transformations

Question

y=f(x) , y=g(x)

f(x)=x^2 and g(x)=-x^2+6x-13

the graph of y=g(x) can be obtained by from the graph y=f(x) by first reflecting the graph of y=f(x) and then translating the graph of y=f(x)

 

describe fully each of the transformations 

Answer 1

If we convert g(x) into vertex form, we can easily see the transformation taking place.

 

A function in vertex form is

 

g(x) = a(x - h)² + k

 

where:

a is the coefficient of the x² term

the vertex has coordinates (h, k)

 

g(x) = -(x² - 6x + 9) - 4

 

g(x) = -(x - 3)(x - 3) - 4

 

g(x) = -(x - 3)² - 4

 

 

This means that after reflecting f(x) about the x-axis, we shift f(x) 3 units to the right.  Then shift f(x) 4 units down.

Answer 2

Hey Aa.

 

You're interested in seeing how the vertex of the parabola for f(x) is different from that for g(x). Essentially you're looking for a parabola whose equation is:

 

y - k = A*(x - h)^2

 

And you want to compare (h,k) for each one.

 

For f(x) it's really simple:

 

y - 0 = 1*(x - 0)^2

 

So (h,k) in this case is (0,0)

 

For g(x) it'll take a little more manipulation.

 

Let's first pull out a -1 from the right side of the equation:

 

g(x) = -(x^2 - 6x + 13)

 

We can rewrite this as:

 

g(x) = -(x^2 - 6x + 9 + 4)

 

We then recognize the first three terms as a completed square.

 

g(x) = -((x - 3)^2 + 4)

 

So, redistributing the -1:

 

g(x) = -(x - 3)^2 - 4

 

g(x) + 4 = -(x - 3)^2

 

Your (h,k) in this case is (3,-4)

 

g(x) is f(x) flipped upside-down, then translated to the right 3 spaces and then down 4 spaces.

 

Hope this helps.