Question

1.The earnings of a worker is shown by the graph below. Using this graph, answer the questions that follow.

a.Write the equation for this graph in the form of y = mx + b
b.This function represents an arithmetic sequence. What does the slope represent?
c. Show that this graph can also be represented by the arithmetic sequence equation an = a1 + (n - 1)d.
d. In this case, what is a10?

Answer 1

Answer:

a. $y = 2x$

b. The slope represents the common difference

c. $a_n = 2n$

d. $a_{10} = 20$

Step-by-step explanation:

Given:

The graph

Solving (a): The equation

First, we pick any two corresponding points on the graph

$(x_1,y_1) = (1,2)$

$(x_2,y_2) = (2,4)$

Next, we calculate the slope using:

$m = \frac{y_1 - y_2}{x_1 - x_2}$

$m = \frac{2-4}{1-2}$

$m = \frac{-2}{-1}$

$m=2$

The equation is then calculated as:

$y - y_1 = m(x - x_1)$

$y - 2 = 2(x - 1)$

$y - 2 = 2x - 2$

Make y the subject

$y = 2x - 2+2$

$y = 2x$

(b) Interpret the slope

In (a) above, the slope is calculated as: $m=2$

This represents the common difference of the sequence;

(c) Represent as an arithmetic sequence

We have the following points from the graph:

$(x_1,y_1) = (1,2)$

$(x_2,y_2) = (2,4)$

$(x_3,y_3) = (3,6)$

This means that: The first term is 2; the second is 4, the third is 6.....

So, we have:

$a_1 = 2$ --- First Term

$d = a_2 - a_1 = 4 - 2 = 2$ --- Difference

The nth term of an AP is:

$a_n = a_1 + (n - 1)d$

This gives:

$a_n = 2 + (n - 1)*2$

$a_n = 2 + 2n - 2$

Collect Like Terms

$a_n = - 2+2 + 2n$

$a_n = 2n$

(d) Find a10

To do this, we simply substitute 10 for n in $a_n = 2n$

So, we have:

$a_{10} = 2 * 10$

$a_{10} = 20$