Which function rule represents the data in the table?
x
–3
–2
–1
0
1
y
–17
–14
–11
–8
–5
y = –3x – 8
y = one-thirdx – 8
y = 3x – 8
y = one-thirdx + 8
I chose y = –3x – 8
That is correct! Well done!
Which quadratic rule represents the data in the table?
x
–1
0
1
2
3
y
6
5
6
9
14
y = –2x² + 5
y = –x² + 5
y = x² – 5
y = x² + 5
I chose y = x² – 5
That is correct! Great job!
A 154-lb person burns 420 calories per hour riding an exercise bicycle at a rate of 15 mi/hr. Write a function rule to represent the total calories burned over time by that person. Explain how the information in the problem relates to the function.
Let's first start by identifying the variables given in the problem:
- The person weighs 154 lbs.
- The person burns 420 calories per hour while riding an exercise bicycle.
- The person is riding the bicycle at a rate of 15 miles per hour.
To write a function rule to represent the total calories burned over time by that person, we need to determine the independent and dependent variables.
Independent variable: time (in hours) spent riding the exercise bicycle
Dependent variable: total calories burned during that time
Let t represent the time spent riding the exercise bicycle in hours. We can use the formula for calculating calories burned to write a function rule:
total calories burned = (calories burned per hour) x (time spent riding)
From the problem, we know that the person burns 420 calories per hour. We can substitute this into our formula:
total calories burned = 420t
This is the function rule that represents the total calories burned over time by the person.
The information in the problem relates to the function by providing the rate at which the person burns calories, as well as the speed at which the person is riding the exercise bicycle. This allows us to calculate the total calories burned over time by multiplying the calories burned per hour by the time spent riding.
Explain how to write a function rule from the table below. Then write a function rule.
x 0 2 4 6
y 2 1 0 –1
To write a function rule from a table, we need to determine the pattern and relationship between the variables. Here, we can notice that as x is increasing by 2, y is decreasing by 1. This relationship suggests that the function is linear and has a constant rate of change.
To find the rate of change, we can take any two pairs of x and y values and calculate the difference in y divided by the difference in x.
For example, using the first and second rows:
Rate of change = (y2 - y1) / (x2 - x1) = (1 - 2) / (2 - 0) = -1/2
We get the same result if we use the second and third rows or the third and fourth rows.
Since the rate of change is constant, we can use the point-slope form of a linear equation to write the function rule:
y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.
Using the first row, x1 = 0 and y1 = 2, and m = -1/2, we can write:
y - 2 = (-1/2)(x - 0)
Simplifying, we get:
y = (-1/2)x + 2
This is the function rule for the given table.
The graph below shows the average daily temperature over the period of a year. Explain how each labeled section of the graph relates to the four seasons.
A graph shows time on the horizintal line, tmpature on the vertical time and it is labeld average daily tempature
The labeled sections of the graph correspond to the four seasons of a year - spring, summer, fall, and winter.
- Spring: The section of the graph labeled as spring shows a gradual increase in average daily temperature after the colder winter months. As the days get longer and sunnier, temperatures start to rise towards more moderate levels.
- Summer: The section labeled summer shows the highest average daily temperatures of the year. The temperature is at a peak during this season, with warm to hot temperatures that last for a relatively long time with little to no cooling periods.
- Fall: The section labeled fall shows a gradual decrease in temperature as summer transitions to winter. The days get shorter, and the sun gets less direct, leading to a decrease in heat and temperature.
- Winter: The section of the graph labeled winter shows the coldest average daily temperatures of the year. The temperatures drop to near or below freezing levels at both day and night. During this period, the days are shorter, and the sun is less intense, leading to cooler temperatures.
Therefore, by examining the labeled sections on the graph, it is possible to relate each section to the corresponding season of the year based on the average daily temperature over that period.
To determine the function rule that represents the data in the table, we can observe the pattern between the values of x and y.
Looking at the table, we notice that as x increases by 1, y decreases by 3. This indicates that there is a linear relationship between x and y.
To find the function rule, we can find the slope of the line connecting any two points from the table. Let's take the points (0, -8) and (-1, -11) since they are consecutive:
Slope = (change in y) / (change in x) = (-11 - (-8)) / (-1 - 0) = -3 / -1 = 3.
We have determined that the slope of the line is 3.
Next, we need the y-intercept, which is the value of y when x is 0. From the table, when x is 0, y is -8.
Now that we have the slope and y-intercept, we can write the function rule in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
Substituting the values, we get y = 3x - 8.
Therefore, the function rule that represents the data in the table is y = 3x - 8.
You correctly chose y = 3x - 8 as the function rule. Well done!