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? Thanks
To determine if the data in a table represents a linear function, we can check if there is a constant rate of change between successive x-values and y-values.
For the first question:
x -3 -2 -1 0 1
y 1 -2 -5 -8 -11
We can calculate the differences between successive y-values: -2 - 1 = -3, -5 - (-2) = -3, -8 - (-5) = -3, -11 - (-8) = -3. The differences between the y-values are all the same (-3), indicating a constant rate of change.
To find the rule for a linear function, we can use the slope-intercept form y = mx + b, where m represents the slope and b represents the y-intercept. To find the slope, we can pick any two points from the table and calculate the change in y divided by the change in x.
Using the points (-3, 1) and (0, -8), we have:
Slope = (y2 - y1) / (x2 - x1) = (-8 - 1) / (0 - (-3)) = -9 / 3 = -3/1 = -3
The y-intercept can be determined by substituting the slope and any point from the table into the slope-intercept form and solving for b. Plugging in the values (-3, 1) and the slope (-3), we get:
1 = (-3)(-3) + b
1 = 9 + b
b = 1 - 9
b = -8
Therefore, the rule for the linear function is y = -3x - 8. So, the answer is "yes; y = -3x - 8".
For the second question:
x -1 0 1 2 3
y 4 5 4 1 -4
We can follow the same process as above to find if there is a constant rate of change. However, in this case, the differences in the y-values are not the same. For example, the differences between the y-values for the first two points are 5 - 4 = 1, but the differences between the y-values for the second two points are 1 - 4 = -3. Since the differences in y-values are not constant, the data in the table does not represent a linear function.
To find the rule for a quadratic function, we can use the general form of a quadratic equation: y = ax^2 + bx + c. To find the values of a, b, and c, we can substitute three points from the table into the equation and solve a system of equations.
Using the points (-1, 4), (0, 5), and (1, 4), we have the following equations:
4 = a(-1)^2 + b(-1) + c
5 = a(0)^2 + b(0) + c
4 = a(1)^2 + b(1) + c
Simplifying and rearranging the equations, we get:
a - b + c = 4
c = 5
a + b + c = 4
Substituting the value of c in the first and third equations, we have:
a - b = -1
a + b + 5 = 4
Solving the system of equations, we find that a = -2, b = 1, and c = 5.
Therefore, the rule for the quadratic function is y = -2x^2 + x + 5. So, the answer is "y = -2x^2 + 5".