Find the sum of the squared errors for the linear model f(x) and the quadratic model g(x) using the given points.
f(x) = −0.7x+3, g(x)=0.06x^2−0.6x+2
(−3, 5), (−1, 3), (1, 2), (3, 1)
linear model-?
quadratic model -?
To find the sum of squared errors for the linear model f(x) and the quadratic model g(x) using the given points, we first need to calculate the error for each point.
The error for each point is the difference between the actual y-coordinate and the predicted y-coordinate for that point. For the linear model f(x), the predicted y-coordinate can be calculated using the equation f(x) = -0.7x + 3. For the quadratic model g(x), the predicted y-coordinate can be calculated using the equation g(x) = 0.06x^2 - 0.6x + 2.
Let's calculate the error for each point:
Point (-3, 5):
For the linear model f(x):
f(-3) = -0.7(-3) + 3
= 2.1 + 3
= 5.1 (predicted y-coordinate)
Error = actual y-coordinate - predicted y-coordinate
= 5 - 5.1
= -0.1
For the quadratic model g(x):
g(-3) = 0.06(-3)^2 - 0.6(-3) + 2
= 0.54 + 1.8 + 2
= 4.34 (predicted y-coordinate)
Error = actual y-coordinate - predicted y-coordinate
= 5 - 4.34
= 0.66
Following the same process, let's calculate the errors for the other points:
Point (-1, 3):
For the linear model f(x):
f(-1) = -0.7(-1) + 3
= 0.7 + 3
= 3.7 (predicted y-coordinate)
Error = 3 - 3.7
= -0.7
For the quadratic model g(x):
g(-1) = 0.06(-1)^2 - 0.6(-1) + 2
= 0.06 + 0.6 + 2
= 2.66 (predicted y-coordinate)
Error = 3 - 2.66
= 0.34
Point (1, 2):
For the linear model f(x):
f(1) = -0.7(1) + 3
= -0.7 + 3
= 2.3 (predicted y-coordinate)
Error = 2 - 2.3
= -0.3
For the quadratic model g(x):
g(1) = 0.06(1)^2 - 0.6(1) + 2
= 0.06 - 0.6 + 2
= 1.46 (predicted y-coordinate)
Error = 2 - 1.46
= 0.54
Point (3, 1):
For the linear model f(x):
f(3) = -0.7(3) + 3
= -2.1 + 3
= 0.9 (predicted y-coordinate)
Error = 1 - 0.9
= 0.1
For the quadratic model g(x):
g(3) = 0.06(3)^2 - 0.6(3) + 2
= 0.54 - 1.8 + 2
= 0.74 (predicted y-coordinate)
Error = 1 - 0.74
= 0.26
Now that we have the error for each point, we can calculate the sum of squared errors for each model. To do this, we square each error, then sum them up.
For the linear model f(x):
Sum of squared errors = (-0.1)^2 + (-0.7)^2 + (-0.3)^2 + 0.1^2
= 0.01 + 0.49 + 0.09 + 0.01
= 0.6
For the quadratic model g(x):
Sum of squared errors = 0.66^2 + 0.34^2 + 0.54^2 + 0.26^2
= 0.4356 + 0.1156 + 0.2916 + 0.0676
= 0.91
Therefore, the sum of squared errors for the linear model f(x) is 0.6, and for the quadratic model g(x) is 0.91.