Two-Variable Data: Tables, Graphs, and Patterns
Represent and interpret relationships between two quantities
Two-Variable Data: Tables, Graphs, and Patterns
Represent and interpret relationships between two quantities
Math - Grade 6-8
- 1
The table shows the cost of notebooks at a school store. 1 notebook costs $2, 2 notebooks cost $4, 3 notebooks cost $6, and 4 notebooks cost $8. What is the rule relating the number of notebooks, n, to the cost, c?
Look at what happens to the cost each time the number of notebooks increases by 1.
The rule is c = 2n. Each notebook costs $2, so the total cost is 2 times the number of notebooks. - 2
A taxi charges a starting fee of $5 plus $3 for each mile. Complete the table for 0, 1, 2, 3, and 4 miles, then write an equation for the total cost, c, in dollars after m miles.
The starting fee is the cost when m = 0.
The completed table is 0 miles: $5, 1 mile: $8, 2 miles: $11, 3 miles: $14, and 4 miles: $17. The equation is c = 3m + 5 because the cost increases by $3 per mile and starts at $5. - 3
The points (1, 4), (2, 7), (3, 10), and (4, 13) follow a pattern. What is the change in y for each increase of 1 in x? Write an equation for the relationship.
The y-value increases by 3 each time x increases by 1. The equation is y = 3x + 1. - 4
A graph of a relationship contains the points (0, 2), (1, 5), (2, 8), and (3, 11). What is the y-intercept, and what does it mean in this relationship?
The y-intercept is the y-value when x equals 0.
The y-intercept is 2 because the graph crosses the y-axis at (0, 2). It means that when x is 0, the value of y is 2. - 5
The table shows the number of cups of flour needed for batches of muffins: 1 batch needs 2 cups, 2 batches need 4 cups, 3 batches need 6 cups, and 4 batches need 8 cups. Is this a proportional relationship? Explain.
Check whether y divided by x is the same for each row.
Yes, this is a proportional relationship because the ratio of cups of flour to batches is always 2 to 1. The equation is y = 2x, and the graph would pass through the origin. - 6
A gym charges a one-time sign-up fee of $20 plus $10 per month. Is the relationship between months and total cost proportional? Explain.
No, the relationship is not proportional because there is a $20 starting cost when the number of months is 0. A proportional relationship must have the form y = kx and pass through the origin. - 7
The ordered pairs (2, 6), (4, 12), (6, 18), and (8, 24) are graphed. Describe the pattern and write the equation.
Divide each y-value by its matching x-value.
The y-value is always 3 times the x-value, so the pattern is multiplication by 3. The equation is y = 3x. - 8
A plant is 6 centimeters tall when it is measured on day 0. It grows 2 centimeters each day. Make a table for days 0 through 5 and write an equation for the height, h, after d days.
Start at 6 centimeters, then add 2 centimeters for each day.
The table is day 0: 6 cm, day 1: 8 cm, day 2: 10 cm, day 3: 12 cm, day 4: 14 cm, and day 5: 16 cm. The equation is h = 2d + 6. - 9
A student says the relationship in the table x: 1, 2, 3, 4 and y: 2, 4, 8, 16 is linear because y increases each time. Is the student correct? Explain.
The student is not correct. The relationship is not linear because the y-values do not increase by the same amount each time. The increases are 2, 4, and 8. - 10
The scatter plot shows hours studied and quiz scores for 10 students. The points generally rise from left to right. What type of association does the scatter plot show, and what does it mean?
A graph that rises from left to right shows that both variables tend to increase together.
The scatter plot shows a positive association. This means that students who studied more hours tended to have higher quiz scores. - 11
The scatter plot shows the age of a used car and its selling price. The points generally fall from left to right. What type of association does the scatter plot show, and what does it mean?
The scatter plot shows a negative association. This means that older used cars tended to have lower selling prices. - 12
For the equation y = 4x - 3, find the y-values when x = 0, 1, 2, and 3. Then describe the pattern.
Substitute each x-value into the equation.
The y-values are -3, 1, 5, and 9. The pattern is that y increases by 4 each time x increases by 1. - 13
A line goes through the points (3, 9) and (7, 17). What is the rate of change of y with respect to x?
The rate of change is 2. The change in y is 17 - 9 = 8, and the change in x is 7 - 3 = 4, so 8 divided by 4 equals 2. - 14
A movie rental service charges $4 per movie plus a monthly membership fee. The table shows that 2 movies cost $14 and 5 movies cost $26. What is the monthly membership fee?
Use the cost of the movies first, then subtract to find the fixed fee.
The monthly membership fee is $6. Since each movie costs $4, 2 movies cost $8, and $14 - $8 = $6. - 15
A graph shows the relationship between time and distance for a bicyclist. The line starts at (0, 0) and passes through (2, 18) and (4, 36). What is the bicyclist's speed in miles per hour?
Speed is distance divided by time.
The bicyclist's speed is 9 miles per hour. The distance increases by 18 miles in 2 hours, and 18 divided by 2 equals 9.