GRADE PRE-CALCULUS UNIT C: QUADRATIC FUNCTIONS CLASS NOTES FRAME. After linear functions, = m + b, and their graph the Quadratic Functions are the net most important equation or function. The Quadratic Function can be written in several forms. Two of the forms are: f ( ) = a + b + c or also f ( ) = a( h) + k. Prerequisites. In order to be comfortable with this unit ou will want to be familiar with: a. plotting points in Cartesian coordinates (Grade 9 and ), b. using a graphing calculator such as the TI-83 Graphing Calculator, c. the concept of Functions and Relations (Grade Unit), d. multipling binomials (aka: F.O.I.L.) (Grade ), e. factoring Perfect Square Trinomials (Grade ), f. interval notation ( eample: < < ), g. Linear Models and how to graph lines (Grade ), and h. the absolute value function ( Grade 9 ) and lots of algebra. 3. Manuall plot the function = 6 + 6 + - 3 4 6 7 GrPrecalc_C_QuadraticFunctions.doc Revised:3
4. The curve formed b a quadratic function is called a parabola. Where have ou seen or eperienced parabolas before?? Characteristics of the Quadratic Function. The quadratic function has five important characteristics: a. Verte. The peak of the curve, where it has a maimum or a minimum either side of which the curve is smmetrical. The verte is a point, eg: (3, 4). We will learn several was to find the verte of a quadratic. b. Ais of Smmetr. The ais of smmetr is the vertical line that passes through the verte. The parabolic curve to the left and to the right of this vertical line are mirror images. A vertical line ou will recall from Grade is of the form: = constant. c. Domain. The values that the (input) of the function can take on. In the case of quadratic functions can be an number { < < } [More in another Chapter] d. Range. The values that the f() (the output) plotted on the can have. will alwas have a limited range in a quadratic because there is alwas a maimum or a minimum value for at the verte. In the eample we did above of = 6 + the range of the function is: { 4 < }. There are several was to show the interval notation, we will eamine others. e. Direction of Opening. The a parameter in the quadratic function governs whether the quadratic curve opens upwards (positive a) or downward (negative a). If the function has a positive a obviousl it also has a minimum in its range, if a negative a then a maimum in its range. f. Intercepts - where the curve crosses the -ais and the -ais are also significant points. () -intercept. A quadratic, like all functions, has no more than one -intercept, and its graph must cross the -ais somewhere and at one value. () -intercept. -intercepts are where the curve crosses the -ais (ie: where = ). The graph of the quadratic function has either zero, one or two values. X- intercepts are also called the roots of the quadratic function. In this unit we will onl find roots graphicall or conceptuall; we will stud how to find roots analticall and more rigorousl using Algebra in Unit A.
EXAMPLES 3 6. Manuall plot the following quadratic function and find all its characteristic values. = 6 6-7 -6-4 -3 -. - -. 7. Plot and label the following quadratic functions as values of on the same graph: (or use TI-83 or an on-line graphing tool) f ( ) = + 6 7 and g ( ) = + 4 + 6 f() g() - -8-6 -4-4 6 8 Scale the vertical -ais to count b twos! f() g() Verte Ais Smmetr Verte: Ais Smmetr: Domain: Range: Y Intercept: X-Intercept(s): Direction of Opening Domain Range -Intcpt -intcpt(s)
EXPLORE CHARACTERISTICS OF QUADRATIC FUNCTION f() = a 4 8. We will begin to look in more detail at quadratics. The a coefficient is with the term. 9. Graph the equations below and complete the table. Using an on-line grapher or a graphing calculator would be the best idea for this eercise. g ( ) = h ( ) = f ( ) = Verte Point (, ) (, ) (, ) Line of Smmetr Equation (Ais of Smmetr) Domain = = = < < Range < < < -intercept (, ) (, ) (, ) -intercept(s) Direction of Opening Maimum or Minimum Values of function Sketch of Graph Sketch: Characteristic points of the curve should be in correct quadrants.. Tr it quickl also for f ( ) =, negative value for the a coefficient do? g( ) =, h( ) =. What does having a. Notes for the simple quadratic = a : a. The Ais of Smmetr is the -ais given b the equation =. b. the coordinate point of the verte is (, ) c. If a>, graph opens up. If a< graph opens down. (what about if a = eactl?) d. If a >, graph is narrower than =, If a <, graph is wider than = *ie: if a is between and the graph is narrower, and if a is more than or less that it is wider. e. If a>, the min value of is. If a<, the ma value of is.
