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
Well, you could say this parabola is a bit "off-balance" with its values! But let's try to find the equation in standard form.
To do that, we'll need to find the values of a, b, and c in the equation y = ax^2 + bx + c.
Let's start with the first point (-1, 12):
12 = a(-1)^2 + b(-1) + c
12 = a - b + c
Now let's move on to the second point (0, 5):
5 = a(0)^2 + b(0) + c
5 = c
Lastly, let's use the third point (2, 15):
15 = a(2)^2 + b(2) + c
15 = 4a + 2b + c
So, we now have the following:
12 = a - b + 5
15 = 4a + 2b + 5
Let's simplify the equations:
a - b = 7 (Equation 1)
4a + 2b = 10 (Equation 2)
To eliminate b, let's multiply Equation 1 by 2:
2a - 2b = 14 (Equation 3)
Now, let's add Equation 2 and Equation 3:
2a - 2b + 4a + 2b = 14 + 10
6a = 24
Dividing both sides by 6, we find that:
a = 4
Substituting this value back into Equation 1, we find:
4 - b = 7
Solving for b:
b = -3
Lastly, substituting a and b into Equation 2, we find:
4(4) + 2(-3) = 10
Simplifying that, we get:
16 - 6 = 10
So, a = 4, b = -3, and c = 5.
Therefore, the equation in standard form that models the values in the table is:
y = 4x^2 - 3x + 5
To find the equation in standard form of a parabola that models the values in the table, we can use the general form of a quadratic equation, which is:
f(x) = ax^2 + bx + c
First, let's substitute the given values from the table into the equation to form three equations:
When x = -1, the corresponding y-coordinate is 12:
12 = a(-1)^2 + b(-1) + c --> (Equation 1)
When x = 0, the corresponding y-coordinate is 5:
5 = a(0)^2 + b(0) + c --> (Equation 2)
When x = 2, the corresponding y-coordinate is 15:
15 = a(2)^2 + b(2) + c --> (Equation 3)
Now, we have a system of three equations with the three unknowns (a, b, c). We can solve this system of equations to find the values of a, b, and c.
Let's start by solving Equation 2 since it is the simplest:
From Equation 2, we have:
5 = c --> (Equation 4)
Next, let's substitute this value of c into Equations 1 and 3:
Equation 1 becomes:
12 = a(-1)^2 + b(-1) + 5
12 = a - b + 5
Equation 3 becomes:
15 = a(2)^2 + b(2) + 5
15 = 4a + 2b + 5
Simplifying these equations, we have:
a - b = 7 --> (Equation 5)
4a + 2b = 10 --> (Equation 6)
Now, we can solve this system of equations (Equations 5 and 6) to find the values of a and b.
By multiplying Equation 5 by 2, we can eliminate b:
2(a - b) = 2(7)
2a - 2b = 14
Adding this equation to Equation 6, we can solve for a:
(2a - 2b) + (4a + 2b) = 14 + 10
6a = 24
a = 4
Substituting the value of a back into Equation 5, we can solve for b:
4 - b = 7
b = -3
Now that we have found the values of a, b, and c, we can write the equation in standard form:
f(x) = ax^2 + bx + c
f(x) = 4x^2 - 3x + 5
Therefore, the equation in standard form of the parabola that models the values in the table is f(x) = 4x^2 - 3x + 5.