I need help with this parabola question.
A parabola with the axis of symmetry in the y-axis passes through the points (1,4) and (-3, -20). Determine its equation algebraically. Could someone give me some help, I havent seen a question like this before, Could you show your work please.
A parabola with the axis of symmetry in the y-axis passes through the points (1,4) and (-3, -20). Determine its equation algebraically. Could someone give me some help, I havent seen a question like this before, Could you show your work please.
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Ok, the best way I can think of to do a problem like this would be to think about the general equation of a parabola y=ax^2+bx+c. If the y-axis is a line of symmetry, then b must be 0. This makes sense because now the equation is just y= ax^2 +c. It's just a parabola moved up or down the y-axis by c places.
So now we can use the two points (1,4) and (-3,-20) to create two equations which we can solve simultaneously to find the values of a and c.
4 = a * 1^2 +c __________________ eqn 1
-20 = a * (-3)^2 +c ________________ eqn 2
eqn 2 - eqn 1
-24 = 8a
a = -3
substitute into equ 1 to find c:
c= 7
Therefore the equation of your parabola is y= -3x^2 +7
If you draw this on the computer or a calculator you'll see it does indeed go through (1,4) and (-3, -20)
Hope that helped!
So now we can use the two points (1,4) and (-3,-20) to create two equations which we can solve simultaneously to find the values of a and c.
4 = a * 1^2 +c __________________ eqn 1
-20 = a * (-3)^2 +c ________________ eqn 2
eqn 2 - eqn 1
-24 = 8a
a = -3
substitute into equ 1 to find c:
c= 7
Therefore the equation of your parabola is y= -3x^2 +7
If you draw this on the computer or a calculator you'll see it does indeed go through (1,4) and (-3, -20)
Hope that helped!
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The general form of the equation of a parabola is
y = f(x) = ax² + bx + c.
Since (1,4) and (-3, -20) belong to the parabola,
f(1) = 4 and f(-3) = -20.
Hence the equations:
1a + 1b + 1c = 4 . . . (1)
9a − 3b + 1c = -20. . (2)
Since the parabola is symmetrical with respect to
the y-axis, f(-x) = f(x) for all x, in particular f(-1) = f(1).
Hence the equation
1a − 1b + 1c = 4 . . . (3)
Solving the system of equations (1), (2) and (3), we get
(a, b, c) = (-3, 0, 7)
The equation of the parabola is
y = -3x² + 7
y = f(x) = ax² + bx + c.
Since (1,4) and (-3, -20) belong to the parabola,
f(1) = 4 and f(-3) = -20.
Hence the equations:
1a + 1b + 1c = 4 . . . (1)
9a − 3b + 1c = -20. . (2)
Since the parabola is symmetrical with respect to
the y-axis, f(-x) = f(x) for all x, in particular f(-1) = f(1).
Hence the equation
1a − 1b + 1c = 4 . . . (3)
Solving the system of equations (1), (2) and (3), we get
(a, b, c) = (-3, 0, 7)
The equation of the parabola is
y = -3x² + 7
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The equation of a parabola with the axis of symmetry in the y-axis is of the form
y = ax² + b
Since (1,4) is on the parabola,
4 = a(1)² + b = a + b
Since (-3,-20) is on the parabola,
-20 = a(-3)² + b = 9a + b
Solving 4 = a + b and -20 = 9a + b silmultaneously yields
a = -3
b = 7
So, y = -3x² + 7 is the equation you are looking for.
y = ax² + b
Since (1,4) is on the parabola,
4 = a(1)² + b = a + b
Since (-3,-20) is on the parabola,
-20 = a(-3)² + b = 9a + b
Solving 4 = a + b and -20 = 9a + b silmultaneously yields
a = -3
b = 7
So, y = -3x² + 7 is the equation you are looking for.
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y = ax^2 + b
Plug in (1, 4) and (-3, -20),
4 = a + b ......(1)
-20 = 9a + b ......(2)
(2)-(1): -24 = 8a
a = -3
b = 7
Answer: y = -3x^2 + 7
Plug in (1, 4) and (-3, -20),
4 = a + b ......(1)
-20 = 9a + b ......(2)
(2)-(1): -24 = 8a
a = -3
b = 7
Answer: y = -3x^2 + 7