Question

The function f(x) = x^2 is shifted up five units. Which of the following function equations represents this translation?

a) f(x) = (x + 5)^2

b) f(x) = (x - 5)^2

c) f(x) = x^2 + 5

d) f(x) = x^2 - 5

Please choose the correct option that represents the translation of the function.

Answer 1

The function f(x) = x^2 shifted upwards by 5 units is correctly represented by the equation f(x) = x^2 + 5.

Step-by-step explanation:

The student's question relates to the vertical translation of the function f(x) = x^2. When a function is shifted upwards by a certain number of units, the change is reflected in the y-values of the function. In this case, f(x) is being shifted up by 5 units, which means we need to add 5 to the existing y-values. This is represented by the equation f(x) = x^2 + 5.

There is a common misconception that changing the x-value inside the function (like in options a and b) is necessary to achieve a vertical shift, but those actually represent horizontal shifts or changes to the function's shape. Vertical shifts only involve a direct addition or subtraction to the entire function itself without altering the x component whatsoever.