Question

Given a graph for the transformation of f(x) in the format g(x) = f(kx), determine the k value.

two parabolas open up with f of x passing through 2 comma 5 and g of x passing through 1 comma 5

k = −2
k = 2
k = negative one half
k = one half

Answer 1

The correct option is 2, i.e., x=2.

Explanation:

The transformation of f(x) is defined as

`g(x)=f(kx)` .... (1)

We need to find the value of k.

It is given that f(x) passing through (2,5) and g(x) passes through (1,5). it means

`f(2)=5` .... (2)

`g(1)=5` ... (3)

On equating (2) and (3) we get

`g(1)=f(2)` .... (4)

On comparing (1) and (4) we get

`x=1`

`kx=2`

Substitute x=1 in this equation.

`k(1)=2`

`k=2`

The value of k is 2. Therefore, the correct option is 2.

Answer 2

Option B (k = 2).

Explanation:

It can be seen the y-intercept of g(x) is y = 9 and the turning point of g(x) is (1, 5). The equation of the quadratic function involving turning point is given by:

y = a*(x - h)^2 + k; where a determines whether the function is a minimum function (a > 0) or a maximum function (a < 0), and (h, k) is the turning point. Put (h, k) = (1, 5) in the equation:

y = a*(x - 1)^2 + 5.

Since f(0) = 9, therefore:

9 = a*(0 - 1)^2 + 5.

9 = a(1) + 5

a = 4.

Putting a = 4 in the equation gives g(x):

y = 4(x - 1)^2 + 5.

Simplifying:

y = 4*(x^2 + 1 - 2x) + 5.

y = 4x^2 + 4 - 8x + 5

y = 4x^2 - 8x + 9.

Therefore, g(x) = 4x^2 - 8x + 9.

In the similar fashion, find f(x).

y = a*(x - h)^2 + k.

Put (h, k) = (2, 5), x = 0, and y = 9 in the above equation:

9 = a*(0 - 2)^2 + 5.

9 = 4a + 5

4a = 4.

a = 1.

Therefore:

f(x) = (x - 2)^2 + 5.

f(x) = x^2 + 4 - 4x + 5.

f(x) = x^2 - 4x + 9.

It is given that g(x) = f(kx).

Therefore:

4x^2 - 8x + 9 = (kx)^2 - 4kx + 9.

4x^2 - 8x = k^2*x^2 - 4kx.

By comparison:

4 = k^2 and -8 = -4k

k = ±2 and k = 2.

Therefore, k = 2 is the correct answer (Option B)!!!