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.
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 .
Recognizing the shift from the equation, examples of shifting the function f(x) =
Vertical shift of 3 units up
Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)
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.