Foundations of Math 11 Section 7.1 Quadratic Functions 31 7.1 Quadratic Functions Mt. Douglas Secondar Quadratic functions are found in everda situations, not just in our math classroom. Tossing a ball in the air is a quadratic function, so is the flow rate of water in a pipe, and the supporting cables of a suspension bridge. Definition of a Quadratic Function A quadratic function is a function that can be written in the form f () = a + b + c where a, b and c are real numbers, and a c. The graph of a quadratic function is called a parabola. The simplest quadratic function is the function =. This is a basic cup-shaped curve that is smmetric to the -ais, which has the equation =. The lowest point of this cup-shaped curve (, ) is called the verte. The ais of smmetr is the line which divides the parabola into two identical parts. Parabola ais of smmetr = 8 = verte (, ) -3 - -1 1 3 Graphing a Parabola Step 1: Find the verte and ais of smmetr. Step : Find the -intercept b evaluating f (), and the -intercept(s) b evaluating f () =. Step 3: Add additional points if needed. Step : Sketch graph. The graph of = a The graph of = a is a parabola with verte at the origin (, ) If a > : opens upward, verte is a minimum point. If a < : opens downward, verte is a maimum point. The parabola will be wide if 1 < a < 1, and narrow if a > 1 or a < 1 (in comparison to the basic parabola = ). The ais of smmetr is =. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
3 Chapter 7 Quadratic Functions Eample 1 Foundations of Math 11 Mt. Douglas Secondar Graph of = a a=1 a= 1 a= The graph of = a + c The graph of = a + c shifts verticall up if c >, and verticall down if c <. The verte of the parabola is (, c). The ais of smmetr is =. Graph of = a + c = + 1 Eample = 1 + 1 = 1 a = 1 1 3 a= a=3 Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Section 7.1 Quadratic Functions 33 Foundations of Math 11 Mt. Douglas Secondar The graph of = a( c) The graph shifts horizontall to the right if c >, and horizontall to the left if c <. The verte of the parabola is (c, ). The ais of smmetr is = c. Eample 3 Graph of = a( c) = ( ) = 1 ( ) = ( + 3) The graph of = a( h) + k This quadratic formula is said to be in standard form. The verte of the parabola is (h, k). The ais of smmetr is = h. The parabola opens upward if a >, and downward if a <. The parabola will have a minimum value if a >, and a maimum value if a <. Eample Graph of = a( h) + k = 1 ( + 1) + = ( 1) 3 = 13 ( + ) Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
3 Chapter 7 Quadratic Functions Foundations of Math 11 Domain and Range Mt. Douglas Secondar Domain and Range from a Set of Ordered Pairs (, ) The set of first components, or -values, in the ordered pairs is the domain. The set of second components, or -values, is called the range. Eample 1 Determine the domain and range of the ordered pairs: A = { (1, ), ( 3, 5), (, ) } B = { ( 3, ), (1, ), (, ), (3, ) } C = { (, 1), (1, ), (3, 3), (1, ) } D = { ( 3, 1), ( 3, 3), (, 3), (, ) } Domain of A = { 3, 1, }, Range of A = {,, 5 } Domain of B = { 3, 1,, 3 }, Range of B = {,, } Domain of C = {, 1, 3 }, Range of C = {, 1, 3, } Domain of D = { 3,, }, Range of D = { 1,, 3 } Eample Determine the domain and range of the following parabolas: a) = ( 1) 3 b) = 1 ( +1) + c) = 1 ( + ) 3 a) domain: all real numbers range: 3 b) domain: all real numbers range: c) domain: all real numbers range: Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7.1 Quadratic Functions 35 Eample 3 Graph the following quadratic equations, then state the verte, ma/min values of the parabola, ais of smmetr, domain and range. Plot at least points besides the verte. a) = ( 1) 3 b) = 1 ( + ) + Mt. Douglas Secondar a) = ( 1) 3 = ( 1) 3 = 1 = ( 1) 3 = 1 = (3 1) 3 = 5 = ( 1 1) 3 = 5 1 1 3 3 5 1 1 5 Verte: (1, 3) Ma/min: minimum value of 3 Ais of smmetr: = 1 Domain: all real numbers Range: 3 b) = 1 ( + ) + = 1 ( + ) + = = 1 ( + ) + = = 1 ( + ) + = = 1 ( + ) + = Verte: (, ) Ma/min: maimum value of Ais of smmetr: = Domain: all real numbers Range: Note: a point horizontall, left or right of the ais of smmetr will alwas be equal distances. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
