Question

Write the expression for the function whose graph is the graph of y = x^3 but shifted down 4 units and left 5 units.

Answer 1

1. Consider y=f(x)

2. let a and b be 2 positive numbers

3. then the graph of y=f(x)-a is the graph of y=f(x) shifted a units down

4. the graph of y=f(x+b) is y=f(x) shifted b units left
                        y=f(x-b)  is y=f(x)  shifted b units right

5. y=f(x+b)-a is the graph of y=f(x) shifted a units down and b units left

6. So $y=x^{3}$ shifted 4 units down and 5 units left is the graph of 
              
       $y=(x+5)^{3}-4$

7. To check: consider $y=x^{3}$ at x=3. We have the point (3, 27)

     Shift this point 4 units down and 5 units left: (3-5, 27-4)=(-2, 23)

     Consider  $y=(x+5)^{3}-4$ for x=-2

    $(-2+5)^{3}-4=(3)^{3}-4=27-4=23$