A parabola has a vertex located at $(6,-5)$ and passes through the point $(8,3)$ . Use the pattern of the parabola to look for a vertical or horizontal stretch. Write the equation of the graph in vertex form.

Full solution

Q. A parabola has a vertex located at

(6,-5)

and passes through the point

(8,3)

. Use the pattern of the parabola to look for a vertical or horizontal stretch. Write the equation of the graph in vertex form.

Understand Vertex Form: First, we need to understand that the vertex form of a parabola is given by $y = a(x-h)^2 + k$ , where $(h,k)$ is the vertex of the parabola. We are given the vertex $(6,-5)$ , so we can substitute $h$ and $k$ into the equation.
Substitute Vertex: Substituting the vertex into the vertex form equation, we get $y = a(x-6)^2 - 5$ . Now, we need to find the value of $a$ using the point $(8,3)$ that lies on the parabola.
Find Value of 'a': Plugging the point $(8,3)$ into the equation, we get $3 = a(8-6)^2 - 5$ . Simplifying the right side, we have $3 = a(2)^2 - 5$ , which simplifies to $3 = 4a - 5$ .
Solve for 'a': To find the value of 'a', we solve the equation $3 = 4a - 5$ for 'a'. Adding $5$ to both sides gives us $8 = 4a$ , and dividing both sides by $4$ gives us $a = 2$ .
Write Final Equation: Now that we have the value of $a$ , we can write the final equation of the parabola. Substituting $a$ into the equation $y = a(x-6)^2 - 5$ , we get $y = 2(x-6)^2 - 5$ .