3 Chapter 7 Quadratic Functions Foundations of Math 11 Finding the Equation of a Parabola from a Graph Finding the equation of a parabola from a graph requires two things: 1. the verte;. the value that determines the shape and direction of the parabola. Mt. Douglas Secondar Eample 1 Determine an equation for the parabola on the right. Verte of the parabola is (, 5) Therefore, = a( + ) + 5. Now we must solve for the a value which determines the shape of the parabola. A point on the graph other than the verte must be found. Three such points are (, 3), (, 3) or (, 3). We will use (, 3). Therefore, equation of the parabola is = 1 ( + ) + 5 = a( + ) + 5 3 = a( + ) + 5 3 = a + 5 a = a = 1 Eample Determine an equation for the parabola on the right. Verte of the parabola is (1, ) Therefore, = a( 1). Now we must solve for the a value which determines the shape of the parabola. A point on the graph other than the verte could be (, ) or (, ). We will use (, ). = a( 1) = a( 1) = 9a 9a = a = 3 Therefore, equation of the parabola is = 3 ( 1) Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7.1 Quadratic Functions 37 7.1 Eercise Set Graphing a Parabola 1. Determine whether the graph of each quadratic function opens upwards or downwards. Mt. Douglas Secondar a) = 1 3 + b) = + 3 c) = 3( 1) + d) = 3 e) + = f) + + =. Graph the quadratic function. Plot at least points other than verte. a) = Verte: Ma/min: Ais of smmetr: Domain: Range: b) = Verte: Ma/min: Ais of smmetr: Domain: Range: Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
38 Chapter 7 Quadratic Functions Foundations of Math 11. c) = Mt. Douglas Secondar Verte: Ma/min: Ais of smmetr: Domain: Range: d) = ( ) Verte: Ma/min: Ais of smmetr: Domain: Range: e) = ( +1) + Verte: Ma/min: Ais of smmetr: Domain: Range: f) = 1 ( + ) Verte: Ma/min: Ais of smmetr: Domain: Range: Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7.1 Quadratic Functions 39. g) = 1 ( ) + 3 Mt. Douglas Secondar Verte: Ma/min: Ais of smmetr: Domain: Range: h) = ( 1) Verte: Ma/min: Ais of smmetr: Domain: Range: i) = ( + ) 3 Verte: Ma/min: Ais of smmetr: Domain: Range: j) + 3 = 3 ( ) Verte: Ma/min: Ais of smmetr: Domain: Range: Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
31 Chapter 7 Quadratic Functions Foundations of Math 11 3. Determine an equation for the parabola. a) b) Mt. Douglas Secondar c) d) e) f) Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7. General Form of a Quadratic Function 311 7. General Form of a Quadratic Function Mt. Douglas Secondar Not all quadratic functions come in the standard form f () = a( h) + k. Therefore, we must derive a wa of changing a general form quadratic function f () = a + b + c to standard form. Onl in standard form can we easil find the verte of a quadratic function. The steps are summarized below. f () = a + b + c = a + b + c c = a + b c = a( + b a + ) c + b a = a + b a + b a = a + b a + c b a f () = a + b a + c b a Given standard form. Replace f() with to simplif notation. Add c to both sides. Factor a out of right-side equation. Add b a to right side to make a perfect square, and a b a = b to left side to balance equation. a (Recall perfect squares from Math 1.) Factor on right sides and add c b a Write in form f () = a( h) + k. to both sides to isolate =. with h = b a and k = c b a Therefore, the general form quadratic formula f () = a + b + c has verte b a, c b a. We can now define the verte formula. The Verte Formula The graph of f () = a + b + c, a is a parabola with verte (h, k) and ais of smmetr = h where h = b b b and k = c or k = f a a a. If a >, the parabola has a minimum value and opens upwards. If a <, the parabola has a maimum value and opens downwards. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
31 Chapter 7 Quadratic Functions Eample 1 Foundations of Math 11 Mt. Douglas Secondar Determine the verte of f () = 3 directl from the equation. f () = + 8 3 has a =, b = and c = 3. b b ( ) Verte, c =, 3 = (1, 5), or a () () a -coordinate is b = = 1. So, f () = 3 f (1) = (1) (1) 3 = 5. a () Therefore, verte is (1, -5). Eample Determine the verte and ais of smmetr for the quadratic function f () = 1. f () = 1 a =, b = and c = 1 b b ( ) ( ) Verte is, c =, 1 = (1, 3), ais of smmetr is = 1, or a () () a -coordinate = b ( ) = = 1. So, f () = 1 f (1) = (1 ) (1) 1 = 3, a () Therefore, verte is (1, 3). Eample 3 Graph the quadratic function f () = + 8 3 then state the verte, ais of smmetr, ma/min, domain and range. f () = + 8 3 a =, b = 8 and c = 3 b b 8 8 Verte is, c =, 3 = (, 5) or a ( ) ( ) a b 8 = =. So, f () = + 8 3 f () = ( ) + 8() 3 = 5, a ( ) Therefore, verte is (, 5). Plot at least points other than verte: -coordinate = f () = + 8 3 f () = ( ) + 