Create a formula for the points (1,5), (2,7), (3,9), and (4,11).(1 point)
I apologize for the previous response. The correct formula for the points (1,5), (2,7), (3,9), and (4,11) can be found using the slope-intercept form of a linear equation, which is y = mx + b.
To find the equation, we need to first find the slope (m) and the y-intercept (b).
Using the formula for slope (m) = (y2 - y1)/(x2 - x1) where (x1, y1) and (x2, y2) are any two points on the line, we can choose two points and calculate the slope.
Let's use the points (1,5) and (2,7):
m = (7 - 5)/(2 - 1) = 2/1 = 2
Now, we can use the point-slope formula y - y1 = m(x - x1) and choose any of the given points to substitute the values. Let's use the point (1,5):
y - 5 = 2(x - 1)
Expanding and simplifying the equation, we get:
y - 5 = 2x - 2
y = 2x - 2 + 5
y = 2x + 3
Therefore, the formula for the points (1,5), (2,7), (3,9), and (4,11) is:
y = 2x + 3
To find a formula for the points (1,5), (2,7), (3,9), and (4,11), we can use the method of finite differences.
The differences between the y-values are:
7 - 5 = 2
9 - 7 = 2
11 - 9 = 2
The differences between the differences are:
2 - 2 = 0
0 - 2 = -2
The pattern of differences is not constant, indicating that the points do not follow a linear equation. However, the differences between the differences are constant, which suggests that the points may follow a quadratic equation.
A quadratic equation can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants.
To find the values of a, b, and c, we can substitute the x and y values of any given point into the equation. Let's substitute the x and y values of (1, 5):
5 = a(1^2) + b(1) + c
5 = a + b + c (equation 1)
Similarly, we can substitute the x and y values of the other three points:
7 = a(2^2) + b(2) + c (equation 2)
9 = a(3^2) + b(3) + c (equation 3)
11 = a(4^2) + b(4) + c (equation 4)
We now have a system of three equations with three variables (a, b, and c).
Solving this system of equations, we find that a = 1, b = 2, and c = 2.
Therefore, the formula for the points (1,5), (2,7), (3,9), and (4,11) is:
y = x^2 + 2x + 2