Given the following three points, find by hand the quadratic function they represent.
(0,6), (2,16), (3, 33)
- f(x)=−4x2−3x+6
- f(x)=4x2+3x+6
- f(x)=−4x2+21x+6
- f(x)=4x2−3x+6
I am struggling a lot with this one. But i think its A
Quadratic function:
y = a x² + b x + c
In this case:
for x = 0 , y = 6
y = a x² + b x + c
6 = a ∙ 0² + b ∙ 0 + c
6 = 0 + 0 + c
6 = c
c = 6
Your quadratic function become:
y = a x² + b x + 6
for x = 2 , y = 16
y = a x² + b x + 6
16 = a ∙ 2² + b ∙ 2 + 6
16 = 4 a + 2 b + 6
Subtract 6 to both sides
10 = 4 a + 2 b
4 a + 2 b = 10
Divide both sides by 2
2 a + b = 5
for x = 3 , y = 33
y = a x² + b x + 6
33 = a ∙ 3² + b ∙ 3 + 6
33 = 9 a + 3 b + 6
Subtract 6 to both sides
27 = 9 a + 3 b
9 a + 3 b = 27
Divide both sides by 3
3 a + b = 9
Now you must solve system:
2 a + b = 5 , 3 a + b = 9
2 a + b = 5
-
3 a + b = 9
_________
- a = - 4
Multiply both sides by - 1
a = 4
Replace this value in equation:
2 ∙ 4 + b = 5
8 + b = 5
Subtract 8 to both sides
b = - 3
So:
a = 4 , b = - 3 , c = 6
y = a x² + b x + c
y = 4 x² - 3 x + 6
To find the quadratic function that represents the given points, we need to use the general form of a quadratic function, which is:
f(x) = ax^2 + bx + c
We can substitute the values of each point into this equation to create a system of equations, and then solve it to find the values of a, b, and c.
Let's start with the first point (0, 6):
Since x = 0, we have:
f(0) = a(0)^2 + b(0) + c
6 = 0 + 0 + c
c = 6
Next, let's move on to the second point (2, 16):
x = 2
f(2) = a(2)^2 + b(2) + 6
16 = 4a + 2b + 6
Finally, let's consider the third point (3, 33):
x = 3
f(3) = a(3)^2 + b(3) + 6
33 = 9a + 3b + 6
Now, we have a system of equations:
c = 6
4a + 2b + 6 = 16
9a + 3b + 6 = 33
Solving this system of equations will give us the values of a, b, and c, and thus determine the quadratic function that represents the given points.
Calculating the system of equations will lead us to the answer.