8() 3 = 3 3 f (1) = (1) + 8(1) 3 = 3 1 3 f (3) = (3) + 8(3) 3 = 3 3 3 f () = () + 8() 3 = 3 3 f () Verte: (, 5) Ais of Smmetr: = Ma/min: maimum value of 5 Domain: all real numbers Range 5 Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7. General Form of a Quadratic Function 313 Eample Mt. Douglas Secondar Given that f () is a quadratic function with minimum f (1) = 3, find the verte, ais of smmetr, domain and range. f (1) = 3 means the point (1, 3), thus the verte is (1, 3). Ais of Smmetr: = 1 Domain : all real numbers Range: 3 Eample 5 Determine a quadratic function with verte (, 1) and -intercept 3. Method 1 The standard form of a quadratic equation is = a( h) + k so = a( ) +1. The -intercept means where the graph crosses the -ais, at = ; it crosses at (, 3). = a( ) +1 3 = a( ) +1 a = a = 1, thus = ( ) +1 Method The general form of a quadratic equation is = a + b + c. -intercept 3 has 3 = a() + b() + c c = 3, thus = a + b 3 If verte is (, 1), then 1 = a() + b() 3 a + b = ora + b = B smmetr, if (, 3) is a point and verte is (, 1), then (, 3) must be a matching point on the quadratic function. ( units to the right of =, same = 3.) 3 = a() + b() 3 1a + b = orb = a a + b = a a = a = 1, b = a = ( 1) = Thus, = + 3 Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
31 Chapter 7 Quadratic Functions Foundations of Math 11 Mt. Douglas Secondar Eample Determine the quadratic function with -intercepts 1 and 3, and -intercept. Method 1 -intercept is where graph crosses -ais =, thus ( 1, ) and (3, ). -intercept is where graph crosses -ais =, thus (, ). = a + b + c = a() + b() + c c = = a + b + c = a( 1) + b( 1) = = a(3) + b(3) + 3(a b = ) 9a + 3b = a b = b = b = 3a 3b = 18 9a + 3b = 1a = a = Thus, quadratic function is = + +. Method If -intercepts are 1 and 3, then ais of smmetr is = 1+ 3 Thus, = a( 1) + k with -intercept of (, ) = a( 1) + k a + k = -intercept of 1 ( 1, ) so = a( 1) + k = a( 1 1) + k k = a = 1 and verte is (1, k). Then a + k = a a = a = + k = k = 8 Thus, quadratic function is = ( 1) + 8. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7. General Form of a Quadratic Function 315 7. Eercise Set Mt. Douglas Secondar 1. Determine another point that must be on a quadratic function whose graph satisfies the given conditions. a) Verte (, 5); point (, 1) b) Verte (, ), point (3, 5) c) Verte (1, ); point (3, ) d) Verte ( 3, 5); point (, ) e) Verte (a, b); point (, ) f) Verte (, ); point (a, b) Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
31 Chapter 7 Quadratic Functions Foundations of Math 11. Determine the equation of the ais of smmetr of a quadratic function, given the following information. a) Verte ( 1, 3) b) Verte (, 1) Mt. Douglas Secondar c) Points: (, 5) and (, 5) d) Points: ( 3, ) and (5, ) e) Points: (, ) and (, ) f) Points: (, ) and ( 3, ) g) Points: (a, b) and (, b) h) Points: (a, b) and (c, d) Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7. General Form of a Quadratic Function 317 3. Find the verte of the following quadratic functions, and state if the verte is a maimum or minimum point. a) = + b) f () = 8 + 3 Mt. Douglas Secondar c) = 3 +1 d) g () = + 3 e) h() = 1 3 + f) i() = 3 5 + Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
318 Chapter 7 Quadratic Functions Foundations of Math 11. Graph the following quadratic functions. Plot at least points other than the verte. Mt. Douglas Secondar a) = b) = + c) = + + 3 d) = + 8. e) = + 9 f) = 1 + Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7. General Form of a Quadratic Function 319. g) = + + 3 h) = Mt. Douglas Secondar i) = +1 j) = 1 + k) = 3 1 8 l) = 1 3 + Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
3 Chapter 7 Quadratic Functions Foundations of Math 11 5. Determine a quadratic function with the given information. Mt. Douglas Secondar a) Verte (, 1), and goes through origin b) Verte (, 5), and -intercept 3 c) Verte ( 1, ), and -intercept d) Verte (, ), and f ( ) = 1 e) -intercepts 1 and 5, minimum value of 1 f) -intercepts and, and -intercept g) Ais of smmetr is = 1, -intercept, with onl one -intercept h) -intercepts 3 and, and f () = Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section 7.3 Quadratic Equations 333 8. Use the graph to solve 3 =. Mt. Douglas Secondar 9. Use the graph to solve + + 3 = 1. Find the mistake in the working. Then find the correct solution. a) ( )( + 3) = = or + 3 = = 8or = 3 b) = 5 = 5 Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.