Question

Given the parent function f(x) = x², define the function g(x) that

transforms f(x) as follows:

Reflection over the x-axis
Horizontal shift left 3 units
Vertical shift up 7 units

Answer 1

g(x) = -(x-4)².

Explanation:

Given : parent function f(x) = x², Reflection over the x-axis , Horizontal shift left 3 units and Vertical shift up 7 units.

To find : define the function g(x) .

Solution : We have given

Parent function f(x) = x²

By the reflection rule over x axis : f(x) →→ - f(x).

So, f(x) = x² →→ - x².

By the translation rule : f(x) →→ →→ f(x+h-k).it mean function shifted horizontally to ward left. and vertically up by k units

Then it would be - x² →→ →→ - (x+3-7)².

f(x)→→ →→ g(x) = -(x-4)².

Therefore, g(x) = -(x-4)².

Answer 2

Part 1)
`g(x)=-x^(2)`

Part 2)
`g(x)=(x+3)^(2)`

Part 3)
`g(x)=x^(2)+7`

Explanation:

we have

The parent function is
`f(x)=x^(2)`

This is a vertical parabola open upward with vertex at origin (0,0)

Part 1) Reflection over the x-axis

The rule of the reflection across the x=axis is equal to

(x,y) ------> (x,-y)

so

f(x) -----> g(x)

The function g(x) will be

`g(x)=-x^(2)`

Part 2) Horizontal shift left 3 units

The rule of the translation is

(x,y) ------> (x-3,-y)

so

f(x) -----> g(x)

The function g(x) will be

`g(x)=(x+3)^(2)`

Part 3) Vertical shift up 7 units

The rule of the translation is

(x,y) ------> (x,y+7)

so

f(x) -----> g(x)

The function g(x) will be

`g(x)=x^(2)+7`