Section 1-4 Shifting, Reflecting, and Stretching Graphs PreCalculus Section 1-4 Shifting, Reflecting, and Stretching Graphs

Objectives Recognize graphs of parent functions. Use vertical and horizontal shifts and reflections to graph functions. Use nonrigid transformations to graph functions.

Parent Function What is a parent function? A parent function is the most basic form of a function. It has the same basic properties as others like it, but has not been transformed in any way. Parent functions allow us to quickly tell certain traits of their “children”.

Parent Functions The six most commonly used parent functions. Constant Function - Linear (Identity) Function - Absolute Value Function - Square root Function - Quadratic Function - Cubic Function - Familiarity with the basic characteristics and graphs of these parent functions will help you analyze more complicated graphs.

Constant Function f(x) = a where a = a constant Characteristics: f(x)=x is a horizontal line. It is increasing and continuous on its entire domain (-∞, ∞).

Linear Function (Identity) f(x) = x Characteristics: f(x)=x is increasing and continuous on its entire domain (-∞, ∞).

Absolute Value Function f(x) = │x │ Characteristics: is a piecewise function. It decreases on the interval (-∞, 0) and increases on the interval (0, ∞). It is continuous on its entire domain (- ∞, ∞). The vertex of the function is (0, 0).

Square Root Function f(x) = x Characteristics: f(x)=√x increases and is continuous on its entire domain [0, ∞). Note: x≥0 for f to be real.

Quadratic Function f(x) = x 2 Characteristics: f(x)=x2 is continuous on its entire domain (-∞, ∞). It is increasing on the interval (0, ∞) and decreasing on the interval (-∞, 0). Its graph is called a parabola, and the point where it changes from decreasing to increasing, (0,0), is called the vertex of the graph.

Cubic Function f(x) = x 3 Characteristics: f(x)=x3 increases and is continuous on its entire domain (-∞, ∞). The point at which the graph changes from “opening downward” to “opening upward” (the point (0,0)) is called an inflection point.

General Transformations

Types of Transformations Reflections: These are like mirror images as seen across a line or a point. Translations ( or slides): This moves the figure to a new location with no change to the looks of the figure. Dilations: This reduces or enlarges the figure to a similar figure.

Reflections You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image. Example: The figure is reflected across line l . l You could fold the picture along line l and the left figure would coincide with the corresponding parts of right figure.

Reflections – continued… Reflection across the y-axis: the y values stay the same and the x values change sign. (x , y)  (-x, y) Reflection across the x-axis: the x values stay the same and the y values change sign. (x , y)  (x, -y) Example: In this figure, line l : n l reflects across the y axis to line n (2, 1)  (-2, 1) & (5, 4)  (-5, 4) reflects across the x axis to line m. (2, 1)  (2, -1) & (5, 4)  (5, -4) m

Reflections across specific lines: To reflect a figure across the line y = a or x = a, mark the corresponding points equidistant from the line. i.e. If a point is 2 units above the line its corresponding image point must be 2 points below the line. Example: Reflect the fig. across the line y = 1. (2, 3)  (2, -1). (-3, 6)  (-3, -4) (-6, 2)  (-6, 0)

Translations (slides) If a figure is simply moved to another location without change to its shape or direction, it is called a translation (or slide). If a point is moved “a” units to the right and “b” units up, then the translated point will be at (x + a, y + b). If a point is moved “a” units to the left and “b” units down, then the translated point will be at (x - a, y - b). Example: A Image A translates to image B by moving to the right 3 units and down 8 units. B A (2, 5)  B (2+3, 5-8)  B (5, -3)

Dilations A dilation is a transformation which changes the size of a figure but not its shape. This is called a similarity transformation. Since a dilation changes figures proportionately, it has a scale factor k. If the absolute value of k is greater than 1, the dilation is an enlargement. If the absolute value of k is between 0 and 1, the dilation is a reduction. If the absolute value of k is equal to 1, the dilation is congruence transformation. (No size change occurs.)

Dilations – continued… In the figure, the center is C. The distance from C to E is three times the distance from C to A. The distance from C to F is three times the distance from C to B. This shows a transformation of segment AB with center C and a scale factor of 3 to the enlarged segment EF. In this figure, the distance from C to R is ½ the distance from C to A. The distance from C to W is ½ the distance from C to B. This is a transformation of segment AB with center C and a scale factor of ½ to the reduced segment RW. C E A F B C R A B W

Transformations of Functions How does the graph of a function change if you add a constant to the input of a function (x)? To the output of a function (f(x))? How does the graph of a function change if you multiply a constant times the input of a function (x)? To the output of a function (f(x))? How does the graph of a function change if you change the sign of the input of a function (x)? Of the output of a function (f(x))?

