1 College Algebra Acosta/Karwowski

2 Quadratic Functions and circlesUnit 3 Quadratic Functions and circles

3 Solving quadratic equationsChapter 4 – section 1

4 A quadratic is any equation with a 2nd degree term and no higher degree termsIn general form ax2 + bx + c = 0 f(x) = ax2 + bx + c is the general form of a quadratic function

5 Finding f(x)=a : solvingIn general the inverse of a power is its root you can cancel square powers with square roots if you account for restrictions on the root function 2 𝑥 2 ≠𝑥 for every x 2 𝑥 2 =|𝑥| thus: if 𝑥 2 =25 𝑖𝑡 𝑓𝑜𝑙𝑙𝑜𝑤𝑠 𝑡ℎ𝑎𝑡 𝑥=±5

6 Reminders : working with square rootsIf the square root is irrational – leave the radical form of the answer unless ASKED to round the answer - This is called the exact form of the number ex: ans: 4 ans: 21 Simplify all radicals 𝑎𝑛𝑠: 21 𝑎𝑛𝑠: 4 5 Square root of a negative is an IMAGINARY number −4 =2𝑖 NOT - no solution or dne −12 =2𝑖 3

7 Examples: solve using square root5x2 – 3 = f(x) = 5x2 – 3 find f(x) = 77 3x2 + 7 = 25 (x – 4)2 = 16 24 + (x – 3)2 = 15 (comment on standard form) x2 +8x = 15

8 Draw back to inversing If x does not appear in the problem only once (in other words the function is a combination of linear and square) then using square root to isolate the x becomes problematic since: A. You cannot combine unlike terms 3x2 + 5x B. You cannot do the square root on an expression that contains addition outside of ( ) 𝑥 −5 2 =𝑥 −5 𝑥 2 −4𝑥 ≠ 𝑥 −2

9 Using factors to solve quadraticsMany quadratics will factor ex: x2 – 7x + 12 = 3x x + 20 = 9x2 – 121 = 15x2 + 35x – 90 =

10 Mathematical fact if ab = 0 either a= 0 or b = 0

11 Thus: zero product rule is bornGiven a quadratic equation, you can SOMETIMES isolate the x by 1. Making the equation = 0 2. Factoring the quadratic 3. Splitting the one equation into 2 and Solving each equation

12 Examples x2 + 6x – 16 = 0 x2 – 7x = -12 (x - 3)(x – 8) = 66Next hurdle: x2 -10x + 18 = 0 ??????

13 Bridging the gap When a problem cannot be done by factoring because they do not have integral factors and cannot be solve by square root because there is an x – term in the problem then: These problems can be re written so that the x only appears in the equation once. The process is called completing the square

14 Completing the square Concept - change a quadratic so that it is a square quadratic ie: so it factors (x +a)(x + a) which is (x + a)2 Given the first 2 terms you can determine the 3rd term Fill in the blanks x2 + 8x + ___ = (x + ___)2 x2 – 10x + ___ =(x + ___)2 x2 + 5x + ____=(x + ___)2 Filling in these blanks is called completing the square

15 Completing the square to solve equations4x2 – 40x = 0 Prepare for completing the square by dividing by 4 and moving the 20 ( which will be a 5) x2 – 10x = 5 Completing the square CHANGES the number so add the same amount to BOTH sides of the equation x2 – 10x + 25 = (x – 5)2 = x = 5± 30

16 Examples : solve the following3x2 – 24x + 6 = 0 x2 – 7x – 3 = 0 3x2 – 11x – 4 = 0 x2 + 5x + 15 =0

17 Interesting to note All quadratics can be solvedAll quadratics are transformations of f(x)= x2

18 Guaranteed test question – 10 ptsMathematics finds the formulas – You find a formula by solving without using any known numbers Solve ax2 + bx + c = for x thereby deriving a formula for x

19 Deriving the quadratic formula Solve ax2 + bx + c = 0 by completing the squareDivide by a : Move constant to the right Divide middle coefficient by 2 /square/ add Get a common denominator and combine fraction Write as a square Isolate the x

20 Quadratic functions Chapter 4 – Section 2

21 General overview All quadratic functions are transformations of the parent f(x) = x2 Significant elements of quadratic function: y-intercept x-intercept (root/zero equation) maximum/minimum (vertex) concavity line of symmetry increasing/decreasing intervals domain/range

22 Quadratic functions- useful formsf(x) = ax2 + bx + c = y is a quadratic function in general form f(x) = a(x – h)2 + k = y can be the SAME quadratic function - in standard form f(x) = a(x – x1)(x – x2) can be the same function also – in factored form

23 Changing forms- can be done but is not necessaryTo general form from factored or standard – simplify ex: g(x) = 3(x – 2)(x – 2/3) k(x) = 5(x – 7)2 - 9 To factored form from general – factor the polynomial if possible ex: m(x) = 2x2 – 3x – 20 To standard form from general ( a variation on completing square) ex: f(x) = 2x2 – 12x + 10

24 Examples: find x and y interceptsf(x) = x2 – 12x +32 g(x) = (x + 2)(x – 9) k(x) = 3(x – 6)

25 An important correlationsolutions to the zero equation – x-intercepts – factors are significantly related x = 5 is a solution to the zero equation if and only if (5,0) is an x – intercept and (x – 5) is a factor of the quadratic If x = …… If x = 4 – 9i ……

