GEOGEBRA RESOURCES https://www.geogebra.org/m/wUdfS5ZG TRANSLATIONS https://www.geogebra.org/material/show/id/1596 Reflections 5.1.2: Transformations As Functions

A function f(x) is defined as f(x) = −8x2. What is f(−3)? Do Now A function f(x) is defined as f(x) = −8x2. What is f(−3)? A Essential Question: How can you translate a preimage by changing the function?

Agenda Do Now + Good Things! Recap of yesterday Intro to transformations Isometric transformations Translations Reflections Guided practice Exit Ticket Independent Practice (due Friday!)

Good things!!!! Quiz Friday!

Recap of Yesterday… 2. Given the graph of f(x), what is f(2)? 1. 2. Given the graph of f(x), what is f(2)? 3. If g(x,y) = (x,-y), and f(x,y) = (x+1, -2y), then what is g(f(1, 2))?

Lesson Intro A function is a relationship between two sets of data where each input has exactly one output We have a function f(x,y) = (x + 1, y – 2). Evaluate f(2,3). Graph the original point (2,3) (let’s call it point A) Graph the point f(2,3) (let’s call it point A’) What happened when we put the point through the function? We just transformed point A!

Lesson Intro *we know it’s the image because of the prime marks ‘ What does the word “transform” mean? A transformation changes the position, shape, or size of a figure on a coordinate plane through a function. The preimage is the original image It is changed or moved though a transformation, and the resulting figure is called an image ‘ *we know it’s the image because of the prime marks

Transformations Today will focus on two isometric transformations: 1. Translations – slide or shift 2. Reflections – mirror image or flip Isometric Transformations are when the image is congruent to the preimage Figures are congruent if they both have the same shape, size, lines, and angles. The new image is simply moving to a new location. We also call these “rigid” transformations

Isometric Transformations Which transformation is isometric? How do you know?

Transformations as Functions Which figure is the image? Looking at point C: Preimage: C is at (1, 3) Image: C’ is at (5, 2) We applied a transformation to this figure  same as putting inputs into a function and getting outputs How many units did the x-coordinate move? Y-coordinate? (x,y)  (x+4, y-1)

Translations Translation = slide The figure does not change size, shape, or direction – it is simply being moved from one place to another All points on the preimage move parallel to a given line

Translations Notation: Th,k(x, y) = (x + h, y + k) ALWAYS MOVE X FIRST *Hint: this is the only transformation where we are adding and subtracting to our x and y ALWAYS MOVE X FIRST We move the “x” by positive h, and the “y” by positive k https://www.geogebra.org/m/wUdfS5ZG

REMEMBER… Positive Positive Negative Negative 5.1.2: Transformations As Functions

What is the transformation from the preimage to the image? Th,k(x,y)  ? Translations Let’s look at one point A: (-5, 3) A’: (2, -2) The x coordinate has moved 7 units to the right (positive) The y coordinate has moved 5 units down (negative) Th,k(x,y)  (x+h, y+k) T7,-5 (x,y)  (x+7, y-5)

Translations Practice Pg. 89 Describe the transformation that has taken place using the form Th,k(x,y) = (x+h, y+k) Using the form Th,k(x,y) = (x+h, y+k), how can we describe a translation that moves point W 5 units to the left and 1 unit down?

Translations – Working Backwards What translation moves point B(-6, -2) to B’(-4, 4)? Write in the form Th,k(x+h, y+k) What is the change in x? What is the change in y? Fill out formula h is change in x k is change in y 5.1.2: Transformations As Functions

First Block: Page 96 Translations practice – your turn! #5: Translate the given quadrilateral to the left 6 units and down 5 units. Using the form Th,k(x,y) = (x+h, y+k), how can we describe the translation? 5.1.2: Transformations As Functions

Translations Practice – your turn Triangle ABC has the following points: Point A: (3,5) Point B: (5, 5) Point C: (4, 7) Apply the transformation T-3,6(x,y) = (x-3, y+6) to each point to find the coordinates of the image A’B’C’. GRAPH the preimage and the image

Reflections Mirror image over an axis (flip)

Reflections Figure is moved along a line perpendicular to the line of reflection What is perpendicular? The reflected image is always the same size, it is just facing a different direction. Is this an isometry?

Reflections *notice that the lines we are drawing are perpendicular to the line of reflection *pg 96 https://www.geogebra.org/m/k77xx27e We will focus on 3 reflections: x-axis, y-axis, and the line y=x

rx-axis(x,y) = (x,-y) Reflections: X-Axis WHY DOES THIS MAKE SENSE?? I want to reflect point A(9, 14) over the x-axis, what will my new coordinates be for point A’?

ry-axis(x,y) = (-x,y) Reflections: Y-Axis WHY DOES THIS MAKE SENSE?? If I want to reflect the point (9, 14) over the y-axis, what will my new coordinates be?

ry=x(x,y) = (y,x) Reflections: Y=X If I want to reflect the point (9, 14) over the line y=x, what will my new coordinates be?

Reflections Practice What coordinates will a point (3, -2) reflected over the line y=x have? Given A (-5, -6), state the coordinates of A’ after a reflection over the x-axis.

Exit Ticket The point J(8,-8) undergoes the translation T-2, -1. What are the coordinates of J’? What translation moves point Q(-7, 5) to (-8, 6)? Write in the form Th,k(x+h, y+k) Triangle ABC has vertices A (-5,4) B (-2,4) C (-2, 2) Triangle ABC is reflected over the y-axis. What are the new coordinates of Triangle A’B’C’?

Composite Transformations We talked about composite functions, such as f(g(x)) Always do the innermost transformation first We do the same for transformations! Example: Given A (-6, -5) and T(x-3, y +2), state A” after a reflection of the x-axis of the point T(A). rx-axis(T-3,2(-6,-5)) First, transform: (-6, -5)  (-6 – 3, -5+2) (-9, -3) Second, reflect over x-axis: (-9, -3)  (-9, 3) Given P(4, -6) and T-5, 3(x-5, y+3), state P” after a reflection over the line y=x of the point T(P).