Objectives Vertical Shifts Up and Down Horizontal Shifts to the Right and Left Reflections of Graphs Stretching and Shrinking of Graphs

Laws for Graphing Shifts All graphs must be placed on graphing paper. We do not use ruled paper for graphing. Graphs are to be drawn using a straight edge. Arrows are to be placed on each end of the x and y-axis. Label the x and y-axis. When graphing two functions on the same graph, use different colored pencils or markers. Label points. Be sure the equation(s) is located on the graph. Keep everything color-coded. Use a straight-edge where needed. Straight lines must not be sketched. Curved lines are sketched to the best of your ability. Be sure your graph is very neat and professional.

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 other functions like it, but it has not been transformed in any way. Parent functions allow us to quickly tell certain traits of their “children” (which have been transformed).

Parent Functions The six most commonly used parent functions are: Constant Function 𝑓 𝑥 =𝑐 𝑦=𝑐 Linear (Identity) Function 𝑓 𝑥 =𝑥 𝑦=𝑥 Absolute Value Function 𝑓 𝑥 =| 𝑥 | 𝑦=| 𝑥 | Square root Function 𝑓 𝑥 = 𝑥 𝑦= 𝑥 Quadratic Function 𝑓 𝑥 = 𝑥 2 𝑦= 𝑥 2 Cubic Function 𝑓 𝑥 = 𝑥 3 𝑦= 𝑥 3

  Constant Function f(x) = c where c is a constant 𝒚=𝟐 (−5, 2) (2, 2)   Characteristics: f(x)=a is a horizontal line. It is increasing and continuous on its entire domain (-∞, ∞). 𝒙

  Linear Function (Identity) f(x) = x 𝒚=𝒙 𝒚 (4, 4) Characteristics: f(x)=x is increasing and continuous on its entire domain (-∞, ∞). 𝒙 𝒚=𝒙  (−5, −5)

  Absolute Value Function f(x) = │x │ 𝒚=| 𝒙 | 𝒚 (−4, 4) (4, 4) 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). 𝒙 𝒚=| 𝒙 |

  Square Root Function f(x) = x 𝒚= 𝒙 𝒚= 𝒙 f(x) = x (4, 2)   (1, 1) Characteristics: f(x)=√x increases and is continuous on its entire domain [0, ∞). Note: x≥0 for f to be real. (0, 0) 𝒙

f(x) = x 2 Quadratic Function 𝒚= 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 𝒚= 𝒙 𝟑 (2, 8) f(x) = x 3 𝒚= 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 the origin.)  (1, 1)  (−1, −1)  (−2, −8)

Vertical Shifts The graph represents the equation f(x) We add or subtract to the output of the function. Translates the function vertically (+ up, - down)

          Vertical Shifts Graph 𝑓 𝑥 = 𝑥 2 +2 x y −3 11 −2 6 2 3       What is the shift of the graph 𝑓 𝑥 = 𝑥 2 −2? 𝒚= 𝒙 𝟐 𝒚= 𝒙 𝟐 +𝟐

Vertical Shifts

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 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

Graphing y=𝑓(𝑥)±𝑐 Neatly graph your parent function (with a colored pencil). Plot some points for reference. Shift each point from your parent function up c units (if c is positive) or down c units (if c is negative). Use a different colored pencil.

Horizontal Shifts Start with f(x) The c is now in parentheses with x. Add or subtract to the input of the function. Translates the function horizontally (+ left, - right)

              Horizontal Shifts Graph 𝑓 𝑥 = 𝑥+3 2 𝒚= 𝒙 𝟐 x y −6 9 −5 4 −4 1 −3 −2 −1           𝒙 𝒚= 𝒙 𝟐 𝒚= 𝒙+𝟑 𝟐 What is the shift of the graph 𝑓 𝑥 = 𝑥−3 2 ?