Translations

Horizontal Translations Start with f(x) Add or subtract to the input of the function. Translates the function horizontally (+ left, - right)

Horizontal Translations Given the graph you get the graph by moving the first graph left k units (for k>0) Adding a constant to the input of the function moves the graph of the function left.

Horizontal Translating

Horizontal Translations (Continued) Given the graph you get the graph by moving the first graph right k units (for k>0) Subtracting a constant from the input of the function moves the graph of the function right.

Horizontal Translating (Continued)

Example y = x2 + 3x + 1

Example y = (x-3)2 + 3(x-3) + 1 y = x2 + 3x + 1 is translated by [ ] 3

Horizontal Shifts If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units. x y -c +c If c is a positive real number, then the graph of f (x + c) is the graph of y = f (x) shifted to the left c units. y = f (x + c) y = f (x) y = f (x – c)

Example: Use the graph of f (x) = x3 to graph g (x) = (x – 2)3 and h(x) = (x + 4)3 . y -4 4 f (x) = x3 h(x) = (x + 4)3 g(x) = (x – 2)3

Horizontal Translation Rules in a Nutshell For any graph y = f(x) The translation y = f(x - a) moves it ‘a’ units to the right The translation y = f(x + a) moves it ‘a’ units to the left [ ] a i.e. for y = f(x - a) is translated by

Your Turn: Describe the transformation f(x) = 4x + 3 f(x) = 4(x+5) + 3 x y Translation of -5 units in the ‘x’ direction [ ] -5

Vertical Translations

Vertical Translations Start with f(x) Add or subtract to the output of the function. Translates the function vertically (+ up, - down)

Vertical Translations Given the graph you get the graph by moving the first graph up k units (for k>0) Adding a constant to the output of the function moves the graph of the function up.

Vertical Translating

Vertical Translations (Continued) Given the graph you get the graph by moving the first graph down k units (for k>0) Subtracting a constant from the output of the function moves the graph of the function down.

Vertical Translating (Continued)

Example y = x2 + 3x + 1

[ ] Example y = x2 + 3x + 1 - 4 y = x2 + 3x + 1 y = x2 + 3x + 1 is translated by [ ] -4 y = x2 + 3x + 1 - 4

Vertical Shifts If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units. If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units. x y f (x) + c f (x) +c f (x) – c -c

Example: Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 4. y -4 4 8 g(x) = |x| + 3 f (x) = |x| h(x) = |x| – 4

Vertical Translation Rules in a Nutshell For any graph y = f(x) The translation y = f(x) + b moves it ‘b’ units up … this can be considered as y - b = f(x) The translation y = f(x) - b moves it ‘b’ units down … this can be considered as y + b = f(x) [ ] b i.e. for y = f(x) + b is translated by

Your Turn: Describe the transformation f(x) = 4x + 3 f(x) = 4x + 7 x y Translation of 4 units in the ‘y’ direction [ ] 4

Combined Vertical and Horizontal Translations

Example: Graph the function using the graph of . First make a vertical shift 4 units downward. Then a horizontal shift 5 units left. -4 x y 4 -4 y 4 x (4, 2) (0, 0) (4, –2) (–1, –2) (0, – 4) (– 5, –4)

Your Turn: Describe the transformation f(x) = x2 - 3 f(x) = (x-2)2 + 2 x y Translation of 5 units in the ‘y’ direction and 2 units in the ‘x’ direction [ ] 2 5

Reflections

Reflections Start with flips graph vertically (across the x-axis) flips graph horizontally (across the y-axis)

Reflections about the x-axis Given the graph you get the graph by reflecting the first graph across the x-axis. Changing y to -y reflects the graph across the x-axis.

Reflections about the x-axis

Reflections about the y-axis Given the graph you get the graph by reflecting the first graph across the y-axis. Changing x to -x reflects the graph across the y-axis.

Reflections about the y-axis

Reflecting Rules in a Nutshell (reflection) For any graph y = f(x) The transformation y = - f(x) negative of the output of a function reflects it about the x-axis The translation y = f(-x) negative of the input of a function reflects it about the y-axis

Reflections x y The graph of the function y = f (–x) is the graph of y = f (x) reflected in the y-axis. y = f (–x) y = f (x) y = –f (x) The graph of the function y = –f (x) is the graph of y = f (x) reflected in the x-axis.

