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