EXAMPLE 4 Write an absolute value function Holograms In holography, light from a laser beam is split into two beams, a reference beam and an object beam.

Slides:

Advertisements

Similar presentations

1.4 – Shifting, Reflecting, and Stretching Graphs

Advertisements

Transforming graphs of functions

EXAMPLE 1 Graph y= ax 2 where a > 1 STEP 1 Make a table of values for y = 3x 2 x– 2– 1012 y12303 Plot the points from the table. STEP 2.

1 Transformations of Functions SECTION Learn the meaning of transformations. Use vertical or horizontal shifts to graph functions. Use reflections.

Use intercepts to graph an equation

EXAMPLE 2 Graph direct variation equations Graph the direct variation equation. a.a. y = x 2 3 y = –3x b.b. SOLUTION a.a. Plot a point at the origin. The.

EXAMPLE 1 SOLUTION STEP 1 Graph a function of the form y = a x Graph the function y = 3 x and identify its domain and range. Compare the graph with the.

EXAMPLE 1 Graph an equation of a parabola SOLUTION STEP 1 Rewrite the equation in standard form x = – Write original equation Graph x = – y.

Graph an equation of a parabola

Standard 9 Write a quadratic function in vertex form

EXAMPLE 1 Find the axis of symmetry and the vertex Consider the function y = – 2x x – 7. a. Find the axis of symmetry of the graph of the function.

Write and graph a direct variation equation

3.6 Graph Rational Functions Part II. Remember Rational functions have asymptotes To find the vertical asymptote, set the denominator = 0 and solve for.

Substitute 0 for y. Write original equation. To find the x- intercept, substitute 0 for y and solve for x. SOLUTION Find the x- intercept and the y- intercept.

Substitute 0 for y. Write original equation. To find the x- intercept, substitute 0 for y and solve for x. SOLUTION Find the x- intercept and the y- intercept.

EXAMPLE 3 Solve a multi-step problem Photography You can take panoramic photographs using a hyperbolic mirror. Light rays heading toward the focus behind.

2.4 Use Absolute Value Functions and Transformations

Lesson 6.5, For use with pages

Translations Translations and Getting Ready for Reflections by Graphing Horizontal and Vertical Lines.

Describe correlation EXAMPLE 1 Telephones Describe the correlation shown by each scatter plot.

SOLUTION STEP 1 Use intercepts to graph an equation EXAMPLE 2 Graph the equation x + 2y = 4. x + 2y = 4 x =  x- intercept 4 Find the intercepts. x + 2(0)

3.6 Solving Absolute Value Equations and Inequalities

Transformations on the Coordinate Plane. Example 2-2a A trapezoid has vertices W(–1, 4), X(4, 4), Y(4, 1) and Z(–3, 1). Trapezoid WXYZ is reflected.

SOLUTION EXAMPLE 1 Find the image of a glide reflection The vertices of ABC are A(3, 2), B(6, 3), and C(7, 1). Find the image of ABC after the glide reflection.

1.What is the absolute value of a number? 2.What is the absolute value parent function? 3.What kind of transformations can be done to a graph? 4.How do.

Graphing Absolute Value Equations How do I make one of those V graphs?

Transformations Review Vertex form: y = a(x – h) 2 + k The vertex form of a quadratic equation allows you to immediately identify the vertex of a parabola.

5.3 Transformations of Parabolas Goal : Write a quadratic in Vertex Form and use graphing transformations to easily graph a parabola.

Solve the equation for y. SOLUTION EXAMPLE 2 Graph an equation Graph the equation –2x + y = –3. –2x + y = –3 y = 2x –3 STEP 1.

2.7 – Use of Absolute Value Functions and Transformations.

How do I graph and write absolute value functions?

2.7: Use Absolute Value Functions and Transformations HW Monday: p.127 (4-20 even) Test , 2.7: Friday.

3.3 Graph Using Intercepts You will graph a linear equation using intercepts. Essential question: How do you use intercepts to graph equations? You will.

Algebra 2. Lesson 5-3 Graph y = (x + 1) 2 – Step 1:Graph the vertex (–1, –2). Draw the axis of symmetry x = –1. Step 2:Find another point. When.

EXAMPLE 1 Compare graph of y = with graph of y = a x 1 x 1 3x3x b. The graph of y = is a vertical shrink of the graph of. x y = 1 = y 1 x a. The graph.

