Quadratic Equations 02/11/12 lntaylor ©

Quadratic Equations Table of Contents Learning Objectives Finding a, b and c Finding the vertex and min/max values Finding the discriminant Finding the roots Factoring Completing the Square Quadratic Formula Quadratics with Function Tables Graphing Quadratics 02/11/12 lntaylor ©

Learning Objectives TOC 02/11/12 lntaylor ©

Learning Objectives LO 1 LO 2 Understand what a Quadratic Equation represents Perform basic operations with Quadratic Equations LO 3Build a Quadratic equation using various techniques TOC 02/11/12 lntaylor ©

Definitions Definition 1 Quadratic Equations are in the form ax² + bx + c = 0 TOC Definition 2 “a” determines the direction and Magnitude (width) of the curve Definition 3 “-b/2a” determines whether the min/max has ± x value : note the OPPOSITE of b! Definition 4“c” determines whether the y intercept has a ± y value Definition 5The Discriminant determines the number of roots (b² - 4ac) Definition 6The Roots (x intercepts) are determined by factoring the equation 02/11/12 lntaylor ©

Previous knowledge PK 1Basic Operations and Properties TOC 02/11/12 lntaylor ©

Rule 1 Rule 2 Know how to find a, b, -b, and c Know how to find the vertex (-b/2a) and the discriminant (b² - 4ac) Rule 3 Know how to find the roots by factoring, completing the square and the formula Basic Rules of Quadratics TOC Rule 4 Always check your work 02/11/12 lntaylor ©

Quadratic Equations Quadratic Equations f(x) = ax² + bx + c –You are given certain information in a function f(x) Width of the curve is determined by (a) Symmetry is determined by -b/2a (note the opposite of b) Y intercept of the curve is determined by (c) Remember all function tables are the same regardless of the equation Go up or down the same amount and look for a pattern TOC 02/11/12 lntaylor ©

Finding a, b, -b and c TOC 02/11/12 lntaylor ©

Q1 TOC 02/11/12 lntaylor ©

Quadratics Finding a, b and c In the quadratic equation below find a, b, -b and c f(x) = 3x² + 5x + 2 a = b = -b = c = TOC 02/11/12 lntaylor ©

Q1 Answer TOC 02/11/12 lntaylor ©

Quadratics Finding a, b and c In the quadratic equation below find a, b and c f(x) = 3x² + 5x + 2 a = b = -b = c = TOC /11/12 lntaylor ©

MathTV.com TOC 02/11/12 lntaylor ©

Now you try f(x) = -2x² - 4x - 8 TOC 02/11/12 lntaylor ©

Quadratics Finding a, b and c In the quadratic equation below find a, b and c f(x) = -2x² -4x -8 a = b = -b = c = TOC 4 02/11/12 lntaylor ©

Now you try f(x) = -¾ x² - ¼x + ½ TOC 02/11/12 lntaylor ©

Quadratics Finding a, b and c In the quadratic equation below find a, b and c f(x) = -¾ x² - ¼x + ½ a = b = -b = c = -¾ + ½- ¼ ¼ 02/11/12 lntaylor © TOC

Changes in a TOC 02/11/12 lntaylor ©

0,0 f(x) = 4x² f(x) = - 4x² Step 1 – Compare Graphs Graph 4x ² Graph - 4x ² Step 2 - Compare a values a = 4 a = - 4 Step 3 – Conclusion When a is +, curve goes up When a is -, curve goes down TOC 02/11/12 lntaylor ©

0,0 f(x) = 4x² f(x) = - 1/4x² Step 1 – Compare Graphs Graph 4x ² Graph - 1/4x ² Step 2 - Compare a values a = 4 a = - 4 Step 3 – Conclusion When a is large, curve narrows When a is small, curve widens TOC 02/11/12 lntaylor ©

Changes in b TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x f(x) = x² - 4x Step 1 – Compare Graphs Graph x² + 2x Graph x ² - 4x Step 2 - Compare b values b =+2 b= - 4 Step 3 – Question If b is +, why is the curve on the negative side of x axis? If b is -, why is the curve on the positive side of x axis? TOC 02/11/12 lntaylor ©

0,0 f(x) = x² - 2x f(x) = x² + 4x Step 1 – Compare Graphs Previous Graph x² + 2x Previous Graph x ² - 4x Step 2 - Compare b values b = -2 b = + 4 Step 3 – Conclusion If b is +, why is the curve on the negative side of x axis? If b is -, why is the curve on the positive side of x axis? Hint: use –b/2a TOC 02/11/12 lntaylor ©

