Grade 10 Academic (MPM2D) Unit 4: Quadratic Relations Quadratic Formula Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved

Completing the Square A quadratic relation in standard form can be rewritten in vertex form by creating a perfect square in the expression, then factoring the square. This technique is called completing the square. Completing the square can be used to find the vertex of a quadratic in standard form without finding the zeros of the relation or two points equidistant from the axis of summetry. Completing the square allows you to find the maximum or minimum value of a quadratic relation algebraically, without using a graph. Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Completing the Square (Steps) Remove the common constant factor from both the x2 and x term. Find the constant that must be added and subtracted to create a perfect square. This value equals the square of half of the coefficient of the x –term in step 1. Rewrite the expression by adding, then subtracting, this value after the x – term inside the brackets. Group the three terms that form the perfect square. Move the subtracted value outside the brackets by multiplying it by the common constant factor. 4) Factor the perfect square and collect like terms. Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

25 + - 16 RECALL: Example 2c from yesterday lesson!!! Convert each of the following into vertex form by completing the square. Determine the vertex, axis of symmetry, opening, intercepts and the mapping rule. Common factor for first two terms only! 25 16 + - Think about: To create Perfect Trinomial, multiply the last term -25/16 to common factor 2! Perfect Trinomial Mapping Rule: Vertex: Last term in standard form Parabola opens (concaves) up, vertical expands by factor of 2, translates 1.25 units right and down 33/8 units. Opens up Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

b2 4a2 + - Creating the Quadratic formula Convert each of the following into vertex form by completing the square. b2 4a2 + - Think about: Mapping Rule: Vertex: Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Summary: Quadratic Formulas Vertex Form: Axis of Symmetry: x-intercepts: y-intercept: Standard Form: Vertex: Axis of Symmetry: x-intercepts: y-intercept: Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: Solving Quadratic equation using the quadratic formula Solve each of the following using the quadratic formula and leave the answer in square root form. Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: Solving Quadratic equation using the quadratic formula Solve each of the following using the quadratic formula and leave the answer in square root form. Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 2: Finding the roots using the quadratic formula Find the roots (x – intercepts) of the following using the quadratic formula and accurate the answer in two decimal places. Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 2: Finding the roots using the quadratic formula Find the roots (x – intercepts) of the following using the quadratic formula and accurate the answer in two decimal places. Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 3: Word Problem using the quadratic formula The height of an object being thrown from the edge of a cliff that is 15 m height, in metres, is given by the relation where t is in seconds. How long is the object in the air? Since time > 0, the object stays in air: 6.21 sec Note: If the ball starts on the ground and the height of the ball after t seconds is given by . Then the ball will be in the air between the two zeros . Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

49 + - 25 Example 4: Quadratic Relation For the quadratic relation (Example 2c) Convert the defining equation of the relation from standard form into vertex form. Use the quadratic formula to approximate the roots. Determine the vertex, axis of symmetry and y – intercept. Sketch the graph using the key the results from b & c State the mapping rule. 49 25 + - Vertex: Think about: Opens down Perfect Trinomial Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 4: Quadratic Relation For the quadratic relation (Example 2c) Convert the defining equation of the relation from standard form into vertex form. Use the quadratic formula to approximate the roots. Determine the vertex, axis of symmetry and y – intercept. Sketch the graph using the key the results from b & c State the mapping rule. Solution Copied from eg 2c Method 1: Formula Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 4: Quadratic Relation For the quadratic relation (Example 2c) Convert the defining equation of the relation from standard form into vertex form. Use the quadratic formula to approximate the roots. Determine the vertex, axis of symmetry and y – intercept. Sketch the graph using the key the results from b & c State the mapping rule. Method 2: Isolate x from Vertex form Recall from part a) Same Results as Method 1 Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 4: Quadratic Relation For the quadratic relation (Example 2c) Convert the defining equation of the relation from standard form into vertex form. Use the quadratic formula to approximate the roots. Determine the vertex, axis of symmetry and y – intercept. Sketch the graph using the key the results from b & c State the mapping rule. y – intercept: Recall from Part A: Standard form: Last constant term in standard form: or Let x = 0 in vertex form Vertex form: Vertex: Opens down Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Example 4: Quadratic Relation For the quadratic relation (Example 2c) Convert the defining equation of the relation from standard form into vertex form. Use the quadratic formula to approximate the roots. Determine the vertex, axis of symmetry and y – intercept. Sketch the graph using the key the results from b & c State the mapping rule. Vertex: Opens down Mapping Rule: x – intercepts: Parabola expands vertically by factor of 5 opens (concaves) down, translates 1.4 units right and up 12.8 units. y – intercept: Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

Homework Work sheet: Quadratic Formula Text: P. 403 #3,5,8,9,11,14,17,20,24 Check the website for updates Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved

End of lesson Quadratic Formula © 2017 E. Choi – MPM2D - All Rights Reserved