what is the equation in standard form of parabola that contains the following points
(-2-20) (0,-4) (4,-20)
The standard form of a parabola is y = ax^2 + bx + c. We can substitute the given points into this equation to find a, b, and c.
For point (-2, -20):
-20 = a(-2)^2 + b(-2) + c,
-20 = 4a - 2b + c ------ (1).
For point (0, -4):
-4 = a(0)^2 + b(0) + c,
-4 = c -------- (2),
For point (4, -20):
-20 = a(4)^2 + b(4) + c,
-20 = 16a + 4b + c ------- (3).
We now have a system of three equations. From equation (2) we know that c = -4. Substituting c = -4 into equations (1) and (3) we get:
-20 = 4a - 2b - 4,
16 = 4a - 2b ------ (4),
-20 = 16a + 4b - 4,
-16 = 16a + 4b ------- (5).
Subtracting equation (5) from equation (4) gives:
32 = -20a,
a = -32/20 = -8/5 = -1.6.
Substituting a = -1.6 into equation (4) gives:
16 = -6.4 - 2b,
20 = 2b,
b = 10.
The equation of the parabola is therefore y = -1.6x^2 + 10x - 4.
To write the equation of the parabola in standard form, we need to use the general equation for a parabola:
y = ax^2 + bx + c
We can plug in the given points into this equation and form a system of equations to solve for the values of a, b, and c. Let's do this step-by-step.
Step 1: Plug in the coordinates of the first point (-2, -20)
-20 = a(-2)^2 + b(-2) + c
-20 = 4a - 2b + c -- equation (1)
Step 2: Plug in the coordinates of the second point (0, -4)
-4 = a(0)^2 + b(0) + c
-4 = c -- equation (2)
Step 3: Plug in the coordinates of the third point (4, -20)
-20 = a(4)^2 + b(4) + c
-20 = 16a + 4b + c -- equation (3)
Step 4: Substitute equation (2) into equations (1) and (3)
-20 = 4a - 2b + (-4)
-20 = 4a - 2b - 4 => 4a - 2b = -16 -- equation (4)
-20 = 16a + 4b + (-4)
-20 = 16a + 4b - 4 => 16a + 4b = -16 -- equation (5)
Step 5: Solve the system of equations (4) and (5) simultaneously.
To do this, we can multiply equation (4) by 4 and equation (5) by 2 to eliminate the b terms:
4(4a - 2b) = 4(-16)
16a - 8b = -64 -- equation (6)
2(16a + 4b) = 2(-16)
32a + 8b = -32 -- equation (7)
Add equations (6) and (7) together:
(16a - 8b) + (32a + 8b) = -64 + (-32)
48a = -96
Divide both sides by 48:
a = -96/48
a = -2
Step 6: Substitute the value of a into equation (4) to solve for b:
4a - 2b = -16
4(-2) - 2b = -16
-8 - 2b = -16
-2b = -8
Divide both sides by -2:
b = -8/-2
b = 4
Step 7: Substitute the values of a and b into equation (2) to solve for c:
c = -4
Step 8: Write the equation of the parabola in standard form:
y = ax^2 + bx + c
y = -2x^2 + 4x - 4
So, the equation of the parabola in standard form is y = -2x^2 + 4x - 4.
To find the equation of a parabola in standard form that passes through three given points, you can use the general equation of a parabola, which is y = ax^2 + bx + c.
Step 1: Plug in the x and y coordinates of one of the points into the equation to get an equation with only a, b, and c variables. Let's take the first point (-2, -20):
-20 = a(-2)^2 + b(-2) + c
Step 2: Repeat Step 1 for the second point (0, -4):
-4 = a(0)^2 + b(0) + c
Step 3: Repeat Step 1 for the third point (4, -20):
-20 = a(4)^2 + b(4) + c
Now, we have three equations with three unknowns (a, b, and c) that can be solved simultaneously to find their values.
Step 4: Simplify the three equations obtained in Steps 1-3:
-20 = 4a - 2b + c (equation 1)
-4 = c (equation 2)
-20 = 16a + 4b + c (equation 3)
Step 5: Substitute equation 2 into equations 1 and 3 to eliminate the variable c:
-20 = 4a - 2b - 4 (equation 1, after substituting -4 for c)
-20 = 16a + 4b - 4 (equation 3, after substituting -4 for c)
Step 6: Simplify the two equations obtained in Step 5:
4a - 2b - 4 = -20 (equation 4)
16a + 4b - 4 = -20 (equation 5)
Step 7: Solve the system of equations 4 and 5 to find the values of a and b:
Multiply equation 4 by 2 to eliminate the coefficient of b:
8a - 4b - 8 = -40 (equation 6)
Add equation 5 and equation 6 to eliminate the variable b:
(16a + 4b - 4) + (8a - 4b - 8) = (-20) + (-40)
24a - 12 = -60
Step 8: Solve equation 7 to find the value of a:
24a = -48
a = -2
Step 9: Substitute the value of a into equation 4 to find the value of b:
4(-2) - 2b - 4 = -20
-8 - 2b - 4 = -20
-2b = -8
Solving for b, we get:
b = 12
Step 10: Substitute the values of a, b, and c (which is -4 from equation 2) into the general equation of a parabola to get the standard form:
y = ax^2 + bx + c
y = -2x^2 + 12x - 4
Therefore, the equation in standard form of the parabola that passes through (-2, -20), (0, -4), and (4, -20) is y = -2x^2 + 12x - 4.