Your Turn: Describe the transformation f(x) = x2 - 3 f(x) = -(x2 - 3) x y Reflection in the x-axis

Your Turn: Describe the transformation f(x) = (x – 3)2 f(x) = (-x-3)2 x y Reflect in the y-axis

Dilations

Nonrigid Transformations Horizontal shifts, vertical shifts, and reflections are called rigid transformations because the basic shape of the graph is unchanged. These transformations change only the position of the graph in the coordinate plane. Nonrigid transformations are those that cause a distortion – a change in the shape of the original graph. These transformations include vertical stretch, vertical shrink, horizontal stretch, and horizontal shrink.

Dilation from the y-axis Vertical ‘stretching’ and ‘shrinking’

Vertical Stretching Given the graph you get the graph by stretching the first graph vertically by a factor of k. Multiplying f(x), the output of the function, by k>1 stretches the graph vertically by a factor of k.

Vertical Stretching

Vertical Shrinking Given the graph you get the graph by shrinking the first graph vertically by a factor of k. Multiplying f(x), the output of the function, by 0<k<1 shrinks the graph vertically by a factor of k.

Vertical Shrinking

Vertical Stretching or Shrinking Rules in a Nutshell The graph of y = af(x) can be obtained from the graph of y = f(x) by stretching vertically for |a| > 1, or shrinking vertically for 0 < |a| < 1. (The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

Vertical Stretching and Shrinking If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c. If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) shrunk vertically by c. – 4 x y 4 y = x2 y = 2x2 Example: y = 2x2 is the graph of y = x2 stretched vertically by a factor of 2. is the graph of y = x2 shrunk vertically by a factor of .

Shrink along the y-axis Stretch along the y-axis Examples Shrink along the y-axis by a factor of 1/10 Stretch along the y-axis by a factor of 2

Your Turn: Describe the transformation f(x) = (x – 3)2-1 f(x) = 3((x-3)2-1) x y Stretch by a factor 3 in the y-axis

Dilations from the x-axis Horizontal ‘stretching’ and ‘shrinking’

Horizontal Stretching Given the graph you get the graph by stretching the first graph horizontally by a factor of 1/k. Multiplying x, the input of the function, by 0<k<1 stretches the graph horizontally by a factor of 1/k.

Horizontal Stretching

Horizontal Shrinking Given the graph you get the graph by shrinking the first graph horizontally by a factor of 1/k. Multiplying x, the input of the function, by k>1 shrinks the graph horizontally by a factor of 1/k.

Horizontal Shrinking

Horizontal Stretching or Shrinking Rules in a Nutshell 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. (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 Stretching and Shrinking If c > 1, the graph of y = f (cx) is the graph of y = f (x) shrunk horizontally by 1/c. If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x) stretched horizontally by 1/c. y = |2x| Example: y = |2x| is the graph of y = |x| shrunk horizontally by 1/2. - 4 x y 4 y = |x| is the graph of y = |x| stretched horizontally by 2 .

Shrink along the x-axis Stretch along the x-axis Examples Shrink along the x-axis by a factor of 1/2 Stretch along the x-axis by a factor of 2

Your Turn: Describe the transformation f(x) = x2 - 3 f(x) = (x/2)2 - 3 x y Stretch by factor 2 along the x-axis

Your Turn: Describe the transformation f(x) = (x – 3)2 f(x) = (3x-3)2 x y Shrink by a factor 1/3 in the x-axis

Summary of Function Transformations

[ ] [ ] [ ] a y = f(x - a) Translation by y = f(x) + b b [ ] a Translation by y = f(x - a) [ ] b Translation by y = f(x) + b [ ] a b Translation by y = f(x - a) + b y = k f(x) stretches by a factor ‘k’ along the y-axis y = f(x/k) stretches by a factor ‘k’ along the x-axis y = - f(x) reflects in the x-axis y = f(-x) reflects in the y-axis

Visualize Transformations

General Function Transformations

Specific Function Transformations

Order of Transforming Functions

Example 1 : translate y and stretch y In reverse ... y = x y = x is translated by [ ] 3 stretch by a factor 2 along the y-axis y = x + 3 DIFFERENT RESULTS, SO THE ORDER MATTERS y = 2x stretch by a factor 2 along the y-axis translated by [ ] 3 y = 2(x + 3) y = 2x + 6 y = 2x + 3

