4.10 Connecting descriptions, tables, equations, and graphs of lines
Consider the equation y = 4 x.
Complete the table of values.
Plot the points in the table of values.
Sketch graph of the line on the same coordinate plane.
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
For each of the following equations and tables of values:
Complete the given table of values.
Plot the points in the table and sketch the graph of the equation.
y = 2x
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
y = - 4 x
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
y = \dfrac{x}{2}
| x | -6 | -4 | -2 | 0 |
|---|---|---|---|---|
| y |
y = - \dfrac{x}{5}
| x | -10 | -5 | 0 | 5 |
|---|---|---|---|---|
| y |
y = - 2 x + 1
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
y = 3 x + 1
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
y = \dfrac{x}{2} + 5
| x | -4 | -2 | 0 | 2 |
|---|---|---|---|---|
| y |
y = - \dfrac{x}{4} + 5
| x | -8 | -4 | 0 | 4 |
|---|---|---|---|---|
| y |
For each of the following equations and tables of values:
Complete the given table of values.
Sketch the graph of the equation.
y = 8 x
| x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y |
y = - 7 x
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
y = \dfrac{x}{5}
| x | -2 | -1 | 0 | 5 |
|---|---|---|---|---|
| y |
y = - \dfrac{x}{7}
| x | -7 | -4 | -3 | 0 |
|---|---|---|---|---|
| y |
y = 9 x + 1
| x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y |
y = - 7 x - 3
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
y = \dfrac{x}{6} - 5
| x | -5 | -3 | 0 | 6 |
|---|---|---|---|---|
| y |
y = - \dfrac{x}{3} - 6
| x | -3 | -2 | -1 | 0 |
|---|---|---|---|---|
| y |
y = \dfrac{5 x + 3}{4}
| x | -3 | -2 | 0 | 1 |
|---|---|---|---|---|
| y |
y = - \dfrac{5 x + 4}{3}
| x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y |
Complete the table of values for the following graphs:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y |
| x | 4 | 5 | 6 | 7 |
|---|---|---|---|---|
| y |
| x | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| y |
| x | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| y |
Consider the following graph:
Complete the table of values:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y |
Does the point \left(3, 8\right) lie on the line?
Consider the following graph:
Complete the table of values:
| x | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| y |
Does the point \left(3, 5\right) lie on the line?
Find the equation of the following lines in slope-intercept form:
Slope of - 4 and crosses the y-axis at - 4.
Slope is - 3 and crosses the y-axis at - 8.
Slope is - 9 and its y-intercept is 2.
Slope is 1 and its y-intercept is -5.
Find the equation of a line that has the same slope as y = 2 - 3 x and the same y-intercept as y = - 6 x - 3.
William is running a 100\text{ km} ultramarathon at an average speed of 8\text{ km/h}.
Write an equation to represent the distance William has left to run, y, as a function of the number of hours since the start, x.
The following graph shows the relationship between the number of cartons and the total number of eggs in them:
Complete the table below.
| Cartons | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Eggs |
Buzz recorded his savings (in dollars) over a few months in the given graph:
Complete the table:
| \text{Months} | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Savings } \left(\$\right) |
Is Buzz correct if he estimates that he will have exactly \$60 in his savings by Month 5?
The graph shows the temperature of a room after the heater has been turned on:
State the slope of the line.
State the y-intercept.
Write a linear equation to represent the temperature of the room, y, as a function of time, t.
Explain the meaning of the slope in this context.
Explain the meaning of the y-intercept in this context.
Find the temperature of the room after the heater has been turned on for 40 minutes.
The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in it's side:
State the slope of the line.
Find the y-intercept.
Write a linear equation to represent the amount of water remaining in the bucket, y, as a function of time, x.
Explain the meaning of the slope in this context.
What does the y-intercept represent in this context?
Find the amount of water remaining in the bucket after 54 minutes.
There are 20 \text{ L} of water in a rainwater tank. It rains for a period of 24 hours and during this time, the tank fills up at a rate of 8 \text{ L} per hour.
