Chapter 4 Quadratics 4.2 Vertex Form of the Equation

Humour Break

4.2 Vertex Form Quadratic Expression Warm-up (Handout – to be collected) What is the same about these three equations. What is different about these three equations y = 2x² -12x + 16 y = 2(x – 4)(x – 2) y = 2(x – 3)² - 2

4.2 Vertex Form Quadratic Expression The equation y = a(x – h)² + k defines a quadratic relation

4.2 Vertex Form Quadratic Expression When a quadratic relation is in the form y = a(x – h)² + k h and k are the coordinates of the vertex (h, k) which is why we write a quadratic written in this form vertex form

4.2 Vertex Form Quadratic Expression When a quadratic relation is in the form y = a(x – h)² + k a tells us the direction of opening

4.2 Vertex Form Quadratic Expression When a quadratic relation is in the form y = a(x – h)² + k a tells us the direction of opening if a > 0, the parabola opens up

4.2 Vertex Form Quadratic Expression

When a quadratic relation is in the form y = a(x – h)² + k a tells us the direction of opening if a < 0, the parabola opens down

4.2 Vertex Form Quadratic Expression

When a quadratic relation is in the form y = a(x – h)² + k What would happen if a = 0?

4.2 Vertex Form Quadratic Expression When a quadratic relation is in the form y = a(x – h)² + k What would happen if a = 0? You would end up with a horizontal line y = k so you would not have a parabola

4.2 Vertex Form Quadratic Expression y = a(x – h)² + k If the coordinates of the vertex (h, k) are known, you can write the quadratic relation in vertex form by substituting h and k and the coordinates of another point (x, y) (that satisfies the relation) and then calculating a

4.2 Vertex Form Quadratic Expression Ex.1 Find the equation in vertex form, given the vertex (3, -2) and a y-intercept of 16 y = a(x – h)² + k h is 3 (given) k is -2 (given) x is 0 (because y intercept is on y axis) y is 16 (given)

4.2 Vertex Form Quadratic Expression Ex. Find the equation in vertex form, given the vertex (3, -2) and a y-intercept of 16 y = a(x – h)² + k 16 = a(0 – 3)² + (-2) (fill in the values) = 9a (move -2 to left of = sign) 18 = 9a (simplify & divide both a = 2 sides by 3)

4.2 Vertex Form Quadratic Expression Ex. (Cont’d) Since a = 2 and h, k are 3 and -2 y = a(x – h)² + k y = 2(x – 3)² + (-2) (fill in the values) y = 2(x – 3)² - 2 (simplify)

4.2 Vertex Form Quadratic Expression If you know the equation in vertex form, you can convert it to standard form by expanding it using FOIL and simplifying it y = 2(x – 3)² - 2

4.2 Vertex Form Quadratic Expression If you know the equation in vertex form, you can convert it to standard form by expanding it using FOIL and simplifying it y = 2(x – 3)² - 2

4.2 Vertex Form Quadratic Expression If you know the equation in vertex form, you can convert it to standard form by expanding it using FOIL and simplifying it collecting like terms y = 2(x – 3)² - 2 y = 2(x² - 3x – 3x + 9) – 2 y = 2(x² - 6x + 9) – 2 y = 2x² - 12x + 18 – 2 y = 2x² - 12x + 16

4.2 Vertex Form Quadratic Expression If you know the equation in vertex form, you can convert it to standard form by expanding it using FOIL and simplifying it y = 2(x – 3)² - 2

4.2 Vertex Form Quadratic Expression Ex. 2 Find the vertex, the AOS, the direction of opening and the number of zeros for the graph of the quadratic relation

4.2 Vertex Form Quadratic Expression Vertex Axis of Symmetry Direction of Opening Number of Zeros

4.2 Vertex Form Quadratic Expression Vertex (5, -3) AOS x = 5 Direction of Opening - down Number of Zeros - none

4.2 Vertex Form Quadratic Expression Ex. 3 A ball is hit into the air. It’s height H (in metres) after t seconds is (a)In which direction does the parabola open? How do you know? (b)What are the coordinates of the vertex? What does the vertex represent in this situation? (c)From what height was the ball hit? (d)Find one other point on the curve and interpret its meaning

4.2 Vertex Form Quadratic Expression Ex. 3 A ball is hit into the air. It’s height H (in metres) after t seconds is (a)In which direction does the parabola open? How do you know? Answer: The parabola opens down because a = -5

4.2 Vertex Form Quadratic Expression Ex. 3 A ball is hit into the air. It’s height H (in metres) after t seconds is (b) What are the coordinates of the vertex? What does the vertex represent in this situation? Answer: The coordinates of the vertex are (4, 120). This means the ball reaches a maximum height of 120m at t = 4s

4.2 Vertex Form Quadratic Expression Ex. 3 A ball is hit into the air. It’s height H (in metres) after t seconds is (c) From what height was the ball hit? Answer: Substitute t = 0 into equation and solve for H

4.2 Vertex Form Quadratic Expression Ex. 3 A ball is hit into the air. It’s height H (in metres) after t seconds is (c) Answer: H = -5(0 – 4)² H = -5 (16) H = H = 40 metres

4.2 Vertex Form Quadratic Expression Ex. 3 A ball is hit into the air. It’s height H (in metres) after t seconds is (d) Find one other point on the curve and interpret its meaning

4.2 Vertex Form Quadratic Expression Ex. 3 A ball is hit into the air. It’s height H (in metres) after t seconds is (d) Find one other point on the curve and interpret its meaning Answer: Lets’ pick another time, say t = 3 into equation and solve for H

4.2 Vertex Form Quadratic Expression Ex. 3 A ball is hit into the air. It’s height H (in metres) after t seconds is (d) Answer: H = -5(3 – 4)² H = -5 (1) H = H = 115 metres At time of 3 seconds, the ball is at a height of 115 metres

Homework Friday, May 13 th, Page 351, #1, 2adef, 3ace, 4ace, 5, 7a, 8a & Monday, May 16 th, Page 352, # , 16, 21 & 24