Question

determine the transformation on $g(t) = 3t + 1$ that results in the function $h(t) = 6(3t + 1)$.

• horizontal dilation by a factor of 6
• vertical dilation by a factor of 6
• horizontal translation of 6 units
• vertical translation of 6 units

Explanation:

To determine the transformation from \( g(t) = 3t + 1 \) to \( h(t) = 6(3t + 1) \), we recall the rules of function transformations. A vertical dilation (stretch or compression) of a function \( y = f(x) \) by a factor of \( a \) (where \( a>0 \)) is given by \( y = a \cdot f(x) \). Here, \( h(t) = 6 \cdot g(t) \), which matches the form of a vertical dilation by a factor of 6. Horizontal dilation would involve a change inside the function (e.g., \( g(kt) \)), and translation involves adding or subtracting a constant inside or outside the function, which is not the case here.

Answer:

B. Vertical dilation by a factor of 6