Write a rule for g. Then identify the vertex.

Question

The graph of g is a vertical stretch by a factor of 4 and a reflection in the x-axis, followed by a translation 2 units up of the graph of f(x)=x2

 

i have 2 blanks to fill.

g(x)= ____

the vertex is (x,y)

algebra 2 work. no idea what the rule for g would be.

Answer 1

While I'm sure you no longer need the answer to this question, perhaps others can benefit from it being answered...

First, consider that the standard form for the quadratic function is f(x)=a(x-h)²+k, where a is the vertical stretch/orientation factor, h represents the horizontal translation, and k represents the vertical translation. Therefore, the standard form for the equation f(x)=x² is actually f(x)=1(x-0)²+0.

Then, to find g(x), all you need to do is enter the correct numbers in the correct places (a, h, k) in the f(x) function. Since there is both a vertical stretch of 4, and vertical "flip" over the x-axis, a=^-4. There is no horizontal translation, so h is still 0. And since there's a vertical translation up 2, k=⁺2. So g(x)=^-4(x-0)²+2; or in simplified form, g(x)=^-4x²+2.

Keep in mind that if there had been a horizontal translation, h would be exactly that number and the expression in parentheses would be x minus the number.

Ex1: h=3 (right 3 units)... h(x)=^-4(x-3)²+2

Ex2: h=^-3 (left 3 units)... j(x)=^-4(x-^-3)²+2=^-4(x+3)²+2

Notice how what's in parentheses seems to reflect opposite of the movement. Really, what's inside parentheses must equal 0 after the translation is applied.

Ex1a: 3-3=0

Ex2a: ^-3+3=0

The vertex is always found at (h,k). So, for f(x) the vertex is (0,0); and for g(x) the vertex is (0,2).

I hope this helps!