Complete the table of values:
| \text{Number of hours passed } (x) | 0 | 1 | 2 | 3 | 4 | 4.5 | 10 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank }(y) |
Write a linear equation linking the number of hours passed, x, and the amount of water in the tank, y.
Plot the points on a coordinate plane.
A baseball is thrown vertically upward by a baseball player when he is standing on the ground, and the velocity of the baseball V (in yards per second) after T seconds is given by V = 120 - 32 T.
Complete the table of values:
| \text{Time} | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Vertical velocity} |
State the slope of the linear function.
Explain the negative value of V when T = 4.
It starts raining and an empty rainwater tank fills up at a constant rate of 2 \text{ gal} per hour. By midnight, there are 20 \text{ gal} of water in the rainwater tank. As it rains, the tank continues to fill up at this rate.
Complete the table of values:
| \text{Number of hours passed since midnight } (x) | 0 | 1 | 2 | 3 | 4 | 4.5 | 10 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank } (y) |
Sketch the graph depicting the situation on a coordinate plane.
Write a linear equation linking the number of hours passed since midnight, x, and the amount of water in the tank, y.
Determine the y-intercept of the line.
At what time prior to midnight was the tank empty?
The following graph shows the relationship between water temperatures and surface air temperatures:
Complete the table of values:
| \text{Water temperature } \left(\degree \text{C} \right) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \text{Surface air temperature } \left(\degree \text{C} \right) |
Write a linear equation representing the relationship between the water temperature, x, and the surface air temperature, y.
Find the surface air temperature when the water temperature is 14 \degree \text{C}.
Find the water temperature when the surface air temperature is 23 \degree \text{C}.
A car travels at an average speed of 47\text{ mi/h}.
Complete the table of values for D = 47 t, where D is the distance traveled in miles and t is the time taken in hours:
| t | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| D |
How far will the car travel in 9 hours?
Sketch the graph of the equation on a coordinate plane.
State the slope of the line.
If the destination is 423\text{ km} ahead, how long would it take for the car to reach the destination at the given speed?
A racing car starts the race with 250 \text{ L} of fuel. From there, it uses fuel at a rate of 5 \text{ L} per minute.
Complete the table of values:
| \text{Number of minutes passed }\left(x\right) | 0 | 5 | 10 | 15 | 20 | 50 |
|---|---|---|---|---|---|---|
| \text{Amount of fuel left in tank }\left(y\right) |
Write a linear equation linking the number of minutes passed, x, and the amount of fuel left in the tank, y.
Describe how the amount of fuel in the car is changing over time.
A racing car starts the race with 150 \text{ L} of fuel. From there, it uses fuel at a rate of 5\text{ L} per minute.
Complete the following table of values:
| \text{Number of minutes passed } (x) | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| \text{Amount of fuel left in the tank } (y) |
Write a linear equation linking the number of minutes passed, x, and the amount of fuel left in the tank, y.
Explain the meaning of the slope in this context.
How many minutes will it take for the car to run out of fuel?
Gas costs a certain amount per liter. The table shows the cost of various amounts of gas in dollars:
| \text{Number of liters }(x) | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of gas }(y) | 0 | 16.40 | 32.80 | 49.20 | 65.60 |
Find the cost of gas per liter.
Write a linear equation linking the number of liters of gas pumped, x, and the cost of the gas, y.
Explain the meaning of the slope in this context.
Calculate the cost of 47 \text{ L} gas.
A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table below shows the depth of the diver, in yards, over 4 minutes:
| \text{Number of minutes passed }\left(x\right) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Depth of diver in yards }\left(y\right) | 0 | 1.4 | 2.8 | 4.2 | 5.6 |
Calculate the increase in depth each minute.
Write a linear equation for the relationship between the number of minutes passed, x, and the depth, y, of the diver.
Calculate the depth of the diver after 6 minutes.
Calculate how long the diver takes to reach 12.6 \text{ yd} beneath the surface.
After Mae starts running, her heartbeat increases at a constant rate.
Complete the following table:
| \text{Number of minutes passed } (x) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate } (y) | 49 | 55 | 61 | 67 | 73 | 79 |
What is the unit change in y for the above table?
