Answer:
Explanation:
To find a quadratic equation with only two points, we need to use the general form of a quadratic equation, which is:
y = ax² + bx + c
where "a", "b", and "c" are constants and "x" and "y" are the variables.
Let's say the two points we have are (x₁, y₁) and (x₂, y₂). We can use these points to form two equations:
y₁ = ax₁² + bx₁ + c
y₂ = ax₂² + bx₂ + c
We can then solve these equations simultaneously to find the values of "a", "b", and "c".
Here's an example:
Suppose we have two points, (2, 3) and (4, 7), and we want to find the quadratic equation that passes through these two points.
We start by plugging in the coordinates of the first point into the equation:
3 = a(2²) + b(2) + c
Next, we plug in the coordinates of the second point:
7 = a(4²) + b(4) + c
We now have two equations with three variables. To solve for the variables, we can use either substitution or elimination. Here, we will use substitution:
From the first equation, we can solve for "c":
c = 3 - 4a - 2b
We can now substitute this expression for "c" into the second equation:
7 = a(4²) + b(4) + 3 - 4a - 2b
Simplifying the equation, we get:
7 = 16a - 2b + 3
4a - b = 2
We can now use the first equation to solve for "b" in terms of "a" and substitute this expression into the second equation:
b = (3 - 4a - c) / 2
b = (3 - 4a - (3 - 4a - 2b)) / 2
b = -2a + 3
Substituting this expression for "b" into the equation 4a - b = 2, we get:
4a - (-2a + 3) = 2
6a = -1
a = -1/6
Now that we know "a", we can use one of the earlier equations to solve for "c":
3 = (-1/6)(2²) + b(2) + c
c = 3 + (1/3) - b
Substituting this expression for "c" into the equation c = 3 - 4a - 2b, we get:
3 + (1/3) - b = 3 + (2/3) + 2(-2a + 3)
b = -2a + (8/3) = 1/3
Therefore, the quadratic equation that passes through the points (2, 3) and (4, 7) is:
y = (-1/6)x² + (1/3)x + 1
Note that this is a quadratic equation in standard form. If we want to write it in vertex form, we can complete the square.