15.
Do the data in the table represent a linear function? If so, write a rule for the function.
x –3 –2 –1 0 1
y 1 –2 –5 –8 –11
(1 point)
yes; y = –3x – 8
yes; y = 1/3x – 8
yes; y = 1/3x + 8
yes; y = 3x + 8
16.
Write a quadratic rule for the data in the table.
x –1 0 1 2 3
y 4 5 4 1 –4
(1 point)
y = –2x^2 + 5
y = –x^2 + 5
y = x^2 – 5
y = x^2 + 5
Can you please also explain how to solve this?
To determine if the data in the table represents a linear function, you need to check if there is a constant rate of change between the x-values and the y-values. In a linear function, the rate of change will remain the same.
For question 15, let's calculate the rate of change between two adjacent points:
From x = -3 to x = -2, the change in y is: -2 - 1 = -3.
From x = -2 to x = -1, the change in y is: -5 - (-2) = -3.
From x = -1 to x = 0, the change in y is: -8 - (-5) = -3.
From x = 0 to x = 1, the change in y is: -11 - (-8) = -3.
As we can see, the rate of change between all adjacent points is -3, which indicates that it is a linear function.
Now, to find the equation of the linear function, we can use the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
From the table, we can see that the slope (m) is -3 because for each increase in x by 1 unit, y decreases by 3 units.
To find the y-intercept (b), we can substitute any x and y values from the table into the equation and solve for b.
Let's take x = 0 and y = -8:
-8 = (-3)(0) + b
-8 = 0 + b
b = -8
Therefore, the equation of the linear function is y = -3x - 8.
Now, let's move on to question 16.
To determine if the data in the table represents a quadratic function, we need to check if the second differences between the y-values are constant. In a quadratic function, the second differences are constant.
Let's calculate the second differences by finding the difference between the differences of the y-values:
From y = 4 to y = 5, the difference is: 5 - 4 = 1.
From y = 5 to y = 4, the difference is: 4 - 5 = -1.
From y = 4 to y = 1, the difference is: 1 - 4 = -3.
From y = 1 to y = -4, the difference is: -4 - 1 = -5.
The second differences, -1 and -3, are not constant. This means that the data in the table does not represent a quadratic function.
Therefore, there is no quadratic rule for the data in the table.