Math Problem Statement

Given the function f(x) = |x|, write the function if f(x) was translated left 1 unit and up 2 units.

Solution

The problem asks you to find the new function g(x)g(x) when the function f(x)=xf(x) = |x| is translated 1 unit to the left and 2 units up.

Translation steps:

  1. Left 1 unit: When you translate a function to the left, you add to the xx-value inside the absolute value. This means the transformation for shifting left 1 unit is x+1|x + 1|.
  2. Up 2 units: When you translate a function up, you add to the entire function. So, adding 2 to the function results in x+1+2|x + 1| + 2.

Thus, the translated function g(x)=x+1+2g(x) = |x + 1| + 2.

The correct answer is:

  • g(x)=x+1+2g(x) = |x + 1| + 2 (Option B)

Would you like further details on transformations?

Here are some follow-up questions:

  1. What is the effect of shifting the function to the right instead of the left?
  2. How do you translate a function down instead of up?
  3. How do reflections affect the graph of an absolute value function?
  4. What happens when you multiply the absolute value function by a negative number?
  5. How does scaling the function vertically affect its shape?

Tip: Always remember, horizontal translations affect the xx-values, while vertical translations affect the entire function.

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Math Problem Analysis

Mathematical Concepts

Function Transformation
Absolute Value

Formulas

g(x) = f(x + h) + k

Theorems

Function Translation Theorem

Suitable Grade Level

Grades 9-12