The function −2𝑓(2𝑥 + 4) − 1 has been transformed from the graph of 𝒇(𝒙) = 𝒙^𝟑
Write the equation for the transformed function.
g(x) = -2 (2x - 4)³ - 1
To find the equation for the transformed function, we need to understand the effect of each transformation.
1. Horizontal Translation:
The expression inside the function, 2𝑥 + 4, represents a horizontal translation. In this case, the graph is being shifted 4 units to the left since the value of 𝑥 has increased by 4.
The original graph of 𝒇(𝒙) = 𝒙^𝟑 would be shifted 4 units left, so the new equation is 𝒇(𝒙 - 4) = (𝒙 - 4)^3.
2. Vertical Stretch/Compression:
The coefficient -2 outside the function represents a vertical stretch or compression. A negative value reflects the graph in the x-axis, but in this case, it also causes a vertical compression.
The original graph of 𝒇(𝒙) = 𝒙^𝟑 would be compressed vertically by a factor of 2, so the new equation is -2(𝒇(𝒙 - 4)) = -2(𝒙 - 4)^3.
3. Vertical Translation:
The constant term -1 in the function translates the graph vertically. In this case, the graph is shifted 1 unit down.
The original graph of 𝒇(𝒙) = 𝒙^𝟑 would be shifted 1 unit down, so the final equation is -2(𝒇(𝒙 - 4)) - 1 = -2(𝒙 - 4)^3 - 1.
Therefore, the equation for the transformed function is -2(𝒙 - 4)^3 - 1.