Math Problem Statement

Translate the function f(x) = x^2 + 4, 3 units to the right and 5 units upwards. What is the resulting function?

Solution

To solve this question:

Problem Breakdown

We start with the given function:

f(x)=x2+4f(x) = x^2 + 4

We are tasked with translating the function 3 units to the right and 5 units upwards.


Transformation Rules:

  1. Rightward translation: To shift a function hh units to the right, replace xx with (xh)(x - h). For this case, h=3h = 3, so replace xx with (x3)(x - 3). (x3)2+4(x - 3)^2 + 4

  2. Upward translation: To shift a function kk units upwards, add kk to the entire function. For this case, k=5k = 5, so add 5: (x3)2+4+5(x - 3)^2 + 4 + 5

Simplify: (x3)2+9(x - 3)^2 + 9


Correct Answer:

The correct choice is d. (x3)2+9(x - 3)^2 + 9.


Would you like more explanation, or do you have further questions?


Related Questions:

  1. How do translations affect the vertex of a parabola?
  2. What happens when you translate a function to the left instead of right?
  3. How does a downward translation differ in terms of the equation?
  4. How can you identify the new vertex of a translated function?
  5. Can you rewrite this function in vertex form after translation?

Tip:

Always remember that translating a function horizontally involves modifying the xx-variable directly, while vertical translations add or subtract from the function itself.

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Math Problem Analysis

Mathematical Concepts

Function Transformation
Horizontal and Vertical Translations
Quadratic Functions

Formulas

f(x) = x^2 + 4
Translation Rule for Horizontal Shifts: f(x) → f(x - h)
Translation Rule for Vertical Shifts: f(x) → f(x) + k

Theorems

Translation Theorem for Functions

Suitable Grade Level

Grades 8-10