Write a linear equation that describes the relationship between the number of minutes passed, x, and Mae’s heartbeat, y.
What does the slope represent in context?
In a study, scientists found that the more someone sleeps, the quicker their reaction time. The table below displays the findings:
| \text{Number of hours of sleep } (x) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Reaction time in seconds } (y) | 6 | 5.8 | 5.6 | 5.4 | 5.2 | 5 |
How much does the reaction time decrease for each extra hour of sleep?
Write a linear equation relating the number of hours of sleep, x, and the reaction time, y.
Calculate the reaction time for someone who has slept 4.5 hours.
Calculate the number of hours someone sleeps if they have a reaction time of 5.5 seconds.
The number of fish in a river is approximated over a five year period. The results are shown in the following table:
| \text{Time in years }(t) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Number of fish }(F) | 4800 | 4600 | 4400 | 4200 | 4000 | 3800 |
Sketch a graph corresponding to the given information.
Calculate the slope of the line.
What does the slope represent in this context?
State the value of F when the line crosses y-axis.
Determine an equation for the line using the given values.
Now, determine the number of fish remaining in the river after 13 years.
Find the number of years, t, until 2000 fish remain in the river.
A ball is rolled down a slope. The table below shows the velocity, V, of the ball after a given number of seconds, t:
| \text{Time in seconds } (t) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Velocity in m/s } (V) | 12 | 13.3 | 14.6 | 15.9 | 17.2 | 18.5 |
Sketch a graph that displays the ball's velocity against time.
Calculate the slope of the line.
What does the slope represent in this context?
State the y-intercept of the line.
What does the y-intercept represent in this context?
Write a linear equation for the line, expressing V in terms of t.
Now, determine the velocity of the ball after 19 seconds.
The cost of a taxi ride is given by C = 5.5 t + 3, where t is the duration of the trip in minutes.
Calculate the cost of an 11-minute trip.
For every extra minute the trip takes, how much more will the trip cost?
What could the constant value of 3 represent in context?
A cellphone salesman earned \$600 in a particular week during which he sold 26 phones and \$540 in another week during which he sold 20 phones.
Write a linear equation to represent the weekly earnings of the salesman, y, as a function of the number of phones sold, x.
State the slope of this function.
Describe what the slope of the line represents in context.
Find the y-intercept.
Describe what the y-intercept represents in context.
Find how much the salesman will earn in a week during which he sells 36 phones.
A particular restaurant has a fixed weekly cost of \$1300 and receives an average of \$16 from each customer.
Write an equation to represent the net profit, y, of the restaurant for the week as a function of the number of customers, x.
Find the slope of the function.
Describe what the slope of the line represents in context.
Find of the y-intercept.
Describe what the y-intercept represents in context.
Find the restaurant's net profit if it has 310 customers for the week.
Mohamad is taking his new Subaru out for a drive. He had only driven 50 \text{ mi} in it before and is now driving it down the highway at 75\text{ mi/h} .
Write a linear equation to represent the total distance, y, that Mohamad had driven in his Subaru as a function of the number of hours, x.
State the slope of the function.
Describe what the slope of the line represents in context.
Find of the y-intercept.
Describe what the y-intercept represents in context.
Find the total distance Mohamad will have driven in his Subaru if his current drive begins at 5:10 pm and finishes at 7:25 pm.
Luigi is running a 100 \text{ mi} ultramarathon at an average speed of 8 \text{ mi/h}.
Write a linear equation to represent the distance Luigi has left to run, y, as a function of the number of hours since the start, x.
State the slope of the function.
Describe what the slope of the line represents in context.
Find of the y-intercept.
Describe what the y-intercept represents in context.
Find the distance Luigi will have left to run after 4.5 hours.
A plumber charges a callout fee of \$60 plus \$25 per hour.
Write a linear equation to represent the total amount charged, y, by the plumber as a function of the number of hours worked, x.
State the slope of the function.
Explain the meaning of the slope in this context.
Find of the y-intercept.
Explain the meaning of the y-intercept in this context.
Find the total amount charged by the plumber for 4 hours of work.