3.1 QUADRATIC FUNCTIONS IN VERTEX FORM

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1 3.1 QUADRATIC FUNCTIONS IN VERTEX FORM PC0 T determine the crdinates f the vertex, the dmain and range, the axis f symmetry, the x and y intercepts and the directin f pening f the graph f f(x)=a(x p) + q withut the use f technlgy. EX #1: Using a table f values, sketch y = x REVIEW: A QUADRATIC FUNCTION is a functin f degree tw: y = x, y = x 5x + 1, y = (x 3) 3, y = (x + 1) The graph f a quadratic functin is in the shape f a The f a parabla is the lwest pint in a parabla that pens upwards r the highest pint f a parabla that pens dwnwards. If the parabla pens upwards then there is a value. If the parabla pens dwn there is a value. The crdinate f the vertex defines the max r min value. The is a line thrugh the vertex that divides the graph f the quadratic functin int tw cngruent halves. The f the vertex defines the equatin f the axis f symmetry. A quadratic functin is written in STANDARD FORM when it is written in the frm and it is written in VERTEX FORM when it is in the frm EX #: State the vertex, the max r min value and the equatin f the axis f symmetry fr the fllwing: REVIEW ACTIVITY: 1. Given the mst basic parabla y = x, rewrite in vertex frm, y = a(x p) + q and identify the values f a, p and q. Graph this parabla using yur graphing calculatr. Des it matter if yu enter it in standard frm r vertex frm? Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 1

2 3. Leaving the first parabla n yur screen, graph the fllwing tw parabla s in the next tw graphing spts. EQUATION Y = x + Y 3 = x - 1 Which variable changed frm step 1? What is the new value f this variable? Hw did the graph change frm step 1 4. Delete the equatins yu have in y and y 3 and use the fllwing equatins instead EQUATION Y = (x + ) Y 3 = (x 1) Which variable changed frm step 1? What is the new value f this variable? Hw did the graph change frm step 1 5. Delete the equatins yu have in y and y 3 and use the fllwing equatins instead EQUATION Y = x Y 3 = 5x 1 y4 x Which variable changed frm step 1? What is the new value f this variable? Hw did the graph change frm step 1 y x y 5 1 x 5 REVIEW: In VERTEX FORM y a( x p) q, the fllwing is true: The base r parent functin is y x and is transfrmed/changed when a, p and q are applied The crdinates f the vertex are at (, ) The equatin f the axis f symmetry is at If the value f a is negative, the graph will pen and have a value at If the value f a is psitive, the graph will pen and have a value at The parabla will be f average width if The parabla will be narrwer if The parabla will be wider if The value f p mves the parabla The value f q mves the parabla The dmain f a parabla (with arrwheads) will be in set ntatin and in interval ntatin The range f a parabla where a>0 (with arrwheads) will be in set ntatin and in interval ntatin while the range f a parabla where a <0 (with arrwheads) will be in set ntatin and in interval ntatin. Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page

3 EX #3: Cmplete the fllwing chart and sketch the last three functins using the transfrmatins f a, p and q Value f a y 3x y x 5 y ( x 5) y 3 x 3 y 1 x 3 Value f p Value f q Directin f pening Width Vertex Max/Min Axis f Symmetry Dmain Range Y intercept Reflectin f the y intercept EX #4: Determine the number f x intercepts f each quadratic functin by visualizing the graph. f x 1 ( ) ( x ) 4 VERTEX DIRECTION OF OPENING VISUALIZE & SKETCH THE GRAPH NUMBER OF X INTERCEPTS f ( x) 3( x 5) f x ( ) ( x 3) 5 Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 3

4 EX #5: Sketch the fllwing graphs. Determine the vertex, the directin f pening, the y intercept & its reflectin, the x intercepts, the dmain and range, the axis f symmetry and the max r min value. Create a table f values t help determine the shape f the graph. a) y ( x1) 8 x y b) y ( x3) 8 x y Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 4

5 3.1 FA#1: P157 #1ad, bc, 4ac (make a table), 6, 7, EXTRA QUESTION 1 BELOW 3.1 ULA#1:P161 #19 1. Fr each f the functins n the right, state the: values f a, p, q vertex x-intercepts y-intercept reflectin pint f the y-intercept dmain & range max/min value sketch using a table f values a) y x d) y x y x b) y x e) 1 c) 1 3 y x y x f) 3 1 Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 5

