Chapter 4 Quadratics 4.3 Using Technology to Investigate Transformations.

Slides:



Advertisements
Similar presentations
6.6 Analyzing Graphs of Quadratic Functions
Advertisements

Parabolas and Modeling
Goal: I can infer how the change in parameters transforms the graph. (F-BF.3) Unit 6 Quadratics Translating Graphs #2.
Chapter 5.1 – 5.3 Quiz Review Quizdom Remotes!!!.
The vertex of the parabola is at (h, k).
Essential Question: In the equation f(x) = a(x-h) + k what do each of the letters do to the graph?
Transforming reciprocal functions. DO NOW Assignment #59  Pg. 503, #11-17 odd.
And the Quadratic Equation……
11.1 Solving Quadratic Equations by the Square Root Property
Topic: U2 L1 Parts of a Quadratic Function & Graphing Quadratics y = ax 2 + bx + c EQ: Can I identify the vertex, axis of symmetry, x- and y-intercepts,
+ Translating Parabolas § By the end of today, you should be able to… 1. Use the vertex form of a quadratic function to graph a parabola. 2. Convert.
Warm-Up: you should be able to answer the following without the use of a calculator 2) Graph the following function and state the domain, range and axis.
Polynomial Function A polynomial function of degree n, where n is a nonnegative integer, is a function defined by an expression of the form where.
Graphing Quadratics.
Graphing Techniques: Transformations
Graphical Transformations
2.4 Use Absolute Value Functions and Transformations
2.2 b Writing equations in vertex form
5.1 Stretching/Reflecting Quadratic Relations
Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities Chapter 5 – Quadratic Functions and Inequalities 5.7 – Analyzing Graphs of Quadratic.
Graph Absolute Value Functions using Transformations
Algebra II Piecewise Functions Edited by Mrs. Harlow.
Consider the function: f(x) = 2|x – 2| Does the graph of the function open up or down? 2. Is the graph of the function wider, narrower, or the same.
Summary of 2.1 y = -x2 graph of y = x2 is reflected in the x- axis
Vertex & axis of Symmetry I can calculate vertex and axis of symmetry from an equation.
GRAPHING QUADRATIC FUNCTIONS
Transformations Transformations of Functions and Graphs We will be looking at simple functions and seeing how various modifications to the functions transform.
 How would you sketch the following graph? ◦ y = 2(x – 3) 2 – 8  You need to perform transformations to the graph of y = x 2  Take it one step at a.
Friday, March 21, 2013 Do Now: factor each polynomial 1)2)3)
Parabolas.
Graphing Parabolas Students will be able to graph parabolas or second degree equations.
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.
QUADRATIC EQUATIONS in VERTEX FORM y = a(b(x – h)) 2 + k.
5.3 Transformations of Parabolas Goal : Write a quadratic in Vertex Form and use graphing transformations to easily graph a parabola.
10.1 Quadratic GRAPHS!.
Math-3 Lesson 1-3 Quadratic, Absolute Value and Square Root Functions
 .
Warm Up Give the coordinates of each transformation of (2, –3). 4. reflection across the y-axis (–2, –3) 5. f(x) = 3(x + 5) – 1 6. f(x) = x 2 + 4x Evaluate.
Vocabulary The function f(x) = |x| is an absolute value function. The highest of lowest point on the graph of an absolute value function is called the.
Section 9.3 Day 1 Transformations of Quadratic Functions
Objectives: Be able to graph a quadratic function in vertex form Be able to write a quadratic function in vertex form (2 ways)
 1.x 2 – 7x -2  2.4x 3 + 2x 2 + 4x – 10  3.3x 4 – 4x 3 + x 2 – x – 6  4.10x – 15  5.6x 3 – x 2 + 8x + 5  6.8x x 2 – 14x – 35  7.x – 7  8.12x.
UNIT 5 REVIEW. “MUST HAVE" NOTES!!!. You can also graph quadratic functions by applying transformations to the parent function f(x) = x 2. Transforming.
Transformations of Functions. The vertex of the parabola is at (h, k).
5-3 Using Transformations to Graph Quadratic Functions.
Unit 2 – Quadratic Functions & Equations. A quadratic function can be written in the form f(x) = ax 2 + bx + c where a, b, and c are real numbers and.
Chapter 3 Quadratic Functions
Grab Interactive Notes Homework Study Guide & Khan Academy
Inequality Set Notation
Interesting Fact of the Day
Use Absolute Value Functions and Transformations
2.6 Translations and Families of Functions
Graphs of Quadratic Functions
3.5 Transformations of Graphs
Translating Parabolas
Objectives Transform quadratic functions.
Unit 5a Graphing Quadratics
Lesson 5.3 Transforming Parabolas
Bellwork.
Unit 6 Graphing Quadratics
Chapter 8 Quadratic Functions.
Lesson 5.3 Transforming Parabolas
Chapter 8 Quadratic Functions.
2.1 Transformations of Quadratic Functions
The vertex of the parabola is at (h, k).
Translations & Transformations
Warm Up (5 Minutes) (-2,-2); Translated: Vertically 4, Horizontally -3
Unit 3 Graphing Quadratics
Unit 5 Graphing Quadratics
Unit 5a Graphing Quadratics
Presentation transcript:

Chapter 4 Quadratics 4.3 Using Technology to Investigate Transformations

Humour Break

4.3 Using Technology to Investigate Transformations The relation y = x² is the simplest quadratic relation. It is the base curve for all relations

4.3 Using Technology to Investigate Transformations y = x²... a = 1 and the graph opens up, standard width This equation is both in vertex form and in standard form Consider... y = 1x² + 0x + 0 (standard form) Consider... Y = a(x – h)² + k Consider... y = 1(x – 0)² + 0 (vertex form) Vertex is (0, 0) and y-intercept is also (0, 0)

4.3 Using Technology to Investigate Transformations xy y = x²

4.3 Using Technology to Investigate Transformations The relation y = -x² is the simplest quadratic relation reflected down about the x axis.

4.3 Using Technology to Investigate Transformations y = -x²... a = -1 and the graph opens down, standard width This equation is both in vertex form and in standard form Consider... y = -1x² + 0x + 0 (standard form) Consider... y = a(x – h)² + k Consider... y = -1(x – 0)² + 0 (vertex form) Vertex is (0, 0) and y-intercept is also (0, 0)

4.3 Using Technology to Investigate Transformations xy y = - x²

4.3 Using Technology to Investigate Transformations Consider… y = 2x² and y = -2x² What impact does the 2 have?

4.3 Using Technology to Investigate Transformations y = 2x²... a = 2 and the graph opens up, more narrow width (double height for any given point) This equation is both in vertex form and in standard form Consider... y = 2x² + 0x + 0 (standard form) Consider... y = a(x – h)² + k Consider... y = 2(x – 0)² + 0 (vertex form) Vertex is (0, 0) and y-intercept is also (0, 0)

4.3 Using Technology to Investigate Transformations xy y = 2x²

4.3 Using Technology to Investigate Transformations y = -2x²... a = -2 and the graph opens down, more narrow width (double height for any given point) This equation is both in vertex form and in standard form Consider... y = -2x² + 0x + 0 (standard form) Consider... y = a(x – h)² + k Consider... y = -2(x – 0)² + 0 (vertex form) Vertex is (0, 0) and y-intercept is also (0, 0)

4.3 Using Technology to Investigate Transformations xy y = -2x²

4.3 Using Technology to Investigate Transformations So… y = 2x² and y = -2x² an “a” of 2 doubles the height of the graph for a given x value an “a” of -2 doubles the height of the graph for a given x value but opening down We can generalize this rule for different values of “a”

4.3 Using Technology to Investigate Transformations Consider… y = 1/2x² and y = - 1/2x² What impact does the a of ½ and -½ have?

