Question

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

Answer 1

Answer:

g(x) = -(x-4)².

Step-by-step 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

Answer:

Part 1) $g(x)=-x^{2}$

Part 2) $g(x)=(x+3)^{2}$

Part 3) $g(x)=x^{2}+7$

Step-by-step 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

$g(x)=x^{2}+7$