2. The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

3. The linear function f(x) = x

4. The quadratic function

5. The square root function

6. The absolute value function

7. The cubic function

8. The hyperbolic/reciprocal function

9. We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations/shifts, (2) reflections/flips, (3) stretching/change of scale.

15. The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right , while numbers added or subtracted outside translate up or down .

17. Example Use the graph of f ( x ) to obtain the graph of h ( x ) = ( x + 1) 2 – 3. Solution Step 1 Graph f ( x ) = x 2 . The graph of the standard quadratic function is shown. Step 2 Graph g ( x ) = ( x + 1) 2 . Because we add 1 to each value of x in the domain, we shift the graph of f horizontally one unit to the left. -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 Step 3 Graph h ( x ) = ( x + 1) 2 – 3. Because we subtract 3, we shift the graph vertically down 3 units.

18. Use the basic graph to sketch the following:

22. Use the basic graph to sketch the following:

29. Graph of Example