4.3 Using Technology to Investigate Transformations y = 1/2x²... a = 1/2 and the graph opens up, but wider (half height for any given point) This equation is both in vertex form and in standard form Consider... y = 1/2x² + 0x + 0 (standard form) Consider... y = a(x – h)² + k Consider... y = 1/2(x – 0)² + 0 (vertex form) Vertex is (0, 0) and y-intercept is also (0, 0)

4.3 Using Technology to Investigate Transformations xy y = 1/2x²

4.3 Using Technology to Investigate Transformations y = -1/2x²... a = -1/2 and the graph opens down, but wider (half height for any given point) This equation is both in vertex form and in standard form Consider... y = -1/2x² + 0x + 0 (standard form) Consider... y = a(x – h)² + k Consider... y = -1/2(x – 0)² + 0 (vertex form) Vertex is (0, 0) and y-intercept is also (0, 0)

4.3 Using Technology to Investigate Transformations xy y = -1/2x²

4.3 Using Technology to Investigate Transformations So… y = 1/2x² and y = - 1/2x² an “a” of ½ halves the height of the graph for a given x value an “a” of -1/2 halves the height of the graph for a given x value but opening down We can generalize this rule for different values of “a”

4.3 Using Technology to Investigate Transformations Consider… y = x² + 1 and y = x² - 1 What impact does adding or subtracting 1 have to the graph?

4.3 Using Technology to Investigate Transformations y = 1x² a = 1 and the graph opens up This equation is both in vertex form and in standard form Consider... y = x² + 0x + 1 (standard form) Consider... y = a(x – h)² + k Consider... y = 1(x – 0)² + 1 (vertex form) Vertex is (0, 1) and y-intercept is also (0, 1)

4.3 Using Technology to Investigate Transformations xy y = x² + 1

4.3 Using Technology to Investigate Transformations y = 1x² a = 1 and the graph opens up This equation is both in vertex form and in standard form Consider... y = x² + 0x - 1 (standard form) Consider... y = a(x – h)² + k Consider... y = 1(x – 0)² - 1 (vertex form) Vertex is (0, -1) and y-intercept is also (0, -1)

4.3 Using Technology to Investigate Transformations xy y = x² - 1

4.3 Using Technology to Investigate Transformations So… y = x² + 1 and y = x² - 1 a “k” of +1 outside the brackets shifts the entire graph up by 1 a “k” of -1 outside the brackets shifts the entire graph down by 1 We can generalize this rule for different values of “k”

4.3 Using Technology to Investigate Transformations Consider… y = (x - 1)² and y = (x + 1)² What impact does adding or subtracting 1 inside the brackets have to the graph?

4.3 Using Technology to Investigate Transformations y = 1(x - 1)²... a = 1 and the graph opens up This equation is both in vertex form and in standard form Consider... y = x² - 2x + 1 (standard form) Consider... y = a(x – h)² + k Consider... y = 1(x – 1)² + 0 (vertex form) Vertex is (1, 0) and y-intercept is also (0, 1)

4.3 Using Technology to Investigate Transformations xy y = (x - 1)²

4.3 Using Technology to Investigate Transformations y = 1(x – (- 1))² or y = 1(x + 1)²... a = 1 and the graph opens up This equation is both in vertex form and in standard form Consider... y = x² + 2x + 1 (standard form) Consider... y = a(x – h)² + k Consider... y = 1(x + 1)² + 0 (vertex form) Vertex is (-1, 0) and y-intercept is also (0, 1)

4.3 Using Technology to Investigate Transformations xy y = (x + 1)²

4.3 Using Technology to Investigate Transformations So… y = (x - 1)² and y = (x + 1)² a “h” of -1 in the brackets (with the subtraction providing the negative) shifts the entire graph to the right by 1 a “h” of +1 in the brackets (with the double negative providing the positive) shifts the entire graph to the left by 1 We can generalize this rule for different values of “h” Unlike k, the general rule shift is counter-intuitive because you move in the opposite direction of the sign