Changes in c TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x f(x) = x² - 2x - 1 Step 1 – Compare Graphs Previous Graph x² + 2x New Graph x ² - 2x - 1 Step 2 - Compare c values c = 0 c = - 1 Step 3 – Conclusion C represents where the curve crosses the y axis + c means positive y intercept No c means the origin - c means negative y intercept TOC 02/11/12 lntaylor ©

Q2 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x - 3 Step 1 – Is this function: Up or down? Wide, Normal or Narrow? Left or right of 0,0? Positive or negative y intercept? Step 2 - Answers: Up Normal Left of 0,0 Y intercept - 3 TOC 02/11/12 lntaylor ©

Q 3 TOC 02/11/12 lntaylor ©

0,0 f(x) = ? Step 1 – What are the values: a is/is not a fraction a 0 b 0 C 0 Step 2 - Answers: a is a fraction a<0 b = 0 c >0 The equation is: f(x) = - 1/4x² + 5 TOC 02/11/12 lntaylor ©

Finding the Vertex TOC 02/11/12 lntaylor ©

Q4 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x - 3 What is the vertex of the equation? TOC 02/11/12 lntaylor ©

Q4 Answer TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x - 3 What is the vertex of the equation? TOC -1, -4 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x - 3 Step 1 – The vertex The vertex is a coordinate value Minimums go with + a values Maximums go with – a values a in this equation is + Step 2 – The formula Locate -b in the equation Use the formula (-b/2a) This is the x value (-1 ) Plug x into equation; find y f(x) = x² + 2x – 3 f(x) = (-1)² + 2(-1) -3 = -4 This is the y value TOC min - 2 2(1) -1, -4 02/11/12 lntaylor ©

Now you try x² + 6x + 8 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 6x + 8 Step 1 – The vertex Minimums go with + a values a in this equation is + Step 2 – The formula Locate b in the equation Reverse the sign (-b/2a) This is the x value (-3 ) Plug x into equation; find y f(x) = x² + 6x + 8 f(x) = (-3)² + 6(-3) + 8 = -1 This is the y value TOC min (1) -3, 02/11/12 lntaylor ©

Now you try - 2x² - 8x - 4 TOC 02/11/12 lntaylor ©

0,0 f(x) = - 2x² - 8x - 4 Step 1 – The vertex Maximums go with - a values a in this equation is - Step 2 – The formula Locate -b in the equation Use the formula (-b/2a) This is the x value (-2 ) Plug x into equation; find y f(x) = - 2x² - 8x - 4 f(x) = -2(-2)² - 8(-2) - 4 = 4 This is the y value TOC max (-2) -2, 4 02/11/12 lntaylor ©

Last one x² - 4 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² - 4 Step 1 – The vertex Minimums go with + a values a in this equation is + Step 2 – The formula Locate b in the equation Use the formula (-b/2a) This is the x value (0 ) Plug x into equation; find y f(x) = x² - 4 f(x) = (0)² - 4 = - 4 This is the y value No middle term means? The vertex is the y intercept! TOC min (1) 0, /11/12 lntaylor ©

Q5 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x - 3 What is the vertex of the equation? Is it a min or max point? Answers: Vertex = (-1, -4) Minimum TOC min 02/11/12 lntaylor ©

Q6 TOC 02/11/12 lntaylor ©

0,0 f(x) = - 2x² - 8x - 4 TOC max What is the vertex of the equation? Is it a min or max point? Answers: Vertex = (-2, 4) Maximum 02/11/12 lntaylor ©

Finding the Discriminant TOC 02/11/12 lntaylor ©

Q7 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x - 3 What is the discriminant? How many roots are there? TOC 02/11/12 lntaylor ©

Q7 Answer TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x - 3 TOC What is the discriminant? How many roots are there? Answers: discriminant is + There are two roots -3, 01, 0 02/11/12 lntaylor ©

0,0 f(x) = - 2x² - 8x - 4 Step 1 – The discriminant Discriminant give # of roots + means 2 roots 0 means 1 root - means no roots Step 2 – The formula b² - 4ac Locate a, b and c a = -2 b = -8 -b = 8 C = -4 Substitute into the formula (-8)² - 4(-2)(- 4) 64 – 32 = + means two roots Roots are x intercepts Watch your signs!!!! TOC 02/11/12 lntaylor ©

Now you try x² - 4 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² - 4 TOC Step 1 – The discriminant Discriminant give # of roots + means 2 roots 0 means 1 root - means no roots Step 2 – The formula b² - 4ac Locate a, b and c a = 1 b = 0 -b = 0 C = -4 Substitute into the formula (0)² - 4(1)(- 4) = + means two roots Roots are x intercepts Watch your signs!!!! 02/11/12 lntaylor ©