FIND EQUATION OF SIMPLE QUADRATIC GIVEN VERTEX AND ANOTHER POINT. Find the equation of the simple parabola of the form = a that has verte at the origin (, ) and goes through point (, 4). = a So put in an other points coordinates 4 = a() a = So the equation is: = You tr for a verte at (, ) and another point ( 3, 8): EXPLORE CHARACTERISTICS OF QUADRATIC FUNCTION IN FORM = f() = a + k 3. Eplore on the TI-83 or using an on-line grapher the following equations: Verte (point) Eqn of Line of Smmetr Domain = + 4 = + = = = 4 (, ) (, ) (, ) (, ) (, ) = = = = = < < Range -intercept (, ) (, ) (, ) (, ) (, ) -intercepts (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Opening Ma or Min Sketch of Graph Notice that the constant parameter k just gives a vertical translation, or vertical shift, to the curve. 4. General Ideas for simple quadratics of form = a + k a. Ais of smmetr is the -ais (ie: vertical line with equation = ) b. the Verte is at point (, k) c. = a + k is the graph of = a shifted (or translated) k units verticall and: If k >, the parabola is shifted upwards, if k < it is shifted down d. If a > then the curve opens upwards and the function has a minimum value at k, if a < then opens downwards and has a maimum value at k.
Mental Math. Which of the following could possibl be the curve = f() = a +3 6-4 -3 - - 3 4-4 -3 - - 3 4 - - a. - b. - -4-3 - - 3 4-4 -3 - - 3 4 - - c. - d. - EXERCISE - FINDING THE EQUATION FROM THE POINTS ON IT. We have been taking equations, then making a table of values, and then plotting curves. Tr to take some points from a curve and find the equation that makes it! (ie: Let s go backwards!) a. Given a function of the simple form = a + k that has a verte at (, 4) and passes through the point (, 4), find the equation. - We know from our studies so far that the verte is at (, k). So = a + 4. - Now we just need to determine the a. - Substitute in the coordinates of an other point on the curve (ie: (, 4) ) ( 4) = a() + 4 - solving with algebra: a = - so the equation that fits the points is = + 4 - do a table of values or graph it if ou don t believe it! b. Now ou tr for a simple quadratic of the form = a + k with a verte of (, ) and passing through a point (4, 3). Ans: =. Graph it to see that this right if ou want.
EXPLORE CHARACTERISTICS OF QUADRATIC FUNCTIONS OF THE FORM: = a( h). 7 6. Graph each of the equations below, start with the plain-jane =. Using an on-line grapher or a Graphing calculator would be quickest unless ou need practice at computing tables of values and making graphs manuall. =( + ) =( + ) =( + ) =( ) =( ) Verte (point) (, ) Eqn of Line of Smmetr Domain Range = -intercept (, ) -intercept(s) (, ) Dir n opening Ma and Min Sketch of Graph 7. Where do ou think the line of smmetr for the equation = ( 4) will be?. BTW: The form we used here is called the Standard form but probabl more accuratel known as the Verte Form. It can be converted to the General form b epanding the brackets (multipling with FOIL). General Ideas for simple quadratics of form = a( h) a. Ais of smmetr is the line = h (ie: with equation = h) b. Verte is at point (h, ). c. = a( h) is the graph of = a shifted (or translated) h units horizontall and: If h >, the parabola is shifted to the right, if h < it is shifted left d. If a > then the curve opens upwards and the function has a minimum of, if a < then opens downwards and has ma value of. Mental Math: a. describe the graph of 3( - ) relative to the plain jane parent function of b. describe the graph of. ( + 4) relative to the plain jane parent function
EXPLORE THE FULL STANDARD FORM OF THE QUADRATIC EQUATION: = a( h) + k. (Also known as Verte Form) 8 Verte (point) Eqn of Line of Smmetr Domain Range -intercept = 3( ) + = 3( ) = ( + ) 3 + = ( + ) 3 -intercept(s) save for Unit AC Direction of opening Ma or Min Values Sketch 4 3 4 3 4 3 4 3-4 -3 - - - 3 4-4 -3 - - - 3 4-4 -3 - - - 3 4-4 -3 - - - 3 4 - - - - -3-3 -3-3 -4-4 -4-4 Eample 8. Given the graph of a quadratic function below; find the characteristic information. (assume integral values. ie: nice round integer numbers for coefficients and points)) 6 State: Verte: 4 3-4 -3 - - 3 4 - Ais of Smmetr: Maimum or minimum value: Y-Intercept: The Equation of this curve : - -3-4 Domain: Range: -6