4.3 Using Technology to Investigate Transformations y = a(x – h)² + k… putting it together “a” opens up & “-a” opens down If a>1, the graph is more narrow & higher by a factor of “a”, so if a = 2, the y value for a given x will be twice as high If a<1, the graph is wider & flatter by a factor of “a”, so if a = 1/2, the y value for a given x will be ½ as high

4.3 Using Technology to Investigate Transformations y = a(x – h)² + k… putting it together (x – h) moves the x of the vertex from 0 right by “h” so x of vertex of (x – 2) would be at x=2 (x + h) moves the x of the vertex from 0 left by “h” so x of vertex of (x + 2) would be at x=-2 + k moves the vertex (and graph) up by k - k moves the vertex (and graph) down by k

4.3 Using Technology to Investigate Transformations Ex. 1 Write the relation for a parabola that satisfies each of the following conditions: Vertex at (4,7), opens downward, same shape as y = x²

4.3 Using Technology to Investigate Transformations Ex. 1 Write the relation for a parabola that satisfies each of the following conditions: Vertex at (4,7), opens downward, same shape as y = x² y = - 1(x – 4)² + 7

4.3 Using Technology to Investigate Transformations y = x² and y = -1(x – 4)² + 7

4.3 Using Technology to Investigate Transformations Ex. 2 Consider a parabola that is congruent to y = x², opens up and has the vertex at (1, -3). Now find the equation of a new parabola, that results if P is: (a) stretched vertically by a factor of 3 (b) compressed by a factor of 3 (c) translated 2 units to the left (d) translated 3 units up (e) reflected about the x-axis and translated 2 units to the left and 4 units down and stretched by a factor of 2

4.3 Using Technology to Investigate Transformations

Ex. 2 Consider a parabola that is congruent to y = x², opens up and has the vertex at (1, -3). Now find the equation of a new parabola, that results if P is: (a) stretched vertically by a factor of 3: Starting point is y= 1(x -1)² - 3 Stretching vertically by a factor of 3 makes “a” 3 y= 3(x -1)² - 3 (transformed equation)

4.3 Using Technology to Investigate Transformations

Ex. 2 Consider a parabola that is congruent to y = x², opens up and has the vertex at (1, -3). Now find the equation of a new parabola, that results if P is: (b) compressed by a factor of 3: Starting point is y= 1(x -1)² - 3 Compressing by a factor of 3 makes “a” 1/3 y= 1/3(x -1)² - 3 (transformed equation)

4.3 Using Technology to Investigate Transformations

Ex. 2 Consider a parabola that is congruent to y = x², opens up and has the vertex at (1, -3). Now find the equation of a new parabola, that results if P is: (c) translated 2 units to the left: Starting point is y= 1(x -1)² - 3 Translating 2 units to the left, moves the x of the vertex from 1 to -1, which changes the (x -1) to (x- (-1)) which is the same as (x + 1) y= 1(x + 1)² - 3 (transformed equation)

4.3 Using Technology to Investigate Transformations

Ex. 2 Consider a parabola that is congruent to y = x², opens up and has the vertex at (1, -3). Now find the equation of a new parabola, that results if P is: (c) translated 3 units up: Starting point is y= 1(x -1)² - 3 Translating 3 units up adds 3 to the k of -3 y= 1(x + 1)² (transformed equation)

4.3 Using Technology to Investigate Transformations

Ex. 2 Consider a parabola that is congruent to y = x², opens up and has the vertex at (1, -3). Now find the equation of a new parabola, that results if P is: (e) reflected about the x-axis and translated 2 units to the left and 4 units down and stretched by a factor of 2 Starting point is y= 1(x -1)² - 3 y= -2(x + 1)²-7 (transformed equation)

4.3 Using Technology to Investigate Transformations

Homework Tuesday, May 17 th, p.363, #1-4, 6-13 & 16