Now you try x² + 4x + 4 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 4x + 4 TOC Step 1 – The discriminant Discriminant give # of roots + means 2 roots 0 means 1 root - means no roots Step 2 – The formula b² - 4ac Locate a, b and c a = 1 b = 4 -b = -4 C = 4 Substitute into the formula (4)² - 4(1)(4) = 0 means one root One root touches the x axis Did you recognize the equation? This is a perfect square (x + 2) ² 02/11/12 lntaylor ©

Last one - x² + x - 1 TOC 02/11/12 lntaylor ©

0,0 f(x) = -x² + x - 1 TOC Step 1 – The discriminant Discriminant give # of roots + means 2 roots 0 means 1 root - means no roots Step 2 – The formula b² - 4ac Locate a, b and c a = - 1 b = 1 -b = -1 C = -1 Substitute into the formula (1)² - 4(-1)(-1) = - means no roots No roots means no x intercepts Did you watch your signs? 02/11/12 lntaylor ©

Q8 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² - 6x + 9 What is the discriminant? How many roots are there? Answers: discriminant is 0 One root TOC 3, 0 02/11/12 lntaylor ©

Q9 TOC 02/11/12 lntaylor ©

0,0 f(x) = - 2x² - 8x - 4 TOC What is the discriminant? How many roots? Answers: Discriminant is + Two roots 02/11/12 lntaylor ©

Finding the Roots TOC 02/11/12 lntaylor ©

Roots of a Quadratic Equation Definition Factor Roots are x intercepts Factoring is the quickest way to finding the roots CTSCompleting the Square is easy – once you get the hang of it TOC QFQuadratic Formula is the most versatile way of finding the roots 02/11/12 lntaylor ©

Factoring Quadratics TOC 02/11/12 lntaylor ©

x 2 + 6x + 8 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by = 3 Step 3 Square the answer and check last term (3)(3) = 9 8 Step 4 If they are both the same you have your factors No match Step 5 If they are not the same subtract one and add one (2)(4) = 8 Step 6 Continue until numbers match Yes Step 7 Add an x to each parenthesis (x + 2) (x + 4) Step 8 Set each () = 0 and solve x + 2 = 0 and x + 4 = 0 x = - 2 and x = - 4 TOC 02/11/12 lntaylor ©

Now you try x 2 – 20x + 96 TOC 02/11/12 lntaylor ©

x 2 – 20x + 96 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 – 20 2 = – 10 Step 3 Square the answer and check last term (-10)(-10) = Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one (-11)(-9) = 99 Step 6 Continue until numbers match Yes (-12)(-8) = 96 Step 7 Add an x to each parenthesis (x – 12)(x – 8) Step 8 Set each () = 0 and solve x - 12 = 0 and x - 8 = 0 x = 12 and x = 8 TOC 02/11/12 lntaylor ©

Now you try x x + 56 TOC 02/11/12 lntaylor ©

x x + 56 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by = 7.5 Step 3 Since the answer includes 0.5 round up and down (8)(7) = Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one Step 6 Continue until numbers match Yes Step 7 Add an x to each parenthesis (x + 8)(x + 7) Step 8 Set each () = 0 and solve x + 8 = 0 and x + 7 = 0 x = - 8 and x = - 7 TOC 02/11/12 lntaylor ©

Now you try x 2 – 19x + 84 TOC 02/11/12 lntaylor ©

x 2 – 19x + 84 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 – 19 2 = – 9.5 Step 3 Since the answer includes 0.5 round up and down (-10)(-9) = Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one (-11)(-8) = 88 Step 6 Continue until numbers match Yes (-12)(-7) = 84 Step 7 Add an x to each parenthesis (x – 12)(x – 7) Step 8 Set each () = 0 and solve x - 12 = 0 and x - 7 = 0 x = 12 and x = 7 TOC 02/11/12 lntaylor ©

Factoring Negative Constants TOC 02/11/12 lntaylor ©

x 2 – 2x – 48 Step 1 Check to see if last term is negative – Step 2 Construct a table for the “difference” of factors for the constant Step 3 Match the middle term coefficient to the last column 2 Step 4 Use the numbers in the 1 st and 2 nd columns; add a variable to each (x 6) (x 8) Step 5 Put the middle term sign next to the largest number – Step 6 Put the opposite sign next to the smallest number!!!! + Step 7 Set each () = 0 and solve (x + 6) = 0 and (x – 8) = 0 x = -6 and x = 8 TOC 02/11/12 lntaylor ©

