Rewriting f(x) as a Transformation of g(x)

In summary, the conversation is about rewriting a quadratic function in vertex form and understanding how it can be thought of as a transformation of the graph of y = x^2. The final equation for f(x) is given as a(x + h)^2 + k, with h and k representing the translations and transformations relative to the graph of y = x^2.
  • #1
halvizo1031
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0

Homework Statement



Can someone assist me with number 3 please...

Homework Equations





The Attempt at a Solution



I went ahead and rewrote f(x) in vertex form in terms of a,b, and c but I'm having a hard time writing down how f can be thought of as a transformation of g.
 

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  • #2


What did you get? You have f(x) = ax2 + bx + c. Complete the square to get this into a form so that you can recognize the translations and other transformations relative to the graph of y = x2.
 
  • #3


i got a[x+(b/2a)]^2 - [(b^2)/4a] + c where h=-(b/2a) and k=c-[(b^2)/4a]
 
  • #4


halvizo1031 said:
i got a[x+(b/2a)]^2 - [(b^2)/4a] + c where h=-(b/2a) and k=c-[(b^2)/4a]
Shouldn't h be + b/(2a)?
So f(x) = a(x + b/(2a))2 + c - b2/(4a)

Or, f(x) = a(x + h)2 + k, with h = b/(2a) and k = c - b2/(4a).

You know what the graph of y = g(x) = x2 looks like, right? How would you need to transform the graph of g to get the graph of f?
 

FAQ: Rewriting f(x) as a Transformation of g(x)

What is the vertex form of a quadratic function?

The vertex form of a quadratic function is f(x) = a(x-h)^2 + k, where (h,k) represents the coordinates of the vertex.

How do you rewrite a quadratic function in vertex form?

To rewrite a quadratic function in vertex form, complete the square by adding and subtracting (b/2)^2, where b is the coefficient of the x term. Then, factor the resulting perfect square trinomial and simplify the expression.

Why is it useful to rewrite a quadratic function in vertex form?

Rewriting a quadratic function in vertex form allows us to easily identify the coordinates of the vertex, which gives us information about the minimum or maximum point of the function. It also helps us graph the function more accurately.

Can any quadratic function be rewritten in vertex form?

Yes, any quadratic function can be rewritten in vertex form by completing the square.

Is the vertex form of a quadratic function unique?

Yes, the vertex form of a quadratic function is unique because the coordinates of the vertex are unique for a given function. However, the coefficients a, h, and k may vary depending on the original function.

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