Objectives Vertical Shifts Up and Down Horizontal Shifts

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 •

Constant Function f(x) = c where c is a constant Characteristics: f(x)=a 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)=x 2 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)=x 3 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. )

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 x y

Vertical Shifts

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

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 x y

Horizontal Shifts

Horizontal Shifts

Example: Use the graph of f (x) = x 3 to graph g (x) = (x – 2)3 and h(x) = (x + 4)3. y f (x) = x 3 4 -4 h(x) = (x + 4)3 4 x 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. y 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. -c +c x y = f (x + c) y = 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? They are the same. What do we know about the y-coordinates in these two graphs? They are the opposite.

Reflections about the y-axis

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

Vertical Stretching of Graphs x y - 2 4 - 2 8 - 1 1 - 1 2 0 0 1 1 1 2 2 4 2 8 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 x y - 2 4 - 2 2 - 1 1 - 1 ½ 0 0 1 1 1 ½ 2 4 2 2 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. y = x 2 Example: y = 2 x 2 is the graph of y = x 2 stretched vertically by a factor of 2. y y = 2 x 2 4 x is the graph of y = x 2 shrunk vertically by a factor of . – 4 4

Horizontal Stretching of Graphs x y - 2 2 - 2 1 - 1 ½ 0 0 1 1 1 ½ 2 2 2 1 ½ is the constant, c. The x-coordinates stay the same. Our graph stretches horizontally. When the constant is greater than zero and less than one, the graph is stretching horizontally.

Horizontal Shrinking of Graphs x y - 2 2 - 2 4 - 1 1 - 1 2 0 0 1 1 1 2 2 4 2 is the constant, c. The x-coordinates stay the same. 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. Example: y = |2 x| is the graph of y = |x| shrunk horizontally by 1/2. y y = |2 x| y = |x| 4 is the graph of y = |x| stretched horizontally by 2. x -4 4

Graphing a New Function from the Original Function New Function Visual Effect Flip over the x-axis. The x-axis acts as your mirror. Flip over the y-axis. The y-axis acts as your mirror.