6 3.1 DAY QUADRATIC FUNCTIONS IN VERTEX FORM PC0 T graph quadratic functins in the frm f(x)=a(x p) + q using transfrmatins. EX #1: Sketch the graph f y = 3(x + ) 4 using transfrmatins. STEP 1: Describe what the numerical change t a is cmpared t its parent PARENT FUNCTION y = x. Hw will this change alter the graph f the PARENT FUNCTION y = x? Will this change affect the x r the y value f the rdered pairs f the parent functin? Let s cmpare the table f rdered pairs between the parent functin and the Step 1 Transfrmed table (which is the parent functin with just the value f a changed we wn t wrry abut the values f p and q yet) PARENT FUNCTION y = x x y STEP 1: If the value f a changes, decide where hw and where that adjusts the table STEP 1 f TRANSFORMED FUNCTION y = x Describe what changes & hw: x y STEP : Describe hw p and q have changed cmpared t the parent functin y = x Hw will this change alter the graph f the PARENT FUNCTION y = x? Hw will p and q affect where the parent functin mves t? Describe hw x mves and hw y mves. Fill in the Step table by mving the x and y values by the apprpriate amunts STEP : If the values f p and/r q changes, decide where and hw they adjust the table STEP f TRANSFORMED FUNCTION y = (x ) Describe what changes & hw: x y Final Ordered Pair Sketch the graph using the Step table abve. Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 6

7 EX #: Sketch the graph f f x 1 using transfrmatins. ( ) ( x 1) 3 PARENT FUNCTION y = x x y STEP 1: If the value f a changes, decide where hw and where that adjusts the table STEP 1 f TRANSFORMED FUNCTION y = x Describe what changes & hw: x y STEP : If the values f p and/r q changes, decide where and hw they adjust the table STEP f TRANSFORMED FUNCTION y = (x ) Describe what changes & hw: x y Final Ordered Pair 3.1 FA#: P157 #3 & Extra Questins 1 & Belw 3.1 ULA#:P158 #10, 4 1. Cmplete the fllwing chart Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 7

8 . Graph the fllwing functins using transfrmatins. Make sure t state transfrmatins, the vertex and shw the new tables f values. It is imperative that yu use graph paper and a ruler!! a) y x d) f ( x) 4 x f ( x) 1 x f ( x) 4 x 7 3 b) y 3 x 4 e) c) y x f) 3.1 DAY 3: QUADRATIC FUNCTIONS IN VERTEX FORM PC0 T write quadratic functins in vertex frm given a graph r situatin and t slve situatinal questins. EX #1: Determine the quadratic functin in vertex frm fr the fllwing graphs. a) b) Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 8

9 EX #: Suppse a parablic archway has a width f 80 cm and a height f 16 cm at its highest pint abve the flr. a) Write a quadratic functin in vertex frm that mdels the shape f this archway. b) Determine the height f the archway at a pint that is 50 cm frm its uter edge. c) Is there mre than ne answer fr part a. Why? EX #3: Determine a quadratic functin with the fllwing characteristics: a minimum f 1 at x = -4 and y-intercept f FA#3: P158 #8abc, 9abd, 1 & at least f the fllwing:13, 16, 17, ULA#3:P157 #5, 0, Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 9

10 3. : QUADRATIC FUNCTIONS IN STANDARD FORM PC0 T determine the crdinates f the vertex, the dmain and range, the axis f symmetry, the x and y intercepts and the directin f pening f the graph f a functin in standard frm y = ax + bx + c A quadratic in standard frm is y = ax + bx + c r f(x) = ax + bx + c where a, b and c are real numbers and a 0 a determines the width f the parabla and whether the parabla pens upwards r dwnwards (the same as it did fr vertex frm) b INFLUENES the psitin f the graph (vertex) c determines the y intercept f the graph In Fundatins 0, we fund the x value f the vertex by calculating the x intercepts and finding the middle x value f thse intercepts. We will nw develp a frmula that we can use the x-crdinate f the vertex can be calculated by using the frmula The y crdinate f the vertex can be fund by substituting the abve answer int the riginal functin The frmula t find the vertex is:, EX #1: Using the graph, determine the vertex, directin f pening, axis f symmetry, max/min value, dmain, range, x intercept(s) andy intercept Find the vertex using the frmula Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 10

