Question

The graphs below have the same shape. What is the equation of the blue
graph?

Answer 1

g(x) = (x + 3)² + 1

Explanation:

Given the function f(x) then f(x + a) represents a horizontal translation of f(x)

• If a > 0 then a shift to the left of a units

• If a < 0 then a shift to the right of a units

The vertex of f(x) is at (0, 0)

While the x- coordinate of the vertex of g(x) is at - 3, that is a shift to the left of 3 units, thus

g(x) = (x + 3)² + c

Given f(x) then f(x) + c represents a vertical translation of f(x)

• If c > 0 then a shift up of c units

• If c < 0 then a shift down of c units

Here g(x) is 1 unit above the x- axis, thus a shift up of 1 unit

Hence

g(x) = (x + 3)² + 1

Answer 2

Blue graph is

`y-1= (x+3)^2`

Explanation:

Given are two graphs one red and one blue.

Blue is got by some transformations on f(x)

Parent function
`f(x) = x^2`

This is the red graph

Comparing blue graph with this, we find that vertex is shifted to (-3,1)

But as far size is concerned both graphs have the same size hence no dilation or shrink. Only shifted vertically and horizontally.

vertical shift is 1 unit up and horizontal shift is 3 units to the left

Hence blue graph would be

`y-1=(x+3)^2`