Example 2 : translate x and reflect in y In reverse ... y = x2 is translated by [ ] 2 y = x2 DIFFERENT RESULTS, SO THE ORDER MATTERS Reflect in the y-axis y = (x - 2)2 y = (-x)2 Reflect in the y-axis y = x2 translated by [ ] 2 y = (-x - 2)2 y = (-(x+2))2 y = (x+2)2 y = (x-2) 2 The translation y = f(-x) reflects in the y-axis

Example 3 : translate x and reflect in x In reverse ... y = x2 SAME RESULTS, SO THE ORDER DOESN’T MATTER is translated by [ ] 1 y = x2 Reflect in the x-axis y = (x - 1)2 y = -x2 Reflect in the x-axis translated by [ ] 1 y = -(x - 1)2 y = -(x -1)2 The translation y = -f(x) reflects in the x-axis

Example 4 : translate x and translate y y = 3x + 1 In reverse ... is translated by [ ] 3 y = 3x + 1 translated by [ ] 1 SAME RESULTS, SO THE ORDER DOESN’T MATTER y = 3x + 4 y = 3(x-1) + 1 y = 3x - 3 +1 y = 3x -2 translated by [ ] 1 translated by [ ] 3 y = 3(x - 1) + 4 y = 3x - 3 + 4 y = 3x + 1 y = 3x -2 + 3 y = 3x + 1

Example 4 : translate x and translate y Where you have 2 translations they can always be combined into one step y = 3x + 1 translated by [ ] 3 then translated by 1 Is the same as ... y = 3x + 1 translated by [ ] 1 3 y = 3(x-1) + 1 + 3 y = 3x - 3 + 1 + 3 y = 3x + 1 Which matches the previous slide

When applying 2 transformations the order can matter Summary When applying 2 transformations the order can matter If they are both translations, they can be combined, so the order does not matter here If there are 2 ‘x’ operations or 2 ‘y’ operations the order always matters e.g. translate in y and stretch in y e.g. translate in x and reflect in y [y = f(-x)] If there is an ‘x’ operation and a ‘y’ operation, then the order won’t matter e.g. translate in x and reflect in x [-y = f(x)]

Order of Transformations Follow order of operations. Or use the following order: 1. Horizontal shift 2. Stretching or shrinking 3. Reflections 4. Vertical shift To graph, select two points (or more) from the original function and move that point one step at a time.

Combinations of Function Transformations

Examples Vertical stretch by a factor of 2 and a Horizontal stretch by reflection about the x-axis Horizontal stretch by a factor of 2 and a reflection about the y-axis

Graphing with More than One Transformation Graph -|x – 2| + 1 First graph f(x) = |x|

Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right.

Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2|

Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2| 4. Perform vertical translation: f(x) = -|x-2| + 1 The graph shifts up 1 unit.

Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2| 4. Perform vertical translation: f(x) = -|x-2| + 1 The graph shifts up 1 unit.

Consider the graph of a function y = f(x) The transformation described by y = f(2x+4) is horizontal stretch by a factor of ½. The translation described by y = f(2x + 4) is horizontal shift of 4 units to the left. y = f(2(x + 2)) Translations must be factored

Your Turn: 3. 1. 6.

Your Turn: 2. 3. 1.

Example: Identify the transformations to the function, in the correct order. Horizontal translation left 3 units. Vertical stretch by a factor of 2. Reflection over the x-axis. Vertical translation down 5 units. Horizontal translation right 1 unit. Vertical stretch by a factor of 3. Vertical translation up 2 units.

Your Turn: Identify the transformations to the function, in the correct order. Horizontal translation right 2 units. Vertical shrink by a factor of 2/3. No reflection. Vertical translation up 9 units. Horizontal translation right 8 units. Horizontal stretch by a factor of 2. Vertical translation up 1 unit.

Your Turn: Identify the transformations to the function, in the correct order. Horizontal translation right 3 units. Horizontal stretch by a factor of 3 and Vertical shrink by a factor of 1/2. Reflection over the y-axis. Vertical translation down 4 units. Horizontal translation right 2 units. Horizontal strink by a factor of 1/4 and Vertical shrink by a factor of 1/2. Vertical translation down 7 units.

Your Turn: Identify the transformations to the function, in the correct order. Horizontal shrink by a factor of ½ and Vertical stretch by a factor of 3. Reflection over x-axis and y-axis. Vertical translation up 1 unit. Horizontal translation left 3 units. Vertical shrink by a factor of 1/4. Reflection over the y-axis. Vertical translation down 4 units.

Homework Section 1.4, pg. 48 – 50: Vocabulary Check #1 – 6 all Exercises: #1-37 odd, 43-55 odd Read Section 1.5, pg. 51 – 57