11 EX #: Withut lking at a graph, determine the same infrmatin as in example 1 fr the fllwing: a) y = x + 6x + 5 b) y = -x + x + 3 REVIEW: T find the y intercept in standard frm, use the value f. T find the x intercepts in standard frm yu must set the value f t, then slve the resulting equatin by r REVIEW: Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 11

12 EX #3: A diver jumps frm a 3-m springbard with an initial vertical velcity f 6.8 m/s. Her height, h, in metres, abve the water t secnds after leaving the diving bard can be mdelled by the functin h(t) = -4.9t + 6.8t + 3. a) Graph the functin by finding the vertex, x intercept(s), y intercept and it s reflectin. b) What des the y-intercept represent? c) What maximum height des the diver reach? When des she reach that height? d) Hw lng des it take befre the diver hits the water? e) What dmain and range are apprpriate in this situatin? f) What is the height f the diver 0.6 s after leaving the bard? Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 1

13 EX #4: At a children s music festival, the rganizers are rping ff a rectangular area fr strller parking. There is 160 m f rpe available t create the perimeter. a) Write a quadratic functin in standard frm t represent the area fr the strller parking. b) What are the crdinates f the vertex? What des the vertex represent in this situatin? c) Sketch the graph fr the functin yu determined in part a). d) Determine the dmain and range fr this situatin. e) Identify any assumptins yu made. 3. FA: P174 #1,, 3, 6, 7a-d, 8, 10, 1, 15, 19*, 3* 3. ULA:P175 #9, 11, 13, 16, 17, 18, 0, 4 Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 13

14 3.3: COMPLETING THE SQUARE PC0 T change the frm f a quadratic functin frm Standard Frm, y = ax² + bx + c, t Vertex Graphing Frm, y = a(x p) + q (nte that it is smetimes called y = a(x h) + k EX #1: Rewrite the fllwing quadratic functin frm vertex frm t standard frm y = (x + ) 5 What frm d yu feel is mre helpful in be able t quickly sketch the graph? Why? EX # : Review! Factr the fllwing perfect trinmial squares: a) x + 6x + 9 b) x + 1x + 36 c) 3. x 4x + 4 d) x + 14x + 49 What is the relatinship between the middle and the last term? e) x 16x + 3 f) 5x + 10x + 5 g) x 8x 16 = h) -3x + 30x 75 EX #3: Hw culd yu use what yu learned abut the relatinship between the middle and last term t test t see if X + 6x 7 is a perfect trinmial square? Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 14

15 EX #3: Rewrite the functin y = x + 6x y t VERTEX frm(frm the frm y = ax + bx + c t the frm y = a(x p) + q) What is the vertex f this functin? STEPS FOR MS. CARIGNAN S METHOD OF WRITING IN VERTEX FORM WHEN a = 1 1. Mve the value f c t the left side f the functin. b. When the value f a = 1, calculate the value f and add this value t bth sides f the functin. 3. Simplify the left side f the functin and factr the right side f the functin. Nte that the functin will nw always factr int a perfect square f the frm x 4. Mve the cnstant n the left side t the right side. EX #4: Rewrite the fllwing in vertex frm. a) y = x + 8x 5 b) y = x + 9x - 1 D yu think this parabla wuld have a maximum r a minimum? What is the max/min f this parabla? D yu think this parabla wuld have a maximum r a minimum? What is the max/min f this parabla? Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 15

16 EX #4: Rewrite the fllwing in Vertex frm: a) y = x 16x + 11 b) y x x STEPS FOR MS. CARIGNAN S METHOD OF WRITING IN VERTEX FORM WHEN a 1 1. Mve the value f c t the left side f the functin.. Factr ut the value f a n the right side (even if it desn t factr ut evenly yu must factr it ut! T factr it ut f b when it isn t divisible by a yu will turn the term int b x ) Leave a shrt space a inside the brackets n the right end. b 3. Calculate the value f and add this value t THE RIGHT SIDE side f the functin nly (add this value where yu left the space inside the brackets) b 4. Nw calculate the value f a and add this value t the LEFT side the functin. 5. Simplify the left side f the functin and factr the right side f the functin. Nte that the functin will nw always factr int a perfect square f the frm a x 6. Mve the cnstant n the left side t the right side. c) 1 9 d) y = -5x 8x 3 y x x Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 16

