Writing Equation in Vertex Form Given Vertex and a Point

Suppose we wanted to find the vertex and the standard form of a quadratic given a point and a vertex. Say that fireworks reach 100 feet after 10 seconds and it starts from the ground. We know 2 points on our parabola, 10,100 and 0,0.

We use the vertex form of the equation y = a(x - h)2 + k
y = a(x-10)2 + 100          fill in our vertex
then fill in out point (0,0) for the x and y
0 = a(0-10)2 + 100
0 = a(100) + 100
100a = -100
a = -1

Then put the a into the original equation

y = -1(x - 10)2 + 100
y = -1(x2 - 20x + 100) + 100
y = -x2 + 20x          which is the equation in standard form



Suppose that we fired a rocket and it reached its vertex of 40 feet after 3 seconds. We also know that after 1 second it was at a height of 20. Write the equation of the line. Given that we have the vetex (3,40) and a point (1,20), we can solve the problem.

y = a(x-h)2 + k

y = a(x-3)2 + 40          fill in the vertex
20 = a(1-3)2 + 40         fill in the known point
20 = a(1-3)2 + 40
20 = 4a + 40
4a = -20
a = -5

Then fill that into the standard equation

y = -5(x-3)2 + 40
y = -5(x2 - 6x + 9) + 40
y = -5x2 + 30x - 45 + 40
y = -5x2 + 30x - 5

We can see from the standard equation that the rocket was shot off from 5 feet below ground level.


Fireworks reach a height of 50 feet after 10 seconds (10,50) (starting on the ground (0,0)).
Write the equation of the paraobola

y = a(x-h)2 + k
0 = a(x - 10)2 + 50
0 = 100a + 50
100a = -50
a = -1/2

y = -1/2(x - 10)2 + 50
y = -1/2(x2 - 20x + 100) + 50
y = -1/2x2 + 10x -40 + 40
y = -1/2x2 + 10x





Given Vertex = {-1,-4} and a point on the graph is {0,-3)
Write in Vertex form of the equation and Standard form

y=a(x-h)2 + k
y=a(x--1)2 - 4
y=a(x+1)2 - 4
-3=a(0+1)2 - 4
-3 = a - 4
a = 1

y = 1(x + 1)2 - 4
y = x2 + 2x + 1 - 4
y = x2 + 2x - 3



A rocket goes up from the ground and after 3 seconds it reaches its peak of 90 feet.

We know vertex 3,90 and a point 0,0

y = a(x-h)2 + k

0 = a(0-3)2 + 90
0 = 9a + 90
9a = -90
a = -10

y = -10(x-3)2 + 90
y = -10(x2 - 6x + 9) + 90
y = -10x2 + 60x - 90 + 90
y = -10x2 + 60x