Question

Select all the transformations of f(x) = x2 that combine to result in the graph of function g below. A parabola with vertex negative 1 comma negative 2 that opens downward. The graph passes through the points negative 3 comma negative 3 and 1 comma negative 3. Group of answer choices reflection across the x-axis translation of 2 units down translation of 1 unit left translation of 2 units up vertical stretch by a factor of 2 horizontal stretch by a factor of 2 translation of 1 unit right

Answer 1

Explanation:

Equation of a parabola is represented by,

f(x) = -a(x - h)² + k [Parabola opening downwards]

where (h, k) is the vertex of the parabola.

Picture attached shows the vertex as (-1, -2)

Therefore, equation of the transformed parabola will be,

f(x) = -a(x + 1)²- 2

Since the given parabola passes through a point (1, -3)

-3 = -a(1 + 1)² - 2

-1 = -4a²

a =
`\sqrt{(1)/(4) }`

a =
`(1)/(2)`

Therefore, equation of the transformed function will be,

f(x) =
`-(1)/(2)(a+1)^2-2`

If the original or parent function is g(x) = x²,

Transformed function will have the following characteristics,

1). Function will have vertex as (-1, -2) opening downwards.

2). Graph passes through (-3, -3) and (1, -3).

3). Reflection across x-axis.

4). Translation of 2 units down and 1 unit to the left.

5). Horizontal stretch by a factor of 2.

Answer 2

Transformations :

1 translation left of 1.

2. a two-unit down translation

3. factor 2 horizontal stretch

3. reflection along the x-axis

Function f(x), after one unit of translation to the left, becomes f(x + 1).

Given transformation is

f(x) = x²

graph of
`x^2` is upward parabola and vertex at (0, 0)

Transformations:

1- translation 1 unit left f(x) = (x + 1)²

2- horizontal stretch by factor 2

`f(x) = ((x+1)/(2) )^2`

3- reflection over x axis
`f(x)=-((x+1)/(2) )^2`

4- translation of 2 unit down
`g(x) =-((x+1)/(2) )^2-2`