write the equation of the parabola y=ax^2+bx+c that passes through the points(0,3), (1,4), and (2,3)
You can solve for each variable (a, b, c) through forming a system of equations with each set of coordinates. You should have three equations after plugging in.
1. Plug in the coordinates for x and y into the general form. Remember y and f(x) represent the same quantity.
2. Simplify. (Remember the order of operations)
3. Repeat steps 1 & 2 for the other two points.
4. Take two equations at a time and eliminate one variable (c works well)
5. Then repeat using two equations and eliminate the same variable you eliminated in #4.
6. Take the two resulting equations and solve the system (you may use any method).
7. After finding two of the variables, select an equation to substitute the values back into.
8. Find the third variable.
9. Substitute a, b, and c back into the general equation.
since (0,3) and (2,3) have the same y-value, the vertex is at x=1. So, (1,4) is the vertex, and we have
y = a(x-1)^2 + 4
At x=0,
a(1)+4 = 3
a = -1
y = -(x-1)^2 + 4 = -x^2+2x+3
To find the equation of the parabola, we need to substitute the given points into the general equation of a parabola, which is y = ax^2 + bx + c. By doing this, we can form a system of equations to solve for the values of a, b, and c.
Let's start by substituting the coordinates of the first point (0,3) into the equation:
3 = a(0)^2 + b(0) + c
3 = 0 + 0 + c
3 = c
Now, substitute the coordinates of the second point (1,4) into the equation:
4 = a(1)^2 + b(1) + 3 [Substituting c = 3]
4 = a + b + 3
a + b = 1 [Equation 1]
Finally, substitute the coordinates of the third point (2,3) into the equation:
3 = a(2)^2 + b(2) + 3 [Substituting c = 3]
3 = 4a + 2b + 3
4a + 2b = 0 [Equation 2]
Now, we have a system of equations with Equation 1 and Equation 2. Solving this system will give us the values of a and b.
Multiply Equation 1 by 2:
2a + 2b = 2
Subtract the new equation from Equation 2:
4a + 2b - (2a + 2b) = 0 - 2
2a = -2
a = -1
Substitute the value of a into Equation 1:
-1 + b = 1
b = 1 + 1
b = 2
Now that we have the values of a and b, we can substitute them back into the equation of the parabola:
y = ax^2 + bx + c
y = -x^2 + 2x + 3
Therefore, the equation of the parabola that passes through the points (0,3), (1,4), and (2,3) is y = -x^2 + 2x + 3.
To find the equation of a parabola that passes through given points, we need to substitute these points into the general form of the equation y = ax^2 + bx + c and solve the resulting system of equations.
Let's substitute the first point (0,3) into the equation:
3 = a(0)^2 + b(0) + c
This simplifies to:
3 = c
Now, let's substitute the second point (1,4) into the equation:
4 = a(1)^2 + b(1) + c
This simplifies to:
4 = a + b + c
Finally, let's substitute the third point (2,3) into the equation:
3 = a(2)^2 + b(2) + c
This simplifies to:
3 = 4a + 2b + c
Now, we have a system of three equations:
Equation 1: 3 = c
Equation 2: 4 = a + b + c
Equation 3: 3 = 4a + 2b + c
Using these equations, we can solve for a, b, and c.
Substituting Equation 1 into Equation 2:
4 = a + b + 3
Rearranging terms:
a + b = 1 -- Equation 4
Substituting Equation 1 into Equation 3:
3 = 4a + 2b + 3
Rearranging terms:
4a + 2b = 0
Simplifying:
2a + b = 0 -- Equation 5
Now, we have a system of two equations:
Equation 4: a + b = 1
Equation 5: 2a + b = 0
Solving this system of equations, we can find the values of a and b.
By subtracting Equation 5 from Equation 4, we have:
a = 1
Substituting this value of a into Equation 4, we get:
1 + b = 1
Simplifying:
b = 0
We have found the values of a and b. Now, let's substitute these values into Equation 1 to find c:
3 = c
So, we have a = 1, b = 0, and c = 3. Substituting these values into the general form of the equation, our equation of the parabola is:
y = 1x^2 + 0x + 3
Simplifying:
y = x^2 + 3