(1 & 2)
The first three points do not fall on a straight line, and there is no value of k that will make that happen.
(4) To find the quadratic
y = a x^2 + bx + c
that passes through the first three points, solve simultaneously
1 = c
3 = 4a + 2b + c
6 = 16 a + 4b + c
2 = 4a + 2b
5 = 16 a + 4b
4 = 8a + 4b
8a = 1
a = 1/8
2 = 1/2 + 2b
b = 3/4
y = (1/8)x^2 + (3/4)x + 1
Given ordered pairs (0,1),(2,3),(4,6),(6,k), answer the following questions
1. Determine a value for k so data points follow a linear model.
2. Find the linear model(equation).
3. Determine a value for k so data points follow a quadratic model.
4. Find the quadratic model(equation).
Dertermine value of a,b,c,d and e
To determine if the data points follow a linear model, we need to check if they lie on a straight line. However, the first three points do not fall on a straight line, so there is no value of k that will make them follow a linear model.
For the linear model equation, we can use the formula y = mx + b, where m is the slope and b is the y-intercept. Since the data points don't follow a linear model, we can't find a linear equation for them.
To determine if the data points follow a quadratic model, we need to check if they lie on a parabolic curve. We can find a quadratic equation that passes through the first three points by solving a system of equations:
1 = c
3 = 4a + 2b + c
6 = 16a + 4b + c
We can also include the fourth point (6,k) in the system of equations:
2 = 4a + 2b
5 = 16a + 4b
4 = 8a + 4b
By solving this system of equations, we can find the values of a, b, and c.
8a = 1
a = 1/8
2 = 1/2 + 2b
b = 3/4
Substituting these values back into the equation y = ax^2 + bx + c, we find:
y = (1/8)x^2 + (3/4)x + 1
So, if we want the data points to follow a quadratic model, the value of k would depend on the x-coordinate of the last point, which is 6. Plugging x=6 into the quadratic equation, we can find k:
k = (1/8)(6)^2 + (3/4)(6) + 1
k = 9/2 + 9/2 + 1
k = 19/2
Therefore, for the data points to follow a quadratic model, the value of k would be 19/2, and the quadratic model equation is y = (1/8)x^2 + (3/4)x + 1.