Presentation on theme: "QUADRATIC FUNCTIONS AND INEQUALITIES"— Presentation transcript:
1 QUADRATIC FUNCTIONS AND INEQUALITIES Integrated Programme/mainstreamSecondary Three MathematicsName : _____________( ) Class:______Date : ___________ to _____________Term 1: Unit 4 NotesQUADRATIC FUNCTIONS AND INEQUALITIESFactoringCompleting SquareGeneral Quadratic FormulaQuadratic GraphsQuadratic InequalitiesDiscriminant and Nature of rootsAt the end of the unit, students should be able tosolve quadratic equations(1) by factorization (recall Sec 2 work)(2) by completing the square,(3) by formula .understand relationships between the roots and coefficients of the quadratic equationform quadratic equations in the product form given two roots and ,apply substitution to solve some higher order algebraic equations,understand and use discriminant to determine the nature of roots of a quadratic equation,use discriminant to determine when is always positive (or always negative)solve intersection problems between line and curve and discuss the nature of roots.find the maximum or minimum value by using completing the square,sketching of graphs of quadratic functions given in the form(1) y = a(x-h)2 + c ,a > 0 or a < 0(2) y = a(x – b)(x – c) , a > 0 or a < 0 .solve quadratic inequalities using algebraic and graphical methods, representing the solution set on the number line.
2 Contains a x2 term Solving Quadratic Equations by Factorising Sec 2 RevisionSolving Quadratic Equations by FactorisingA quadratic equation is an equation like:y = x2y = x2 + 2y = x2 + x – 4y = x2 + 2x – 3Contains a x2 termThere are several methods of solving these but one methods that you must know is called FACTORISING
3 3 x 2 = x 2 = x 0 = 0A x B = 0 What can you say about A or B(x + 3)(x – 2) = 0 means (x + 3) x (x – 2)What can you say about (x + 3) or (x – 2)x + 3 = 0x = -3x - 2 = 0x = 2or
4 (x + 3)(x + 2) x(x + 2) + 3(x + 2) x X (x + 2) + 3 X (x + 2) You try(x + 5)(x + 2)(x – 2)(x + 3)(x + 2)(x – 4)(x – 3)(x – 2)x(x + 2) + 3(x + 2) x X (x + 2) + 3 X (x + 2) x X x + x X X x + 3 X 2 x2 + 2x + 3x + 6 x2 + 5x + 6
5 (x )(x ) What goes with the x? Solve by factorising: 0 = x2 + 7x + 12 Write down all the factor pairs of 12.(x )(x )What goes with the x?From this list, choose the pair that adds up to 73 + 4 = 70 = (x + 3)(x + 4)x = – 3 and – 4Put these numbers into brackets
6 Solve by factorising: 0 = x2 + x - 6 Write down all the factor pairs of – 6From this list, choose the pair that adds up to 1(3) + (-2) = 13 – 2 = 10 = (x + 3)(x - 2)x = – 3 and 2Put these numbers into brackets
7 Copy and fill in the missing values when you factorise x2 + 8x + 12 = 0 Find all the factor pairs of x 12 = 122 x _ = 123 x 4 = 12From these choose the pair that add up to 8_ + 6 = 8Put these values into the brackets (x + _)(x + _) = 0x = -2 and - 6Solve by factorising1.x2 + 3x + 2 = 02.x2 + x – 12 = 03.x2 – 12x – 20 = 04.x2 – x – 12 = 0
11 This is often called the difference between two squares -1 x 4 = -4-2 x 2 = -44 x -1 = -4= 0x2 – 4x2 + 0x – 4(x – 2)(x + 2)Notice that x2 – 4 could be written asx2 – 22(x – 2)(x + 2)This is often called the difference between two squaresx2 – 25(x + 5)(x – 5)
14 Completing SquareFor quadratic equations that are not expressed as an equation between two squares, we can always express them as If this equation can be factored, then it can generally be solved easily.
15 If the equation can be put in the form then we can use the square root method described previously to solve it.“Can we change the equation from the form to the form ?”