17 EX #5: Cmplete the fllwing table: Vertex y -3x 18x y x x 9 3 y x 0x y = x + 5x - 3 Axis f Symmetry Directin f Opening Max/Min Dmain Range Y intercept Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 17

18 3.3 #1 FA: The fllwing Questins (d them first) fllwed by P193 #5d, 7e 1. Rewrite the fllwing functins in Vertex Frm by Cmpleting the Square. a) y x 4x 5 b) y x 6x 16 c) d) g) j) y x 10x e) y x x 4 1 h) 1 4 k) 3 y x x y x 9x f) y x x i) 1 8 y x x y x x 8 18 y x x 3 10 y x x 5 3. Find each f the fllwing answers fr each questin in #1 abve: a) Vertex b) Axis f Symmetry c) Directin f Opening d) Max/Min e) Y intercept d) Dmain e) Range 3.3 # ULA: P193 8bc, 9, 10, 1, 16 Answers t #1: 1a) y x 9 b) y x 3 5 c) y x 4 d) 9 81 e) y x i) y x f) y x j) y x k) y x 1 y x 5 5 g) y x 16 h) y x Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 18

19 3.3: MAX/MIN WORD PROBLEMS PC0 T slve situatinal questins invlving maximums and minimums f quadratic functins. EX #1: At a children's music festival, the rganizers are rping ff a rectangular area fr strller parking. There is 160 m f rpe available t create the perimeter. a) Write a quadratic functin in standard frm t represent the area fr the strller parking. b) What is the maximum area that can be rped ff? What are the dimensins f the rectangular with the maximum area? What tw methds can yu use t find the maximum? Use bth methds! Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 19

20 EX #: Tw numbers have a sum f 9 and a prduct that is a maximum. Determine the tw numbers and the maximum prduct. Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 0

21 3.3 # FA: The fllwing Questins (d first) fllwed by P 195 #18 1. Slve the fllwing prblems algebraically using the vertex frmula and cmpleting the square. Then, check yur answer n the graphing calculatr. a) What is the maximum prduct that tw numbers can have if their sum is 100? What are the tw numbers? b) One number is 10 larger than anther. What is the smallest pssible value fr the sum f their squares? What are the tw numbers?. What is the maximum rectangular area that can be enclsed with 100 m f fencing material? What are the dimensins f the rectangle? 3. A farmer wishes t fence in a rectangular pen using his barn as ne f the sides f the rectangle. If the farmer has 40 m f fencing, what is the largest area that can be enclsed? What are the dimensins f the rectangle? 4. The managers f a business are examining csts. It is mre cst-effective fr them t prduce mre items. Hwever, if t many items are prduced, their csts will rise because f factrs such as strage and verstck. Suppse that they mdel the cst, C, f prducing n thusand items with the functin: C( n) 75n 1800n Determine the number f items that will minimize their csts. 5. A gymnast is jumping n a trampline. His height, h, in metres, abve the flr n each jump is rughly apprximated by the functin h( t) 5t 10t 4, where t represents the time, in secnds, since he left the trampline. Determine his maximum height n each jump. 6. Sandra is practicing at an archery club. The height, h, in feet, f the arrw n ne f her shts can be mdelled as a functin f time, t in secnds, since it was fired using the functin h( t) 16t 10t 4. What is the maximum height f the arrw, in feet, and when des it reach that height? Answers: 1. a) Max prduct is 500 when the numbers are 50 and 50 b) Min sum is 50 when the numbers are -5 and 5. Max area is 65 m when each side f the rectangle is 5 m. 3. Max area is 00 m when the rectangle is 10m by 0m items 5. 9m 6. Max height is 5.56ft after 0.31 secnds being sht 3.3 ULA: P195 #19, 0,, 4, 5, 8, 31 Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page 1

22 VIDEO LINKS THAT MAY AIDE IN UNDERSTANDING Sectin 3.1 Review f SET Ntatin Vs Interval Ntatin: (Nte: On the Set Ntatin part f this vide, we typically als add x R t the end just befre the curly bracket ie we wuld prefer the first answer t lk like x x 3, x R ) Sectin 3. Review f Windw/Bx Methd fr Factring: Sectin Pre AP PreCalculus 0(Ms. Carignan) P0.7: Chapter 3 Quadratic Functins Page