9 FINDING CHARACTERISTIC INFORMATION STRAIGHT FROM THE STANDARD FORM (VERTEX FORM) EQUATION 9. What if ou don t have a graph of the function; just an equation, and can t be bothered to make a graph. You should be able to jump back and forth directl from tables, to graphs, to equations since the are all reall the same thing anwa.. Given the following equation: = ( + ) - 4. (Verte Form =a( h) + k ) =a( + ) 4 Verte (point) (h, k) = (, 4) since ( + ) is reall same as ( ( )) Eqn of Line = of Smmetr Domain All such that ( < < + ) Range All such that ( 4 < ) -intercept When the equation is evaluated for =, so the -intercept = 3 -intercept(s) To be discussed in unit A, not eas!! Direction of opening Up since there is an implied + in front of the brackets. Ma and Min Values Min = 4 STANDARD (VERTEX) FORM VS GENERAL FORM OF QUADRATIC EQUATION. We have eamined the quadratic function and its graph when it is given in the form: = a( h) + k. This form is called the STANDARD FORM or VERTEX FORM As suggested at the start of the unit there is another form to write the equation, called the GENERAL FORM, = a + b + c. It is easil found b just performing the FOIL multiplication of the Standard Form.. Eample: = ( ) + = ( )*( ) + = 4 + 4 + = 4 + 9. Tr it!!, graph it, do the tables. The are eactl the same, ( ) + is eactl the same as 4 + 9,.just different was to write the equation. 3. You Tr! Convert from the STANDARD form = a( h) + k to the GENERAL form = a + b + c. Graph both forms of each to confirm the are identical functions. a. = ( + ) + 3 b. = ( 3)
APPLICATION 4. A thrown ball (on earth) obes the Standard Form or Verte Form equation h = (t ) + ; where h is height in meters above the ground, and t is time in seconds assuming the ball is thrown at a speed of meters/sec. Notice this is also the same as the epanded equation: h = t + t in the General form. a. to what height does the ball reach? (what is the ma of the verte?) b. at what time is the ball at the maimum height? (use the verte again) c. at what time is the ball at ground level? (-intercepts, solved graphicall) d. at what time does the path become a mirror image of a previous path? e. at t = seconds at what height is the ball (-intercept)? f. is this mathematical equation a perfect model for the motion of a ball?. Finding the -intercept(s) that make a function have a value of zero is rather important in man situations. Knowing when our bank account is zero, knowing when our sliding car comes to a stop, etc. Knowing when a quantit becomes zero is indeed ver important. We will learn how to do this in Unit A: Quadratic Equations. Presentl, we can onl find the -intercepts b graphing (or b luck guess). Maimum f() Maimum -Intercept X-Intercept X-Intercept - Minimum Minimum -
COMPLETING THE SQUARE 6. Completing the Square is the method to convert from General Form of a quadratic to the better Standard (Verte) Form. The form = ( ) + 3 is much easier to picture and to derive useful information. It is easier than the equivalent general form = 4 +. We will also use completing the square in Unit A: Quadratic Equations to find -intercept roots. 7. There is a phsical interpretation of the process using algebra tiles. You ma be familiar with Algebra tiles from man ears ago or from nephews/nieces or our own children. The are a useful wa to visuall eplain algebra in the simplest sense. You will have a demonstration in class on Algebra tiles, but a ver quick summar follows here. + + + 3 8. The above left shows that the epression ( + 3)*( + ) which can be multiplied to give + 4 + 3 which is reall the same as saing one unit short of (+) on the right or in other words (+). Check it out, the are all the same. 9. The method of Completing the Square (for the leading coefficient a = ). a. write the quadratic in the General Form: a + b + c. Eg: = +4 +7 b. Isolate the -terms (get them b themselves): 7 = + 4 (subtract 7 both sides) + c. now we want to make the right side a perfect square trinomial so it requires that there be some number that added to itself (ie: doubled b ) gives 4 and which multiplied b itself (ie: squared) is a square number so we can easil factor it. The number that doubles b to give the b of 4 is simpl 4/ (or b/), and then squared is. Missing d. Complete the square using the constant 7 + (4/) = ( + 4) + (4/) 7 + 4 = + 4 + 4 3 = + 4 + 4 b to add to both sides. e. Factor the trinomial on the right: 3 = (+)
f. Restore the function to solve for b itself: = (+) +3 g. the function is now in Standard Form and man of its characteristics are readil evident. h. Check b epanding using FOIL back to general form i. Teacher will show ou a slightl shorter method of doing all this. EXAMPLE COMPLETING THE SQUARE (leading coefficient a NOT equal to ): a. write the quadratic in the general form: a + b + c. Eg: =3 + +6 b. Isolate the -terms (get them b themselves): - 6 = 3 + c. Factor out the leading coefficient a. - 6 = 3( + 4) b d. Complete the square using the constant a to add to both sides. a 6 + 3(/6) = 3( + 4) + 3(/6) = 3( + 4 +4) e. Factor the trinomial on the right: 6 + = 3( + ) f. Restore the function to solve for b itself: 4 = 3( + ) g. Check b epanding back to general form so = 3( + ) + 4 3. Completing the square will alwas work. Sometimes it ma get a bit ugl with fractions, but it still works.