Now you try x 2 – 11x – 12 02/11/12 lntaylor © TOC

x 2 – 11x – 12 Step 1 Check to see if last term is negative – Step 2 Construct a table for the “difference” of factors for the constant Step 3 Match the middle term coefficient to the last column 11 Step 4 Use the numbers in the 1 st and 2 nd columns; add a variable to each (x 1) (x 12) Step 5 Put the middle term sign next to the largest number – Step 6 Put the opposite sign next to the smallest number!!!! + Step 7 Set each () = 0 and solve (x + 1) = 0 and (x - 12) = 0 x = - 1 and x = 12 TOC 02/11/12 lntaylor ©

Special cases x 2 – 49 TOC 02/11/12 lntaylor ©

x 2 – 49 Step 1 There are only two terms and the last term is negative Step 2 Square root the last term 49 ½ =± 7 Step 3 Write one factor with + and one factor with – (x + 7) (x – 7) Step 4 Set each () = 0 and solve (x + 7) = 0 and (x – 7) = 0 x = - 7 and x = 7 TOC 02/11/12 lntaylor ©

Now you try x 2 – 225 TOC 02/11/12 lntaylor ©

x 2 – 225 Step 1 There are only two terms and the last term is negative Step 2 Square root the last term 225 ½ =± 15 Step 3 Write one factor with + and one factor with – (x + 15) (x – 15) Step 4 Set each () = 0 and solve (x + 15) = 0 and (x – 15) = 0 x = - 15 and x = 15 TOC 02/11/12 lntaylor ©

Completing the Square TOC 02/11/12 lntaylor ©

Completing the Square Step 1 Move the constant Step 2 Take b/2; square the answer and add to both sides Step 3 Factor out the perfect square; square root and solve 02/11/12 lntaylor © TOC

x² + 6x Step 1 Set the equation = 0 Step 2Move c to the opposite side Step 3 Divide the middle term coefficient by = + 3 x² + 6x + 5 (x+3)² = 4 x + 3 = 2 and x + 3 = -2 x = -1 and x = /11/12 lntaylor © = Step 4 Square the number and add to both sides ² = = Step 5 Factor the perfect square Step 6 Square root and solve + 9 TOC

Now you try! f(x) = x² + 2x - 15 TOC 02/11/12 lntaylor ©

x² + 2x Step 1 Set the equation = 0 Step 2Move c to the opposite side Step 3 Divide the middle term coefficient by = + 1 x² + 2x - 15 (x+1)² = 16 x + 1 = 4 and x + 1 = - 4 x = 3 and x = 3 02/11/12 lntaylor © = 15 0 Step 4 Square the number and add to both sides ² = = Step 5 Factor the perfect square Step 6 Square root and solve + 1 TOC

Last one! f(x) = x² -10x + 18 TOC 02/11/12 lntaylor ©

x² - 10x Step 1 Set the equation = 0 Step 2Move c to the opposite side Step 3 Divide the middle term coefficient by = - 5 x² - 10x + 18 (x - 5)² = 7 x - 5 = + √ 7 and x - 5 = - √ 7 x = 5 + √ 7 and x = 5 - √7 02/11/12 lntaylor © = Step 4 Square the number and add to both sides ² = = Step 5 Factor the perfect square Step 6 Square root and solve + 25 TOC

Quadratics The Quadratic Formula _______ -b ± √b² - 4ac 2a TOC 02/11/12 lntaylor ©

Quadratic Formula Step 1 Train yourself to write down a, b, -b, and c Step 2 Set up the equation slowly; make sure signs are correct Step 3 Set up and Solve Slowly!!!! Make sure the signs are correct!!!! 02/11/12 lntaylor © TOC

x² + 2x - 15 Step 1 Write down a, b, -b and c Step 2 Write down the formula Step 3 Set up the equation to solve Watch your signs!!!! a = 1 b = 2 -b = -2 c = -15 ______ -b ± √b² - 4ac 2a __________ -2 ± √2² - 4(1)(-15) 2(1) Step 4 Solve SLOWLY Watch your signs!!!! (wait for the program) _____ -2 ± √ __ -2 ± √ ± x = 3 x = - 5 The curve crosses the x axis at 3,0 and -5,0 02/11/12 lntaylor © TOC

Now you try f(x) = x² + 6x + 9 TOC 02/11/12 lntaylor ©

x² + 6x + 9 Step 1 Write down a, b, -b and c Step 2 Write down the formula Step 3 Set up the equation to solve Watch your signs!!!! a = 1 b = 6 -b = -6 c = 9 ______ -b ± √b² - 4ac 2a __________ -6 ± √6² - 4(1)(9) 2(1) Step 4 Solve SLOWLY Watch your signs!!!! (wait for the program) _____ -6 ± √ __ -6± √ ± x = - 3 The curve touches the x axis at -3,0 02/11/12 lntaylor © TOC

