Write the equation, in standard form, of the parabola containing the following points: (0,1), (1,5), (2,3)
if y = ax^2+bx+c, then plug in your points to get
0a+0b+c = 1
a+b+1 = 5
4a+2b+1 = 3
Solve to get a = -3, b=7
y = -3x^2+7x+1
Why did the parabola go to therapy? Because it had too many points of inflection and couldn't find its focus!
To find the equation of a parabola in standard form, we need to use the general equation of a parabola, which is y = ax^2 + bx + c.
First, let's substitute the given points into the equation to create a system of equations:
For the point (0,1):
1 = a(0)^2 + b(0) + c
1 = c
So, we have c = 1.
For the point (1,5):
5 = a(1)^2 + b(1) + c
5 = a + b + 1
And for the point (2,3):
3 = a(2)^2 + b(2) + c
3 = 4a + 2b + 1
Now, we have a system of equations:
c = 1
a + b = 4 (from the second point)
4a + 2b = 2 (from the third point)
From the equation c = 1, we know c = 1.
To solve for a and b, we can use the second and third equations. Multiply the second equation by 2 to get rid of the b coefficient:
2a + 2b = 8
Now we can subtract this equation from the third equation to eliminate b:
4a + 2b - (2a + 2b) = 2 - 8
2a = -6
a = -3
Substitute the value of a into the equation a + b = 4 to solve for b:
-3 + b = 4
b = 7
So, now we know that a = -3 and b = 7.
Substitute these values back into the general equation:
y = ax^2 + bx + c
y = -3x^2 + 7x + 1
Therefore, the equation of the parabola in standard form is:
y = -3x^2 + 7x + 1.
To find the equation of a parabola given three points, we can use the standard form of the equation of a parabola: y = ax^2 + bx + c.
Let's start by substituting the x and y values from one of the given points into the equation. Let's choose the first point (0,1):
1 = a(0)^2 + b(0) + c
1 = c
So we now know that c = 1.
Next, let's substitute the x and y values from the second point (1,5) into the equation:
5 = a(1)^2 + b(1) + 1
5 = a + b + 1
4 = a + b
Finally, let's substitute the x and y values from the third point (2,3) into the equation:
3 = a(2)^2 + b(2) + 1
3 = 4a + 2b + 1
2 = 4a + 2b
Now we have a system of two equations:
4 = a + b
2 = 4a + 2b
We can solve this system of equations to find the values of a and b.
One way to do this is by substitution: you can solve one equation for one variable and substitute that value into the other equation. However, let's use the elimination method.
Multiply the second equation by 2 to make the coefficients of the variable b the same:
4 = a + b
4 = 8a + 4b
Now we can subtract the first equation from the second equation:
4 = 8a + 4b
-4 = -a - b
0 = 7a + 3b
Solve this new equation for a:
7a + 3b = 0
7a = -3b
a = -3/7b
Substitute this value of a back into the first equation:
4 = (-3/7b) + b
4 = (4b - 3b) / 7
4 = b / 7
b = 28
Now substitute the value of b back into the first equation to find a:
4 = a + 28
a = -24
So the values of a and b are a = -24 and b = 28. We also know that c = 1, as we found earlier.
Now we can write the equation of the parabola in standard form:
y = ax^2 + bx + c
y = -24x^2 + 28x + 1
Therefore, the equation, in standard form, of the parabola containing the points (0,1), (1,5), and (2,3) is y = -24x^2 + 28x + 1.