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 .
16.
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.)
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.
x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)
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