Question

which phrase best describes the translation from the graph $y = 2x^2$ to the graph of $y = 2x^2 + 5$? 5 units right 5 units up 5 units left 5 units down

Explanation:

Step1: Recall vertical translation rule

For a function \( y = f(x) \), the transformation \( y = f(x)+k \) (where \( k>0 \)) shifts the graph of \( f(x) \) up by \( k \) units. If \( k < 0 \), it shifts down by \( |k| \) units.

Step2: Analyze the given functions

We have the original function \( y = 2x^{2} \) and the transformed function \( y=2x^{2}+5 \). Comparing with the vertical translation rule \( y = f(x)+k \), here \( f(x)=2x^{2} \) and \( k = 5>0 \). So the graph of \( y = 2x^{2} \) is shifted up by 5 units to get the graph of \( y=2x^{2}+5 \).

Answer:

5 units up