can be negative, positive or zero. In the case of a line this slope is constant between any pair of x and y values on the line.

We can also find the equation of a line when we know its slope and any point that lies on the line.

If its slope is m and (x1, y1) is a point on the line then the equation of the line is given by:

(y – y 1)=m (x - x1).

This is called point slope form and it can easily be transformed to represent the line in slope intercept form by using simple mathematical operations.

**Example**

For the graph shown above, we can find the equation of the line for company P by finding any two points on the line. The two points that we choose should be accurate. We can take points (2003, 20) and (2008, 30).

The third form of expressing a line is the standard form, for any linear equation, ax + by= c. As an example take the equation y = ¾ x -2.

This line expressed in the 3 forms of equations is as follows:

Slope intercept form: y = ¾ x -2; it was given in this form.

Standard form: 3x -4y = 8; to get to this form multiply the equation by 4: 4y = 3x – 8

Then move the x and y components to the same side of the equation:

3x – 4y – 8 = 0

Add 8 to both sides of the equation: 3 x – 4y – 8 + 8 = 8

3x – 4y = 8

Point slope form: y – 1 = ¾(x – 4); Find a point on the line: (4, 1) is on the line, so x1 = 4, y1 = 1

Slope = ¾ from the slope intercept form of the line

- For the function given below, what are the possible values of y?

Y = 8x -29; x is a real number greater than or equal to 0- Only positive real numbers
- Only negative real numbers
- Real numbers >- 29
- Real numbers ≥ -29

**Answer: D**

For the data table given below, answer questions 2 and 3**Year**No. of students1995

3000

1996

3400

1997

3600

1998

3750

1999

4000

2000

4500

- The speed of a car initially is 10 km/hr. The speed gradually increases at the rate of 5km/hr every 0.2 hrs till it has reached a speed of 50 km/hr. Identify the function for the speed of the car in km/hr (s) with time t (hour)
- 10 + 25x
- 10 + 2.5x
- 10 + 5x
- 10 + 0.2x

**Answer: D** - During their first year of life, swordfish increase in weight at a regular rate. A swordfish weighed 14 pounds at the age of one month and 28 pounds at the age of 2 months. What was its weight at the age of 6 months?

- 76
- 84
- 94
- None of the above

**Answer: B** - For an increment of 1 in the value of x, the value of y at least quadruples. Which of the following best represents this?

- Y= 2x+2
- Y=4X +1
- Y= 4x
- Both B and C

**Answer: D**

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