EXAMPLE 1 Graph a function of the form y = | x – h | + k Graph y = | x + 4 | – 2. Compare the graph with the graph of y = | x |. SOLUTION STEP 1 Identify.

Chapter 4 Section 2. EXAMPLE 1 Graph a quadratic function in vertex form Graph y = – (x + 2) SOLUTION STEP 1 Identify the constants a =

Warm-Up Exercises 1. Identify the axis of symmetry for the graph of y = 3x 2. ANSWER x = 0 2. Identify the vertex of the graph of y = 3x 2. ANSWER (0,

Quadratic Functions Unit Objectives: Solve a quadratic equation.

Transformations of Functions

Lesson 2-6 Families of Functions.

Transformations of Functions

Module 2 Lesson 2 Reflections Reflections.

Unit 5 – Quadratics Review Problems

Solve an equation by multiplying by a reciprocal

Graph Absolute Value Functions using Transformations

Absolute Value Functions

Graph Absolute Value Functions using Transformations

4.1 Quadratic Functions and Transformations

Section 2.5 Transformations.

Graph Absolute Value Functions using Transformations

Graph Absolute Value Equations

Graph Absolute Value Functions using Transformations

Transformations of graphs

Lesson 5-1 Graphing Absolute Value Functions

and 2 units to the left of point B. Count down 5 units from point B

Chapter 2: Analysis of Graphs of Functions

8.4 - Graphing f (x) = a(x − h)2 + k

y x Lesson 3.7 Objective: Graphing Absolute Value Functions.

2.7 Graphing Absolute Value Functions

Copyright © Cengage Learning. All rights reserved.

Transformation rules.

2-6 Families of Functions

2.7 Graphing Absolute Value Functions

Graphing Absolute Value Functions

Transformations of Functions

Jeopardy Final Jeopardy Solving Equations Solving Inequalities

Maps one figure onto another figure in a plane.

Transformations to Parent Functions

1.3 Combining Transformations

Presentation transcript:

EXAMPLE 4 Write an absolute value function Holograms In holography, light from a laser beam is split into two beams, a reference beam and an object beam. Light from the object beam reflects off an object and is recombined with the reference beam to form images on film that can be used to create three-dimensional images. Write an equation for the path of the reference beam.

EXAMPLE 4 Write an absolute value function SOLUTION The vertex of the path of the reference beam is (5, 8). So, the equation has the form y = a x – Substitute the coordinates of the point (0, 0) into the equation and solve for a. 0 = a 0 – Substitute 0 for y and 0 for x. – 1.6 = a Solve for a. An equation for the path of the reference beam is y = – 1.6 x – ANSWER

Apply transformations to a graph EXAMPLE 5 The graph of a function y = f (x) is shown. Sketch the graph of the given function. a. y = 2 f (x) b. y = – f (x + 2) + 1

Apply transformations to a graph EXAMPLE 5 The graph of y = 2 f (x) is the graph of y = f (x) stretched vertically by a factor of 2. (There is no reflection or translation.) To draw the graph, multiply the y-coordinate of each labeled point on the graph of y = f (x) by 2 and connect their images. a. SOLUTION

Apply transformations to a graph EXAMPLE 5 The graph of y = – f (x + 2) +1 is the graph of y = f (x) reflected in the x- axis, then translated left 2 units and up 1 unit. To draw the graph, first reflect the labeled points and connect their images. Then translate and connect these points to form the final image. b. SOLUTION

GUIDED PRACTICE for Examples 4 and 5 4. WHAT IF? In Example 4, suppose the reference beam originates at (3, 0) and reflects off a mirror at (5, 4). Write an equation for the path of the beam. y = –2| x – 5 | + 4 ANSWER

GUIDED PRACTICE for Examples 4 and 5 Use the graph of y = f (x) from Example 5 to graph the given function. 5. y = 0.5 f (x) SOLUTION

GUIDED PRACTICE for Examples 4 and 5 6. y = – f (x – 2) – 5 Use the graph of y = f (x) from Example 5 to graph the given function. SOLUTION

GUIDED PRACTICE for Examples 4 and 5 7. y = 2 f (x + 3) – 1 Use the graph of y = f (x) from Example 5 to graph the given function. SOLUTION