Write a rule for g

Question

Let the graph of g be a translation 5 units up, followed by a reflection in the y-axis of the graph of f(x)=x4−2x3−12

Write a rule for g.

Answer 1

a vertical translation 5 units up would be

f(x) + 5

reflection in the y axis would be f(−x)

g(x) = (−x)⁴ – 2 (−x)³ – 12 + 5

g(x) can be simplified by combining like terms and simplifying x:

g(x) = x⁴ –2 × (−x³) − 7

So (−x)⁴ = -x¹ × -x¹ × -x¹ × -x¹ = x² × x² = x⁴ (You must add exponents because they're the same base. And it will become positive when you multiply an even amount of negative numbers)

(-x)³ = (-x)¹ × (-x)¹ × (-x)¹ = x² × (-x)¹ = -x³ (this remain negative because you multiply an odd amount of negative numbers)

Finally:

g(x) = x⁴ –2(−x³) − 7 Since -2×-1 = 2 , we can turn the middle term into positive 2x³

g(x) = x⁴ +2x³ − 7

Answer 2

To translate the graph 5 units up how do you change the equation? This means that each y value will be 5 units greater than the original. To do this, you just change y to (y-5). Think about it. If originally f(x) = y = x⁴− 2x³ −1 = say 0. Now, in order for y to equal 5, you must add 5 to the right hand side of the equation. You can rearrange both sides so you get y-5 = x⁴− 2x³ −1 (this is just an example for you to see the new change)

To reflect the graph across the y axis, this means that all x values are going to change sign. This can easily be done by changing all x to (-x).

The new equation should be y-5 = x⁴ + 2x³ -1