The function 𝑄(𝑥) has been transformed from the function 𝑓(𝑥) = 𝑥^4
. The mapping rule for the
function can be written as (2𝑥 + 𝒄, 4𝑦 + 𝒃). If the coordinate(1/2 ,1/16) on the function 𝑓(𝑥) is
transformed to (3,5) on Q(x), determine the value of c and 𝒅.
To determine the values of 𝒄 and 𝒅, we need to match the transformations from the equation (2𝑥 + 𝒄, 4𝑦 + 𝒅) with the given points.
Let's start with the initial point on the function 𝑓(𝑥): (1/2, 1/16)
The equation 𝑓(𝑥) = 𝑥^4 tells us that for the input x, the output y is given by y = (x^4).
So, for the point (1/2, 1/16) on 𝑓(𝑥), we have:
x = 1/2
y = 1/16
Now, let's move on to the transformed point on 𝑄(𝑥): (3, 5)
According to the mapping rule (2𝑥 + 𝒄, 4𝑦 + 𝒅), we have:
x = 3
y = 5
Comparing the coordinates, we can equate the transformations:
2𝑥 + 𝒄 = 3 (Equation 1)
4𝑦 + 𝒅 = 5 (Equation 2)
Now, let's solve these equations to find the values of 𝒄 and 𝒅.
From Equation 1:
2𝑥 + 𝒄 = 3
2(3) + 𝒄 = 3
6 + 𝒄 = 3
𝒄 = 3 - 6
𝒄 = -3
From Equation 2:
4𝑦 + 𝒅 = 5
4(5) + 𝒅 = 5
20 + 𝒅 = 5
𝒅 = 5 - 20
𝒅 = -15
Therefore, the values of 𝒄 and 𝒅 are 𝒄 = -3 and 𝒅 = -15, respectively.