Final answer:
The vertex form calculator can be used to find the vertex of a quadratic equation when the vertex and another point are plugged in.
Step-by-step explanation:
The vertex form of a quadratic equation is given by y = a(x-h)^2 + k, where (h,k) is the coordinates of the vertex. To find the vertex form calculator, we need to input the coordinates of the vertex and another point into the equation and solve for the value of a.
Let's take an example to better understand this process. Suppose we have a quadratic equation y = 2x^2 + 12x + 16 and we want to find the vertex form calculator when the vertex is (-3, 4) and another point is (1, 10).
Step 1: Input the coordinates of the vertex and another point into the equation.
4 = 2(-3)^2 + 12(-3) + 16
10 = 2(1)^2 + 12(1) + 16
Step 2: Simplify the equations.
4 = 18
10 = 30
Step 3: Subtract the two equations to eliminate the variable a.
4 = 18
-6 = 12
Step 4: Solve for a by dividing both sides by -6.
a = -2
Step 5: Substitute the value of a into the original equation to get the vertex form.
y = -2(x-h)^2 + k
Therefore, the vertex form calculator for the given quadratic equation is y = -2(x+3)^2 + 4.
Explanation: In this process, we first plugged in the coordinates of the vertex and another point into the equation and simplified it to eliminate the variable a. By solving for a, we were able to find the value of a and substitute it back into the original equation to get the vertex form. This form is useful in graphing quadratic equations as it gives us the coordinates of the vertex, which is the highest or lowest point on the graph.
It is important to note that this process works for any quadratic equation, not just the one given in the example. By following the same steps, we can find the vertex form calculator for any given equation. This calculator can be a useful tool in solving quadratic equations and understanding their properties. It allows us to easily find the vertex and plot it on a graph, giving us a better understanding of the behavior of the equation. In conclusion, the vertex form calculator is a helpful tool in analyzing and graphing quadratic equations.
Complete Question:
How can the vertex form calculator be utilized to find the vertex of a quadratic equation, and what steps are involved in this process, as demonstrated in the explanation using an example equation and specific coordinates for the vertex and another point?