In this section, we are going to see, how to describe a relationship between two variables x and y by interpreting the graph.

**Example 1 :**

A square tile has a side length of x inches. The equation y = x^{2} gives the area of the tile in square inches. Determine whether the rate of change between the side length and area of a square is constant using the graph. And also describe whether the relationship between the two variables is linear.

**Solution : **

**Step 1 :**

Choose several values for the input x. Plug these values for x in the equation to find the output y.

**x**

1

2

3

4

**x ^{2}**

1^{2}

2^{2}

3^{2}

4^{2}

**y**

1

4

9

16

**(x, y)**

(1, 1)

(2, 4)

(3, 9)

(4, 16)

**Step 2 :**

Graph the ordered pairs. Then draw a line through the points to represent the solutions of the function.

**Step 3 : **

Describe the relationship between x and y.

The graph is not a straight line. So the rate of change between the side length and area of a square is not constant.

Only a straight line graph will represent a linear relationship. Since the graph is not a straight line, the relationship between the two variables x and y is non linear.

**Example 2 :**

The temperature of a fish tank was 82°F and decreased steadily 2°F every hour. The equation y = -2x + 82 gives the temperature y after x hours. Determine whether the rate of change between the time and temperature is constant. And also describe whether the relationship between the two variables is linear.

**Solution : **

**Step 1 :**

Choose several values for the input x. Plug these values for x in the equation to find the output y.

**x**

0

1

2

3

4

5

**-2x + 82**

-2(0) + 82

-2(1) + 82

-2(2) + 82

-2(3) + 82

-2(4) + 82

-2(5) + 82

**y**

82

80

78

76

74

72

**(x, y)**

(0, 82)

(1, 80)

(2, 78)

(3, 76)

(4, 74)

(5, 72)

**Step 2 :**

Graph the ordered pairs. Then draw a line through the points to represent the solutions of the function.

**Step 3 :**

Since the graph is a straight line, the rate of change between the time and temperature is constant.

Since the rate of change is constant and the graph is a straight line, the relationship between the two variables x and y is linear.

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