26 Concavity- value of a Oriented up(a >0)/ down a<0Width (rate of change) - as |a | increases the parabola stretches vertically and becimes narrower ex: f(x) = 3x2 – 5x + 12 g(x) = 5 – 2x – 3x2 k(x) = (x + 2)2 – 9 m(x) = (2 – x)(x + 5) Related concept – max/min and range

27 Find vertex- g(x) = 3(x + 5)2 + 3 Related concepts - line of symmetryrange intervals of inc/dec

28 Finding vertex etc. f(x) = -2x2 + 4x – 7method one – easiest – uses information from quadratic formula Method two – change to vertex form – (not recommended) Related concepts - line of symmetry range intervals of inc/dec

29 Find vertex k(x) = (x – 9)(x + 3)typical method: put into general form Food for thought - line of symmetry (midpoint is found by average) Related concepts - line of symmetry range intervals of inc/dec

30 Sketching a graph for a quadratic by handFind vertex and at least one other point- use symmetry or table of solutions to find a third point Ex: f(x) = 2x2 – 3x – 20 Ex: g(x) = 3(x – 5)(x + 3) Ex: k(x) = -(x – 5)2 – 2

31 significance A ball is thrown into the air has a height given by the function h(x) = -16x2 + 34x wherw x is the time that has elapsed since throwing the ball Find the y intercept and interpret its meaning? When will the ball hit the ground? Where will the ball be in 1.2 seconds? When will the ball be 17 feet above ground? How high did the ball go? Find h(3) and explain its significance..

32 Writing equations – word problemsTo write an equation for a quadratic you need either 3 random points or the vertex and one other solution point Or factors and one other point or a formula which has already been derived by someone else.

33 Physics formula the quadratic function h(x) = ax2 + bx + chas a physics application where a = acceleration of an object after it is released (usually this is gravitional acceleration) b = its initial velocity (force with which it is released) c = the original location of the object Write a function for an object with gravitational acceleration (known to be -16 ft/sec2) an initial velocity force of 12 ft/sec and an initial location of 29 feet.

34 Banking formula- compound interestP(r) = p0 (1 + r)t where p0 is original amount in bank t is a set amount of time r is the rate of interest note: t must be the same units as r if r is 6% per month then t = 7 is 7 months Ex. Mark invests $4500 for 3 years. Write a function for his account as a function of r.

35 Given 3 point (2,13) (-3, 38) ( 1,6)

36 Given vertex and one point(2,4) is the vertex and the graph goes through the point (3,10)

37 examples Given the points (5,0) (-2,0) and the point (3,-2) write an equation – you could do a system of equations since this is 3 points Or - a(x – 5) (x + 2)= 0 must be true Thus a(x -5)(x +2) = y is the basis for the equation and a(3 – 5)(3 +2) = -2 so -10a = -2 and a = 1/5 answer g(x) = 1/5 (x – 5)(x + 2)

38 Examples Irrational and imaginary solutions come in pairsgiven intercept , 0 You know there is another intercept – 2− 3 , 0 You have factors [x -(2+ 3 )] [x- (2− 3 )] You can simplify the quadratic and find the stretch factor given a point (-4,3)

39 example Given solution - x = 3i and point (3, - 2)You know another solution is x = - 3i You have factors You simplify and find the stretch factor

40 Quadratic inequalities in one variableChapter 4 – Section 3

41 What we already know The solutions of inequalities in one variable are intervals on the number line The solutions to inequalities are directly related to the solution of equations There are 2 solutions to quadratic equations.

42 What the graph of the quadratic function tells usfind f(x) = 0 find f(x)>0 find f(x)<0 So quadratic inequalities work like absolute value inequalities

43 Examples find x2 – 9x >0 Find 2x2 + 7x – 30 < 0

44 circles Distance formula 𝑎 2 + 𝑏 2 = 𝑐 2𝑥 1 − 𝑥 𝑦 1 − 𝑦 = 𝑑 2 A circle is the set of all points that are the same distance from a given point

45 standard form Given (h,k) is the center and r is the radius the distance formula gives us (x – h)2 + (y – k)2 = r2 This textbook calls this standard form for the circle equation With (h, k) = (0,0) you have a circle centered on the origin so standard form is a transformation of 𝑥 2 + 𝑦 2 =1 (called the unit circle)

46 Graphing circles (x – 5)2 + (y + 2)2 = 16

47 Writing the equation (x – h)2 + (y – k)2 = r2Given center and radius simply fill in the blanks Example: A circle with radius 5 and center at (-2, 5) Given center and a point - find radius and fill in blanks A circle with center at (4,8) that goes through (7, 12)

48 General form Standard form of circle (transformation form)Get standard form from general form. 𝑎𝑥 2 +𝑏𝑥+𝑎 𝑦 2 +𝑐𝑦=𝑟 3 𝑥 2 +12𝑥+3 𝑦 2 −30𝑦=6

49 Quadratic equations: factoring /square root/completing square/quadratic formulaDerive quadratic formula Quadratic function: standard form a(x-h)2+k = y general form ax2 +bx + c =y factored form a(x- x1)(x-x2) Concavity Find x-int(roots) y-intercepts Find vertex (h,k) x = -b/2a Domain range intervals of increase, or decrease Write quadratic equations Quadratic inequalities Circles – standard form general form