GRAPHING LINEAR FUNCTIONS

The table shows the total amount of rain that falls in various amounts of time during a heavy rain.

If 'x' denotes time and 'y' denotes total amount of rain, then the relationship between time and total amount of rain can be represented by the equation

y = 1.5x

The graph of the relationship will be a line, so the equation is a linear equation. Since there is exactly one value of y for each value of x, the relationship is a linear function.

Example 1 :

The temperature at dawn was 8 °F and increased steadily 2 °F every hour. The equation y = 2x + 8 gives the temperature y after x hours. State whether the relationship between the time and the temperature is proportional or non proportional. Then graph the function.

Solution :

Step 1 :

The given equation y = 2x + 8 is in slope-intercept form linear equation. That is, y = mx + b.

When we compare the equation y = 2x + 8 with y = mx + b, we get m = 2 and b = 8.

Therefore, the equation is a linear equation. Since b ≠ 0, the relationship is non proportional.

Step 2 :

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

Step 3 :

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

Problem 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. State whether the relationship between the time and the temperature is proportional or non proportional. Then graph the function.

Solution :

Step 1 :

The given equation y = -2x + 82 is in slope-intercept form linear equation. That is, y = mx + b.

When we compare the equation y = -2x + 82 with y = mx + b, we get m = -2 and b = 82.

Therefore, the equation is a linear equation. Since b ≠ 0, the relationship is non proportional.

Step 2 :

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

Step 3 :

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

Problem 3 :

Ellyn has a part-time job in a pizza restaurant. She receives $12 per hour for her work and is required to work between 1 and 4 hours in a shift.

a) Complete the following table of values:

b) Graph the points on a number grid.

c) Does it make sense to join the points? Why or why not?

d)  Use your graph to find:

i) how much Ellyn earns if she works 1.5 hours

ii) how long Ellyn must work to earn $30.

Solution :

a) Wages per hour = $12

Total pay be x and number of hours working is x.

y = 12x

b)

c) Yes, it makes sense to join the points because it is  possible for Ellyn to work part of an hour

d)  i) When x = 1.5

y = 12(1.5)

If Ellyn works 1.5 hours she earns $18.

When y = 30

30 = 12x

x = 30/12

y = 2.5

To earn $30 Ellyn must work for 2.5 hours.

Problem 4 :

Daniel has $8 to spend on flowers for his garden bed. He buys x punnets of marigolds and y punnets of petunias. Each of the punnets costs $1, and he spends all of his money. 

a) Copy and complete this table which shows the different combinations he could buy

b) Graph the information in the table on a number grid.

c) Are the points collinear?

d) Write a relationship between x and y.

e) Is it meaningful to join the points with straight line segments?

Solution :

a) Amount spent for x punnets of marigolds and y punnets of petunias = 8

x + y = 8

b)

c) Yes, the points are collinear.

d) x + y = 8Write a relationship between x and y.

e) Yes, it creates a linear relationship. That's why we get the straight line as graphical form.

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