Question

Write the equation of the function
please help

Answer 1

This is very similar to the previous problem.

Again, the shape of the parent function resembles a square-root function, sqrt(x).

We will examine the values, first translating the beginning of the graph (-3,0) to the origin, i.e. translating to the right by 3 units.

We will extract a few key points,

(you don't have to physically move the curve, just "pretend" (-3,0) is the origin, and count the squares)

(0,0),

(1,2),

(4,4),

(5,4.4)

It looks like the graph is a vertical stretch of the square-root function by a scale factor of two. We will check the points again by dividing the y-values by two:

(0,0)

(1,1)

(4,2)

(5,2.2)

Which corresponds exactly with a square-root function (at least for the points we chose).

What have we done so far: translate to the right, reduced by a factor of two.

So if we start from the parent function, f(x)=sqrt(x), we will just do the inverse transformations, namely, stretch vertically by two, and translate to the LEFT by 3.

The transformation

g(x)=a*f(x-h)+k

will stretch f(x) vertically by a scale factor of a, translates to the RIGHT by h, and translate UP by k.

Substituting

a=2

h=-3 (to the left)

k=0 (no vertical translation), we get

g(x)=2f(x-(-3))+0=2sqrt(x+3), which is the required equation of the graph.

Answer 2

This is a square root function that has the general form of
`y=a√(x-h)+k`. We can find the equation for this particular function by picking a point on the graph (I used (1, 4)) and the vertex of the function as our h and k. The vertex of the function (the starting point of the function) is (-3, 0). Filling in accodingly, we have
`4=a√((1-(-3)))+0`. Simplifying gives us
`4=2a` a=2. Now, filling that in we get
`y=2√(x+3)` is our equation.