To determine which statement is true regarding the transformation from [tex]\( f(x) = x^2 \)[/tex] to [tex]\( m(x) = x^2 + 5 \)[/tex], let's analyze the changes between the two functions step by step.
1. Parent Function [tex]\( f(x) = x^2 \)[/tex]:
The parent function is a simple quadratic function where the graph is a parabola opening upwards with its vertex at the origin, (0, 0).
2. Transformation to [tex]\( m(x) = x^2 + 5 \)[/tex]:
The function [tex]\( m(x) \)[/tex] is obtained by adding 5 to [tex]\( f(x) \)[/tex]. This means each value of the quadratic function [tex]\( f(x) \)[/tex] is increased by 5 for the corresponding [tex]\( x \)[/tex]-values.
3. Effect of Adding a Constant Term:
When a constant [tex]\( c \)[/tex] is added to a function [tex]\( f(x) \)[/tex] (i.e., [tex]\( f(x) + c \)[/tex]), the graph of the function is shifted vertically. Specifically:
- If [tex]\( c \)[/tex] is positive, the graph shifts up by [tex]\( c \)[/tex] units.
- If [tex]\( c \)[/tex] is negative, the graph shifts down by [tex]\( |c| \)[/tex] units.
Given [tex]\( m(x) = x^2 + 5 \)[/tex], the addition of [tex]\( 5 \)[/tex] to [tex]\( x^2 \)[/tex] means the entire graph of the parent function [tex]\( f(x) = x^2 \)[/tex] is shifted upwards by 5 units.
Thus, the correct statement is:
- "Function [tex]\( f \)[/tex] was translated 5 units up to create function [tex]\( m \)[/tex]."
Therefore, the true statement is that the function [tex]\( f \)[/tex] was translated 5 units up to create function [tex]\( m \)[/tex].
