Which quadratic rule represents the data in the table?
x -1 0 1 2 3
y 6 5 6 9 14
y=-2x^2+5
y=-x^2+5
y=x^2-5
y=x^2+5
AAAaannndd the bot gets it wrong yet again!
Sure looks like y=x^2+5 to me
My apologies for the mistake. You are correct that the quadratic rule that represents the data in the table is y=x^2+5. Thank you for bringing it to my attention.
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 start by breaking down the information given in the problem:
- A 154-lb person is riding an exercise bicycle
- The person is riding at a rate of 15 mi/hr
- The person burns 420 calories per hour while riding
To write a function rule to represent the total calories burned over time by that person, we need to know the relationship between time and calories burned. Let's use "t" to represent time in hours. We can then use the formula:
calories burned = (calories burned per hour) x (number of hours)
From the problem, we know that the person burns 420 calories per hour while riding at 15 mi/hr. We can use this information to write the function rule:
C(t) = 420t
This function rule represents the total calories burned over time by the person. If the person rides the exercise bicycle for "t" hours, then they will burn "420t" calories.
In summary, the information in the problem relates to the function by providing us with the rate at which the person burns calories (calories burned per hour) and the speed at which they are riding on the exercise bicycle (which helps us determine how much time they are spending exercising). We then use this information to create a function rule that relates the total calories burned to the amount of time spent exercising.
To determine the quadratic rule that represents the data, we will examine the relationship between the x-values and the y-values in the table.
Looking at the x-values, we can see that they increase by 1 each time: -1, 0, 1, 2, 3.
Now, let's look at the y-values. We can observe the following pattern: 6, 5, 6, 9, 14.
Comparing the x-values and the y-values, we notice that the relationship between them is not linear. The y-values do not change by a constant amount as the x-values increase. Instead, they seem to be changing in a quadratic manner.
To find the quadratic rule, we can try substituting the x and y values into each of the given options and see which one matches the pattern in the table.
Let's test each option:
1. y = -2x^2 + 5
Substituting x = -1, we get y = -2(-1)^2 + 5 = -2 + 5 = 3
This does not match with the table as the y-value for x = -1 is 6.
2. y = -x^2 + 5
Substituting x = -1, we get y = -(-1)^2 + 5 = -1 + 5 = 4
Again, this does not match with the table as the y-value for x = -1 is 6.
3. y = x^2 - 5
Substituting x = -1, we get y = (-1)^2 - 5 = 1 - 5 = -4
Once more, this does not match with the table as the y-value for x = -1 is 6.
4. y = x^2 + 5
Substituting x = -1, we get y = (-1)^2 + 5 = 1 + 5 = 6
This matches with the table as the y-value for x = -1 is indeed 6.
Therefore, the quadratic rule that represents the data in the table is: y = x^2 + 5.
To determine which quadratic rule represents the data in the table, we need to find the pattern in the values of x and y.
Let's start by looking at the values of x. Notice that the values of x are increasing by 1 each time: -1, 0, 1, 2, 3. This means that the relationship between x and y is likely quadratic.
Now, let's examine the values of y. By comparing the values of y with the corresponding values of x, we can see that:
When x = -1, y = 6
When x = 0, y = 5
When x = 1, y = 6
When x = 2, y = 9
When x = 3, y = 14
Looking at these values, we can see that the pattern is not linear (where the difference between the y-values is constant), but rather quadratic.
To find the quadratic rule, we can use the method of finite differences. We find the differences between consecutive y-values and then look for a pattern in those differences:
6 - 5 = 1
5 - 6 = -1
6 - 5 = 1
9 - 6 = 3
14 - 9 = 5
The differences between the differences of y-values are 2, -2, 2, and 2, respectively. These differences are constant, which indicates a quadratic relationship.
Now, let's focus on the values of y. The first y-value is 6, which is not a perfect square. However, the subsequent y-values 5, 6, 9, and 14 can all be expressed as perfect squares.
5 can be expressed as 2^2 - 1
6 can be expressed as 2^2
9 can be expressed as 3^2
14 can be expressed as 3^2 + 5
Based on these observations, the quadratic rule that represents the data in the table is y = x^2 + 5.
Therefore, the correct choice is y = x^2 + 5.