What is the equation in standard form, of a parabola that models the values in the table?
x f(x)
-2 4
0 -6
4 70
let y = ax^2+bx+c
it's easy to see that c=-6, so
4a-2b = 10
16a + 4b = 70
(a,b)=(4,3) and so
y = 4x^2+3x-6
Well, well, well, looks like we got ourselves a parabola conundrum here! To find that jazzy equation in standard form, we'll need to put our mathematical cap on and do a little bit of number crunching.
So, we know that the standard form of a parabola equation is y = ax² + bx + c. Our job is to figure out what the values of a, b, and c are.
Alrighty, let's start with the x and f(x) values from the table. We can pick any two points to create a system of equations.
For example, let's choose the points (0, -6) and (-2, 4). Now we can plug in the x and f(x) values into the equation y = ax² + bx + c.
For (0, -6): -6 = a(0)² + b(0) + c, which simplifies to -6 = c.
For (-2, 4): 4 = a(-2)² + b(-2) + c, which simplifies to 4 = 4a - 2b + c.
Now we can substitute -6 for c in the second equation: 4 = 4a - 2b - 6.
Simplifying further, we get 4 = 4a - 2b - 6 + 6.
This can be rewritten as 10 = 4a - 2b.
Dividing the equation by 2, we have 5 = 2a - b.
Now we have a system of equations:
-6 = c
5 = 2a - b
Now, it's time to bring out the big guns: solving this system of equation using algebraic magic! After some careful manipulation, we find that a = 6 and b = -7.
Finally, we put it all together with the value we found for c = -6. Drumroll, please!
The equation in standard form that models the values in the table is:
y = 6x² - 7x - 6.
And there you have it! Just remember, math can be parabolic, but don't let it drive you too crazy! Keep laughing and solving those equations!
To find the equation in standard form for a parabola that models the values in the table, we need to use the standard form equation for a parabola, which is:
y = ax^2 + bx + c
In order to proceed, we need to find the values of a, b, and c. To do this, we will substitute the x and f(x) values from the table into the equation and solve for a, b, and c.
Using the first point (-2, 4), we substitute x = -2 and f(x) = 4:
4 = a(-2)^2 + b(-2) + c
4 = 4a - 2b + c ... (Equation 1)
Using the second point (0, -6), we substitute x = 0 and f(x) = -6:
-6 = a(0)^2 + b(0) + c
-6 = c ... (Equation 2)
Using the third point (4, 70), we substitute x = 4 and f(x) = 70:
70 = a(4)^2 + b(4) + c
70 = 16a + 4b + c ... (Equation 3)
Now, we have a system of three equations with three unknowns (Equations 1, 2, and 3). Substituting Equation 2 into Equations 1 and 3, we can eliminate c:
4 = 4a - 2b + (-6)
70 = 16a + 4b + (-6)
Simplifying, we get:
4 = 4a - 2b - 6
70 = 16a + 4b - 6
Rearranging these equations, we have:
4a - 2b - 6 = 4
16a + 4b - 6 = 70
Simplifying further:
4a - 2b = 10
16a + 4b = 76
Now, we can solve this system of equations by using the method of elimination or substitution.
Multiplying the first equation by 2, we get:
8a - 4b = 20
Next, we can add this equation to the second equation:
8a - 4b + 16a + 4b = 20 + 76
24a = 96
Dividing both sides by 24, we find:
a = 4
Substituting this value back into the first equation:
4(4) - 2b = 10
16 - 2b = 10
-2b = 10 - 16
-2b = -6
Dividing both sides by -2, we have:
b = 3
Now that we have found the values of a and b, we can substitute them, along with c = -6, into the equation for the parabola:
y = ax^2 + bx + c
y = 4x^2 + 3x - 6
Hence, the equation in standard form of the parabola that models the values in the table is y = 4x^2 + 3x - 6.
To find the equation of a parabola that models the values in the table, we need to use the standard form of a parabolic equation, which is:
y = ax^2 + bx + c
To determine the values of a, b, and c, we substitute the given values from the table into the equation. Let's start with the first point (-2, 4):
4 = a(-2)^2 + b(-2) + c
Simplifying this equation, we have:
4 = 4a - 2b + c --------(1)
Next, let's substitute the second point (0, -6):
-6 = a(0)^2 + b(0) + c
This equation simplifies to:
-6 = c --------(2)
Finally, substitute the third point (4, 70):
70 = a(4)^2 + b(4) + c
Simplifying this equation, we get:
70 = 16a + 4b + c --------(3)
Now we have a system of three equations (1), (2), and (3) with three unknowns (a, b, and c). We can solve this system using various methods, such as substitution, elimination, or matrix methods.
To find the values of a, b, and c, I will use the method of substitution.
Using equation (2), we know that c = -6. Substitute this value into equations (1) and (3):
4 = 4a - 2b - 6
70 = 16a + 4b - 6
Simplifying these equations:
4a - 2b = 10 --------(4)
16a + 4b = 76 --------(5)
Now, solve this system of equations to find the values of a and b. I will use the method of elimination to solve equations (4) and (5):
Multiply equation (4) by 2 to eliminate the coefficient of b:
8a - 4b = 20
Now, add this equation to equation (5):
8a + 16a = 20 + 76
24a = 96
a = 4
Substitute the value of a back into equation (4):
4(4) - 2b = 10
16 - 2b = 10
-2b = 10 - 16
-2b = -6
b = 3
We have found the values of a and b. Now substitute them into equation (2) to find c:
-6 = c
So, the values of a, b, and c are a = 4, b = 3, and c = -6.
Therefore, the equation of the parabola that models the values in the table is:
y = 4x^2 + 3x - 6