Can you use algebra to find a quadratic equation if your know the coordinates of just ONE point on its graph?
Can you use algebra to find a quadratic equation if your know the coordinates of just TWO points on its graph?
If so, what information is needed?
You mus know 3 coordinats if you want find a quadratic equation.
In general, you need three points, but if you know that one of the points is the vertex, then that gives you extra information. If you know (h,k), then the parabola is
y-k = a(x-h)^2
so only "a" is unknown. The second point will provide that.
Yes, you can use algebra to find a quadratic equation if you know the coordinates of just one point on its graph. To find the equation, you need to use the general form of a quadratic equation, which is in the form of:
y = ax^2 + bx + c
Substituting the known coordinates (x, y) into the equation will give you an equation with three unknowns: a, b, and c. However, with just one point, you can only solve for one of these unknowns, which is not enough to determine the entire equation.
On the other hand, if you know the coordinates of two points on the graph, you can use algebra to find a quadratic equation. To do this, you need to set up a system of two equations using the known coordinates.
Let's say the coordinates of the two points are (x₁, y₁) and (x₂, y₂). Substituting these values into the general form of a quadratic equation, you get the following equations:
y₁ = ax₁^2 + bx₁ + c
y₂ = ax₂^2 + bx₂ + c
Now, you have two equations with three unknowns (a, b, and c). To solve for these unknowns, you need one additional piece of information. This can be either the value of one of the unknowns or an extra condition. For example, if you know the value of one of the coefficients, you can substitute it into one of the equations to get a system of two equations with two unknowns, which is solvable.
Alternatively, you can use the values of the x-coordinates of the points to set up a system of equations involving the differences between the x₁, x₂, and x₃ values, allowing you to eliminate the c term. By solving this system of equations simultaneously, you can find the values of a and b and then substitute them back into the general form of the quadratic equation to get the specific equation.