Graphing Functions using Parent Functions In this lesson, students will be able to identify and use parent functions of linear functions, absolute value functions, and quadratic functions.  Students will be presented with functions and asked to graph them by first identifying the basic function and then using transformations.  The transformations that will be highlighted in this lesson are translation and reflection. 

Counterexample of a function What is a function? Example of a function. Counterexample of a function Watch this video to refresh your knowledge on functions… http://www.youtube.com/watch?v=uJyx8eAHazo&feature=related

Visually and Graphically Know the “Parent” graphs Visually and Graphically

Let’s Learn to Translate (shift) A function has been “translated” when it has been moved in a way that does not change its shape or rotate it in any way. Imagine a graph that has been drawn on tracing paper that was loosely lain over a printed set of axes and moves. A translation is a slide.

Vertical Translation Vertical Translation For b > 0, the graph of y = f(x) + b is the graph of y = f(x) shifted up b units; the graph of y = f(x)  b is the graph of y = f(x) shifted down b units.

Horizontal Translation For d > 0, the graph of y = f(x  d) is the graph of y = f(x) shifted right d units; the graph of y = f(x + d) is the graph of y = f(x) shifted left d units.

Vertical shifts Horizontal shifts Moves the graph up or down Impacts only the “y” values of the function No changes are made to the “x” values Horizontal shifts Moves the graph left or right Impacts only the “x” values of the function No changes are made to the “y” values

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.)

Points represented by (x , y) on the graph of f(x) become If the point (6, -3) is on the graph of f(x), find the corresponding point on the graph of f(x+3) + 2

Use the basic graph to sketch the following:

Combining a vertical & horizontal shift Example of function that is shifted down 4 units and right 6 units from the original function.

Try a and b

Now let’s talk about reflections A reflection is a transformation in which each point of the original figure (pre-image) has an image that is the same distance from the line of reflection as the original point but is on the opposite side of the line.  Remember that a reflection is a flip.  Under a reflection, the figure does not change size.

Reflections The graph of f(x) is the reflection of the graph of f(x) across the x-axis. The graph of f(x) is the reflection of the graph of f(x) across the y-axis. If a point (x, y) is on the graph of f(x), then (x, y) is on the graph of f(x), and (x, y) is on the graph of f(x).

Reflecting Across x-axis (y becomes negative, -f(x)) Across y-axis (x becomes negative, f(-x))

Use the basic graph to sketch the following:

Try c and d

Transformation of Functions Recognize graphs of common functions Use shifts to graph functions Use reflections to graph functions Use stretching & shrinking to graph functions Graph functions w/ sequence of transformations

Vertical Stretching and Shrinking The graph of af(x) can be obtained from the graph of f(x) by stretching vertically for |a| > 1, or shrinking vertically for 0 < |a| < 1. For a < 0, the graph is also reflected across the x-axis. (The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

Now let’s talk about Stretching & Compressing A stretch or compression is a translation in which the size and shape of the graph of a function is changed.  Remember this transformation is a shrink or stretch. .

VERTICAL STRETCH (SHRINK) y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

This notepage can be found at: http://www. regentsprep

This notepage can be found at: http://www. regentsprep

Horizontal Stretching or Shrinking The graph of y = f(cx) can be obtained from the graph of y = f(x) by shrinking horizontally for |c| > 1, or stretching horizontally for 0 < |c| < 1. For c < 0, the graph is also reflected across the y-axis. (The x-coordinates of the graph of y = f(cx) can be obtained by dividing the x-coordinates of the graph of y = f(x) by c.)

Horizontal stretch & shrink We’re MULTIPLYING by an integer (not 1 or 0). 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.)

Try “e”

We now have a better understanding of how certain transformations of a function change its graph. This will allow us to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, (3) stretching. http://www.youtube.com/watch?v=3Q5Sy034fok http://www.youtube.com/watch?v=R3fePYOACLE http://www.youtube.com/watch?v=5oQgzup9nx4 To review using video lesson, click on the links above

Putting it all together http://www.youtube.com/watch?v=h5HEs6krrE8 http://www.youtube.com/watch?v=AgbW1_RnSOU

Let’s Try… http://www.purplemath.com/modules/fcntranq.htm https://www.msu.edu/~osstutor/15_t_103.htm Do together… Do for homework…