How do you find the equation of a parabola?

Question

I know that the x-intercepts are (0,0) and (2.4, 0 ), and that the vertex is (1.22, 7.3 ), and now I need to find the eqaution of this parabola using this information, except I have no clue. Help would be great, thanks :) 

Answer 1

Since we know the vertex, we can put this quadratic function in vertex form.

 

y = a(x - h)² + k

 

where:
a is the coefficient of the x² term

Vertex has coordinate (h, k).

 

Plug in the vertex into this form.

 

y = a(x - 1.22)² + 7.3

 

One of the x-intercepts is a y-intercept.  That x-intercept is (0, 0).  The other is (2.4, 0).  We can create two equations.  Plug in these x and y values into the equation.

 

0 = a(0 - 1.22)² + 7.3               eq1

0 = (-1.22)² a + 7.3

0 = 1.4884a + 7.3

 

0 = a(2.4 - 1.22)² + 7.3              eq2

0 = 1.3924a + 7.3                      

 

 

Solve the system for a in both equations.

 

From eq1, we get

 

-7.3 = 1.4884a

 

-4.905 = a

 

 

From eq2, we get

 

-7.3 =  1.3924a

 

-5.243 = a

 

 

You will have two equations that satisfy these conditions.

 

y₁ = -4.905(x - 1.22)² + 7.3

 

y₂ = -5.243(x - 1.22)² + 7.3

 

 But since both values of a are close to the integer -5, we can go with
 
y = -5(x - 1.22)² + 7.3