Last one! f(x) = -2x² + 3x - 15 TOC 02/11/12 lntaylor ©

-2x² - 3x - 15 Step 1 Write down a, b, -b and c Step 2 Write down the formula Step 3 Set up the equation to solve Watch your signs!!!! a = -2 b = -3 -b = 3 c = -15 ______ -b ± √b² - 4ac 2a ____________ 3 ± √(-3)² - 4(-2)(-15) 2(-2) Step 4 Solve SLOWLY Watch your signs!!!! (wait for the program) _____ 3 ± √ ____ -2 ± √ Cannot square root a negative! The curve does not cross the x axis 02/11/12 lntaylor © TOC

Build a Function Table TOC 02/11/12 lntaylor ©

f(x) = x² - 5x - 6 Step 1 – Construct Table Build 3 column Table Build Headings Step 2 – Choosing x values Start with x = 0 Plug 0 into equation Solve for y Build x column Build middle column Build y column Check your work! Note that every x has one y x and y together are called ordered pairs (coordinates and a single point on a graph) TOC x² - 5x - 6 xf(x) or y 00² -5(0) (-3)² -5(-3) - 6 (-2)² -5(-2) - 6 (-1)² -5(-1) - 6 (0)² -5(0) - 6 (1)² -5(1) - 6 (2)² -5(2) - 6 (3)² -5(3) /11/12 lntaylor ©

Now you try! f(x) = x² + 2x + 1 TOC 02/11/12 lntaylor ©

f(x) = x² + 2x + 1 Step 1 – Construct Table Build 3 column Table Build Headings Step 2 – Choosing x values Start with x = 0 Plug 0 into equation Solve for y Build x column Build middle column Build y column Check your work! Note that every x has one y x and y together are called ordered pairs (coordinates and a single point on a graph) TOC x² + 2x + 1 xf(x) or y 00² + 2(0) (-3)² + 2(-3) + 1 (-2)² + 2(-2) + 1 (-1)² + 2(-1) + 1 (0)² + 2(0) + 1 (1)² + 2(1) + 1 (2)² + 2(2) + 1 (3)² + 2(3) /11/12 lntaylor ©

Now you try! f(x) = x² - 4 TOC 02/11/12 lntaylor ©

f(x) = x² - 4 Step 1 – Construct Table Build 3 column Table Build Headings Step 2 – Choosing x values Start with x = 0 Plug 0 into equation Solve for y Build x column Build middle column Build y column Check your work! Note that every x has one y x and y together are called ordered pairs (coordinates and a single point on a graph) TOC x² - 4 xf(x) or y 00² (-3)² - 4 (-2)² - 4 (-1)² - 4 (0)² - 4 (1)² - 4 (2)² - 4 (3)² /11/12 lntaylor ©

Graphing a Quadratic TOC 02/11/12 lntaylor ©

0,0 f(x) = x² + 2x - 3 Step 1 – Graph Write down: a = 1 b = 2 -b = -2 c = -3 Step 2 – Key measures: -b/2a = -2/2(1) = -1 Plug in x to find y = -4 Min = (-1, -4) b² - 4ac = + two roots Factor (x+3) (x-1) Solve x = -3 and x = 1 Y intercept - 3 TOC 02/11/12 lntaylor ©

Now you try! f(x) = x² - 4 TOC 02/11/12 lntaylor ©

0,0 f(x) = x² - 4 Step 1 – Graph Write down: a = 1 b = 0 -b = 0 c = -4 Step 2 – Key measures: -b/2a = 0/2(1) = 0 Plug in x to find y = -4 Min = (0, -4) b² - 4ac = + two roots Factor (x+2) (x-2) Solve x = -2 and x = 2 Y intercept - 4 TOC 02/11/12 lntaylor ©

Now you try! f(x) = - x² + x - 1 TOC 02/11/12 lntaylor ©

0,0 f(x) = -x² + x - 1 TOC Step 1 – Graph Write down: a = - 1 b = 1 -b = - 1 c = - 1 Step 2 – Key measures: -b/2a = - 1/2(- 1) = 1/2 Plug in x to find y = - 3/4 Min = (1/2, - 3/4) b² - 4ac = - no roots Pick another x value and solve x = 2; y = -3 (2, -3) Y intercept /11/12 lntaylor ©

Post Test TOC 02/11/12 lntaylor ©