Write a rule for g described by the transformation of the graph of f. Then identify the vertex.

Question

f(x)=(x-5)^2-6; horizontal shrink by a factor of 1/3 and a translation 3 units down, followed by a reflection in the x-axis.

Answer 1

Let's pick a point on our parabola and see how we can move it.

 

f(3) = (3-5)² - 6 = (-2)² - 6 = 4 - 6 = -2

 

We have a point of (3,-2). How can we move it?

 

Horizontal shrink by 1/3? Our new point will be (1,-3), right? How can we plug in 1 and still get the same result? Multiply it by 3. f(1) = -2

 

y = (3x-5)² - 6

 

3 units down? Now we have (1,-5). We need to make our y-value 3 less. We subtract 3 so f(1) = -5

 

y = (3x-5)² - 9

 

Reflect on the x-axis? Now, our point is (1,5). The y-coordinate has to switch signs. f(1) = 5

 

y = -(3x-5)² + 9

 

Note, the negative does not multiply to make (-3x+5)² + 9. We want to square first, then make our result negative.

 

y = -(3x-5)² + 9