Math Problem Statement
Solution
Let's break down the given problem step by step to solve for :
Function Definition
You are given a transformation of a function , such that:
Three transformations are applied to :
- Horizontal dilation by a factor of 2: This implies , since the dilation factor is applied inversely in transformations.
- Vertical dilation by a factor of 3: This means .
- Vertical translation by 5 units: This means .
Thus, the transformation becomes:
Find
Now, we need to evaluate :
From the table, we know that:
Substituting this value into the equation for :
Thus, the value of is 20, which corresponds to answer choice D.
Would you like further explanation or have additional questions?
Here are 5 related questions to expand your understanding:
- How do horizontal and vertical dilations affect the graph of a function?
- What is the impact of a vertical translation on the range of a function?
- How can you generalize the transformation formula for different dilation and translation factors?
- How would the result change if the vertical dilation factor were 4 instead of 3?
- What is the significance of the horizontal dilation being applied inversely in the transformation?
Tip: When dealing with transformations, always adjust horizontal changes with inverse factors (e.g., dividing by the dilation factor instead of multiplying).
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Math Problem Analysis
Mathematical Concepts
Function transformations
Horizontal and vertical dilations
Vertical translations
Formulas
g(x) = a f(bx) + c
Horizontal dilation: f(bx) where b = 1/2
Vertical dilation: a * f(x) where a = 3
Vertical translation: f(x) + c where c = 5
Theorems
Function Transformation Theorem
Suitable Grade Level
Grades 10-12