16 The procedure for changing is as follows The procedure for changing is as follows. First, divide by , this gives Then subtract from both sides. This gives
23 The Quadratic formula allows you to find the roots of a quadratic equation (if they exist) even if the quadratic equation does not factorise.The formula states that for a quadratic equation of the form :ax2 + bx + c = 0The roots of the quadratic equation are given by :
24 Example :Use the quadratic formula to solve the equation :x 2 + 5x + 6= 0Solution:x 2 + 5x + 6= 0a = 1 b = 5 c = 6x = - 2 or x = - 3These are the roots of the equation.
25 Example :Use the quadratic formula to solve the equation :8x 2 + 2x - 3= 0Solution :8x 2 + 2x - 3= 0a = 8 b = 2 c = -3x = ½ or x = - ¾These are the roots of the equation.
26 Example :Use the quadratic formula to solve for x to 2 d.p :2x 2 +3x - 7= 0Solution:2x 2 + 3x – 7 = 0a = 2 b = 3 c = - 7x = or x =These are the roots of the equation.
34 b2 - 4ac > 0 : not a perfect square – real, irrational, unequal. b2 - 4ac < 0 : imaginary, complex, no solution.
35 Quadratic Graphs The graph of is a parabola. The graph looks like if a > if a < 0
36 Key features of the graph: The maximum or minimum point on the graph is called the vertex. The x-coordinate of the vertex is:
37 The y-intercept; the y-coordinate of the point where the graph intersects the y-axis. The y-intercept is:When x = 0, y = cThe x-intercepts; the x-coordinates of the points, if any, where the graph intersects the x-axis. To find the x-intercepts, solve the quadratic equation
38 Example:Sketch the graph ofvertex: min. pointy-intercept:x-intercepts:
44 Quadratic Inequalities What do they look like?Here are some examples:
45 Quadratic Inequalities When solving inequalities we are trying to find all possiblevalues of the variable which will make the inequality true.Consider the inequalityWe are trying to find all the values of x for which thequadratic is greater than zero or positive.
46 Solving a quadratic inequality We can find the values where the quadratic equals zeroby solving the equation,
47 Solving a quadratic inequality You may recall the graph of a quadratic function is a parabolaand the values we just found are the zeros or x-intercepts.The graph of is(-2,0)(3,0)
48 Solving a quadratic inequality From the graph we can see that in the intervals around thezeros, the graph is either above the x-axis (positive) or belowthe x-axis (negative). So we can see from the graph theinterval or intervals where the inequality is positive.But how can we find this out without graphing the quadratic?We can simply test the intervals around the zeros in thequadratic inequality and determine which make the inequalitytrue.
49 Solving a quadratic inequality For the quadratic inequality, we foundx = 3 and x = –2 by solving the equation .Put these values on a number line and we can see threeintervals that we will test in the inequality. We will test onevalue from each interval.-23
50 Solving a quadratic inequality IntervalTest PointEvaluate in the inequalityTrue/False
51 Solving a quadratic inequality Thus the intervals make up the solutionset for the quadratic inequality,In summary, one way to solve quadratic inequalities is to findthe x-intercept/s and test a value from each of the intervalssurrounding the zeros to determine which intervals make theinequality true.
52 Example : SolveStep 2:Sketch the quadratic graphStep 1: Solve
53 Quadratic with linear Solve: x2 – 8x + 16 > 2x +7 Estimate ? y = 2x + 10y = x2 – 8x +16
54 Example: Solve: x2 – 8x + 16 > 2x +7 Algebraically: 1. Rearrange first2. Solve like the othersx2 – 8x + 16 > 2x +7x2 – 10x + 16 > 7(-2x)(-7)x2 – 10x + 9 > 0Like the ones we did(x-9)(x-1) > 0x>9 or x<1
55 Try this one Solve: x2 + x + 4 > 4x +14 Algebraically: First: try a sketchAlgebraically:1. Rearrange first2. Solve like the othersx2 + x + 4 > 4x +14x2 – 3x + 4 > 14(-4x)(-14)x2 – 3x - 10 > 0(x+2)(x-5) > 0x<-2 or x>5
56 Summary In general, when solving quadratic inequalities Find the zeros by solving the equation you get when you replace the inequality symbol with an equals.Find the intervals around the zeros using a number line and test a value from each interval in the number line.The solution is the interval or intervals which make the inequality true.