Horizontal Shifts

Horizontal Shifts

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 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)

Graphing 𝑓(𝑥±𝑐) Horizontal shifts don’t make sense when looking at the equation. Everything is in parentheses. Think opposite direction in your shift. Neatly graph your parent function (with a colored pencil). Plot some points for reference. Shift each point from your parent function to the left c units (if c is positive) or to the right c units (if c is negative). Use a different colored pencil.

Reflections about the x-axis What do we know about the x-coordinates in these two graphs? 𝒚= 𝒙 𝟐 (1, 1)  They are the same. What do we know about the y-coordinates in these two graphs?  (1, −1) 𝒚=− 𝒙 𝟐 They are the opposite.

When the right side of a function 𝑦=𝑓(𝑥) is multiplied by −1, the graph of the new function 𝑦=−𝑓(𝑥) is the reflection about the x-axis of the graph of the function 𝑦=𝑓(𝑥). The x-axis acts as a mirror of these two functions.

Reflections about the y-axis When x is replaced with −𝑥, the graph of the new function is the reflection about the y-axis. The y-axis acts as a mirror of the two functions.

What do we observe about the y-coordinates? What do we observe about the x-coordinates? (9, 3) (-9, 3) (-4, 2) (4, 2) (-1, 1) (1, 1)

Vertical Stretching of Graphs 𝒚= 𝒙 𝟐 𝒚=𝟐 𝒙 𝟐 x y - 2 4 8 - 1 1 2 The x-coordinates stay the same. The y-coordinates increase by a factor of 2. 2 is the constant c. Our graph stretches vertically. That is, it moves away from the x-axis to become more vertical.

Vertical Shrinking of Graphs 𝒚= 𝒙 𝟐 𝒚= 𝟏 𝟐 𝒙 𝟐 x y - 2 4 2 - 1 1 ½ The x-coordinates stay the same. The y-coordinates decrease by a factor of ½. ½ is the constant c. Our graph shrinks vertically. That is, it moves more towards the x-axis to be less vertical.

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. (Remember: c is the constant) 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 .

Horizontal Stretching of Graphs 𝒚=| 𝒙 | 𝒚=| 𝟏 𝟐 𝒙 | x y - 2 2 1 - 1 ½ ½ is the constant, c. The x-coordinates stay the same. The y-coordinates decrease by a factor of ½, which is the same as 1 𝑐 . Our graph stretches horizontally. When the constant is greater than zero and less than one, the graph is stretching horizontally.

Horizontal Shrinking of Graphs 𝒚=| 𝒙 | 𝒚=| 𝟐𝒙 | x y - 2 2 4 - 1 1 2 is the constant, c. The x-coordinates stay the same. The y-coordinates increase by a factor of 2, which is the same as 1 𝑐 . Our graph shrinks horizontally. When the constant is greater than one, the graph is shrinking horizontally.

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 .

How Points in Graph of 𝒇 𝒙 become points in new graph Graphing a New Function from the Original Function Original Function New How Points in Graph of 𝒇 𝒙 become points in new graph Visual Effect 𝑓 𝑥 𝑓 𝑥 +𝑐 (𝑥, 𝑦)→(𝑥, 𝑦+𝑐) Shift up by 𝑐 units 𝑓 𝑥 −𝑐 (𝑥, 𝑦)→(𝑥, 𝑦−𝑐) Shift down by 𝑐 units 𝑐𝑓(𝑥) (𝑥, 𝑦)→(𝑥, 𝑐𝑦) Stretch vertically by 𝑐 1 𝑐 𝑓(𝑥) (𝑥, 𝑦)→(𝑥, 1 𝑐 𝑦) Shrink vertically by 1 𝑐 −𝑓(𝑥) (𝑥, 𝑦)→(𝑥, −𝑦) Flip over the x-axis. The x-axis acts as your mirror. 𝑓(−𝑥) (𝑥, 𝑦)→(−𝑥, 𝑦) Flip over the y-axis. The y-axis acts as your mirror.

Homework: Pages 216 – 217 1, 3, 7, 17, 19, 27, 53, 55, 67, 69, 81, 83, 87, 95, 97