what is the equation in standard form of a parabola that contain the following points?
(-2,-20),(0,-4),(4,-20)
Please Help
you can probably save some time by looking at the points. You know that parabolas are symmetric, so, since y(-2) = y(4), the axis of symmetry is at x=(-2+4)/2 = 1.
So, with x=1 the axis of symmetry (and hence at the vertex), you will have
y = a(x-1)^2+k
at x=0, y= -4, so
-4 = a+k
-20 = 9a + k
k=-2
a=-2
so, y = -2(x-1)^2 - 2
or, y = -2x^2 + 4x - 4
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or, if you let y = ax^2 + bx + c,
-20 = 4a - 2b + c
-4 = c
-20 = 16a + 4b + c
a = -2
b = 4
c = -4
y = -2x^2 + 4x - 4
Thanks for helping I have posted two other questions if you might be able to help
To find the equation in standard form of a parabola that passes through the given points, we can use the standard form equation of a parabola: y = ax^2 + bx + c.
Step 1: Plug in the coordinates (-2, -20)
-20 = a(-2)^2 + b(-2) + c
-20 = 4a - 2b + c
Step 2: Plug in the coordinates (0, -4)
-4 = a(0)^2 + b(0) + c
-4 = c
Step 3: Plug in the coordinates (4, -20)
-20 = a(4)^2 + b(4) + c
-20 = 16a + 4b + c
Since we know c = -4 (from step 2), we substitute this value into equations from step 1 and 3:
-20 = 4a - 2b - 4 (Equation 1)
-20 = 16a + 4b - 4 (Equation 2)
Simplifying Equation 1:
4a - 2b = -16 (Equation 1')
Simplifying Equation 2:
16a + 4b = -16 (Equation 2')
Step 4: Solve the system of equations (Equation 1' and Equation 2').
Multiply Equation 1' by 2:
8a - 4b = -32
Add this equation to Equation 2':
(8a - 4b) + (16a + 4b) = -32 + (-16)
Simplifying:
24a = -48
Divide by 24:
a = -48/24
a = -2
Substitute the value of a into Equation 1':
4(-2) - 2b = -16
-8 - 2b = -16
Subtract -8 from both sides:
-2b = -8
Divide by -2:
b = 4
Step 5: Write the equation in standard form.
Using the values of a = -2, b = 4, and c = -4 (from Step 2), the equation of the parabola in standard form is:
y = -2x^2 + 4x - 4
To find the equation in standard form of a parabola that passes through these three points, we can use the general form of a parabola equation: y = ax^2 + bx + c. We need to determine the values of a, b, and c.
Step 1: Plug in the given points into the equation to form a system of three equations:
For the point (-2, -20):
-20 = a(-2)^2 + b(-2) + c
For the point (0, -4):
-4 = a(0)^2 + b(0) + c
For the point (4, -20):
-20 = a(4)^2 + b(4) + c
Step 2: Simplify the equations:
For the point (-2, -20):
-20 = 4a - 2b + c
For the point (0, -4):
-4 = c
For the point (4, -20):
-20 = 16a + 4b + c
Step 3: Since we have obtained the value of c from the equation for the point (0, -4), we can substitute it into the other two equations:
For the point (-2, -20):
-20 = 4a - 2b - 4
For the point (4, -20):
-20 = 16a + 4b - 4
Step 4: Simplify further to eliminate c:
For the point (-2, -20):
-16 = 4a - 2b
For the point (4, -20):
-16 = 16a + 4b
Step 5: Solve the system of equations. There are multiple methods to solve this system, such as substitution or elimination. Let's use the elimination method:
Multiply the equation for the point (-2, -20) by 2 and add it to the equation for the point (4, -20) to eliminate b:
-32 = 8a - 4b
-16 = 16a + 4b
--------------
-48 = 24a
Solving for a:
a = -2
Substitute the value of a back into one of the previous equations to solve for b:
-16 = 16(-2) + 4b
-16 = -32 + 4b
16 = 4b
b = 4
Step 6: Substitute the values of a, b, and c back into the equation:
y = ax^2 + bx + c
y = -2x^2 + 4x - 4
Thus, the equation in standard form of the parabola passing through the given points (-2, -20), (0, -4), and (4, -20) is:
y = -2x^2 + 4x - 4