To find out how many magnets Joyceanne initially had when she turned 13, we need to determine the value of \( m \) (the total number of magnets) when \( p \) (the number of new places she has visited) is 0.
We can use the linear model given:
\[ m = 2p + 7 \]
Now, let's substitute \( p = 0 \) into the equation:
\[ m = 2(0) + 7 \] \[ m = 0 + 7 \] \[ m = 7 \]
Therefore, Joyceanne initially had 7 magnets when she turned 13.