What is the equation in standard form, of a parabola that models the values in the table?
x -1, 0, 2
f(x) 12, 5, 15
y = a x^2 + b x + c
12 = a (1) + b(-1) + c
5 = a(0) + b(0) + c so c = 5
15 = a(4) + b(2) + 5
7 = a - b so a = b + 7
10 = 4 a + 2 b
10 = 4(b+7) + 2 b
10 = 4 b + 28 + 2 b
6 b = -18
b = -3 then a = b+7 = +4
so
y = 4 x^2 - 3 x + 5
You are welcome.
Thank you very much good sir!
Well, it seems that this parabola needs to hire a personal trainer because it has a lot of ups and downs! But don't worry, I got you covered with the equation in standard form.
To find the equation, let's first make use of the vertex form of a parabolic equation: f(x) = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
Now we need to find the vertex. Looking at the table, we see that the highest point of the parabola is when x = 2, f(x) = 15. So, our vertex coordinates are (2, 15).
Next, plug in the vertex coordinates into the vertex form: f(x) = a(x-2)^2 + 15.
Now, let's use one of the other points on the parabola to solve for 'a'. When x = -1, f(x) = 12. Substituting these values, we get:
12 = a(-1 - 2)^2 + 15
12 = a(-3)^2 + 15
12 = 9a + 15
-3 = 9a
a = -1/3
Plugging in the value of 'a' into our equation, we get the final standard form equation:
f(x) = (-1/3)(x - 2)^2 + 15
Hope this equation puts a smile on your face, even if the parabola itself is a little bit of a clown!
To find the equation of a parabola in standard form, we need to identify the values of a, b, and c in the equation:
f(x) = ax^2 + bx + c
Given the values in the table, we can use the method of "substituting into the equation" to find a system of equations:
1. For x = -1:
f(-1) = a(-1)^2 + b(-1) + c = 12
2. For x = 0:
f(0) = a(0)^2 + b(0) + c = 5
3. For x = 2:
f(2) = a(2)^2 + b(2) + c = 15
Now, let's simplify these equations:
1. a - b + c = 12
2. c = 5
3. 4a + 2b + c = 15
We now have a system of three equations with three variables (a, b, c).
To solve this system, a common method is to use substitution or elimination. However, in this case, it is more efficient to use the method of Gaussian elimination, which I will explain step by step:
1. Multiply equation 1 by 2:
2a - 2b + 2c = 24
2. Subtract equation 2 from equation 1:
2a - 2b + 2c - c = 24 - 5
2a - 2b + c = 19 (Equation 4)
3. Subtract equation 3 from equation 2 and multiply equation 2 by 2:
8a + 4b + 2c - 4a - 2b - c = 30 - 15
4a + 2b + c = 15 (Equation 5)
Now, we have two new simplified equations:
4. 2a - 2b + c = 19
5. 4a + 2b + c = 15
We can subtract equation 5 from equation 4:
(2a - 2b + c) - (4a + 2b + c) = 19 - 15
-2a = 4
This simplifies to:
a = -2
We can substitute the value of a into equation 4:
2(-2) - 2b + c = 19
-4 - 2b + c = 19
-2b + c = 23 (Equation 6)
Since we already know the value of c (from equation 2), we can substitute c = 5 into equation 6:
-2b + 5 = 23
-2b = 23 - 5
-2b = 18
b = -9
Now, we have the values of a and b. To find the value of c, we substitute a = -2 and b = -9 into any of the original equations:
f(0) = a(0)^2 + b(0) + c = 5
-2(0)^2 - 9(0) + c = 5
0 + 0 + c = 5
c = 5
Therefore, the equation of the parabola in standard form that models the values in the table is:
f(x) = -2x^2 - 9x + 5