Question

Determine the equation of g(x) that results from translating the function f(x)=x^2+9 upward 10 units.

Answer 1

The simple/ common sense method:
The typical lay out of a quadratic equation is ax^2+bx+c 
'c' represents where the line crosses the 'y' axis. 
The equation is only translated in the 'y' (upwards/downwards) direction, therefore only the 'c' component of the equation is going to change. 
A translation upwards of 10 units means that the line will cross the 'y' axis 10 places higher. 
9+10=19, 
therefore c=19. 
The new equation is: y=x^2+19 

The most complicated/thorough method: 
This is useful for when the graph is translated both along the 'y' axis and 'x' axis.
ax^2+bx+c 
a=1, b=0, c=9
Find the vertex (the highest of lowest point) of f(x). 
Use the -b/2a formula to find the 'x' coordinate of your vertex.. 
x= -0/2*1, your x coordinate is therefore 0. 
substitute your x coordinate into your equation to find your y coordinate.. 
y= 0^2+0+9
y=9. 
Your coordinates of your vertex f(x) are therefore (0,9) 
The translation of upward 10 units means that the y coordinate of the vertex will increase by 10. The coordinates of the vertex g(x)  are therefore: 
(0, 19) 
substitute your vertex's y coordinate into f(x) 
19=x^2+c
19=0+c
c=19
therefore g(x)=x^2+19

Answer 2

$g(x)=x^2+9+10=x^2+19$