How to find the equation if given the graph?

 

Well, it's really easy to determine the equation!

 

Let's start with the vertex, meaning the "h" and "k" values of the equation!

The point for the vertex will be shown in the graph. 

• The horizontal translation (how far left or right is the parabola located) is the "h" value (the x-value).

• The vertical translation (how far up or down is the parabola located) is the "k" value (y-value).

 

Once you have the the vertex, substitute the x-value for "h" in the equation (remember to switch the sign), and substitute the y-value for "k" in the equation (keep the same sign).

 

Now how do we find the "a" value of the equation?

You will be provided with atleast one other point of the parabola (other than the vertex).

• Substitute the x-value of that point for the "x" value in the equation.

• Substitute the y-value of that point for the "y" value in the equation. 

 

 

 

 

 

 

 

 

 

 

Let's try an example...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For this parabola, the vertex point is (3,-2), since that is minimum value that the parabola reaches.

• Now substitute this in the equation, y=a(x-h)^2+k.

• The "3" in the vertex is the "h" value, so it will become y=a(x-3)^2+k (remember to switch the sign, so if it is positive, like in this case, the sign in front of "h" becomes negative.

• The "-2" in the vertex is the "k" value, so it will become y=a(x-3)^2-2 (the sign for "k" remains the same as it was in the vertex).

 

Time to find the "a" value!

• Choose any other point, except the vertex.

• For now, let's go with (1,6).

• Substitute the "1" for the "x" in the equation, so it will become y=a(1-3)^2-2.

• Substitute the "6" for the "y" in the equation, making it 6=a(1-3)^2-2.

• Now solve for "a"

 

6=a(-2)^2-2    (6=a(1-3)^2-2)

 

6=a(4)-2         (6=a(-2)^2-2)

 

6+2=a(4)-2+2    (6=a(4)-2)   

 

8=a(4)           (6+2=a(4)-2+2)

 

8/4=a(4)/4    

 

2=a          (8/4=a(4)/4)

 

 

The equation is y=2(x-3)^2-2

 

 

Video for better understanding!!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Want to try one by yourself?

 

Sure!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer is...

 

 

y=-(x-2)^2+3

 

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