3 EXAMPLE 3 COMPLETING THE SQUARE = + 3 Write in GENERAL FORM 3 = Isolate the -terms (move the constant) 3 = ( + ) Factor out the a leading coefficient of the 3 ( ) ( ) ( ) ( ) ( ) + = + + b add a to both sides a 3 = + + group and form a perfect trinomial for the s 6 6 49 = + factor the perfect trinomial 8 4 = + 4 49 + 8 isolate to solve for again Epand to check: 49 49 48 = + + + = + = + = + 3 4 6 8 6 8 6 Yes! It works. = + 3 is the same as THE LESS ONEROUS WAY! = + 4 Ouch! So that above is the wa most sources will eplain it!! So I can t not show ou that! But lets be a little more realistic, here is the wa we will do it in our class! Divide both sides b a. We will alwas arrange that the leading a coefficient is even if we have to divide both sides b the a. = + 8 becomes: = + 4 b Now we add and subtract 4. In this case or4. = + 4 + 4 4. You are probabl wondering wh we did +4 4. Notice the first three terms are now a perfect square trinomial now. + 49 8
= [ + 4 + 4] 4, factoring the first three terms gives: = ( + ) 4 so adding the constants gives the result: = ( + ). Which is half of the function (if it were verticall squished), so we still need to restore the function back to = ; not =. So multipl both sides b the (which was the a leading coefficient of the original function). * = *[( + ) ] = *( + ) * = ( + ) is the STANDARD or VERTEX FORM. Both the STANDARD (VERTEX ) FORM and the GENERAL FORM give the same graph and table of values. 4
Finding the Verte Directl from the General Form 3. Given that ou probabl do not want to do the completing the square ever time, just realize now that ou have found a wa to find the verte directl from the general form. 3. Given a quadratic in the standard form = a + b + c; the -coordinate of the verte of b the curve is given b. So it tells ou the -coordinate (and line of smmetr) of the verte. a The of the verte is easil found b substituting in the found verte value back into the b function to find the that goes with that. In other words, evaluate the function as f to a find the -coordinate of the verte. The result is that the point: b a, f b a is the verte of the general form equation. Eample: find the verte of the equation 6 + 4. The verte is given b b b, f as shown below. a a b The -coordinate of the verte is simpl or or a (6) And the coordinate of the verte (which is the ma or min value) is found b evaluating for that verte, or = f ( ) = 6( ) + ( ) 4= Sot he verte of 6 + 4 is at the point (, ) 33. Or if ou want to be reall brave, b evaluation of the general equation given the coordinate of the verte above: b 4ac b the verte is at,. (this equation allows ou to find the ma or min without first a 4a having to find the ; to be proved in class) You will be allowed a page of notes on tests, this might be a hand formula.
EXAMPLE: 6 34. Find the verte of the equation = + 4 b 4ac b the verte is at, a 4a 4 4 () 6 or :, = ( 4, 8) FACTORED FORM OF QUADRATIC 3. We have skilfull avoided one other form of a quadratic which is more appropriate to stud when we consider Algebra; but we will introduce it here. It is the factored form : = a*( r )*( r ) It turns out that man quadratics can be factored (well technicall the all can be if ou have an imaginar world ). So for eample: 4 6 can be factored into: ( 4) ( + ). It has readil identifiable -intercepts of +4 and. Its line of smmetr is halfwa between 4 and so at a line of smmetr of =. Of course ou could also have called this function in the Verte Form: = ( ) 34. The are all the same thing! Check and see! Graph all three. = 4 6; = ( 4) ( + ); and = ( ) 8
APPLICATION AND WORD PROBLEMS - OPTIMIZING 7 36. What is the maimum rectangular area that can be enclosed b meters of fencing, if one of the sides is up against a long barn? (hint: maimum verte.and Area = Length * width) Solution: When in doubt, make a table and find some tpical values! Remember, ou onl have meters of fence though! Width 3 4 Length 8 6 4 Area 6 8 6 Plot the points, ou will see that it looks quadratic. From smmetr ou can tell that a width of 3m gives the maimum value of area of 8 square meters of enclosure. You could even state the equation to be = a( 3) + 8 and then find the a if ou wanted a mathematical model of the situation. Special Advanced Question. What eact dimensions would ou use if ou were told the enclosure had to be eactl 87. square meters? 37. Or ou might have just found the mathematical equation directl as follows: You know that Area = Length * Width (or A = L * W) You figure that the Length ou can make the enclosure is just: (L = * width) So reall A = L*W = ( *W)*(W) = W W or W + W Solving that graphicall or mathematicall tells ou that the best width to choose is b or or 3 a ( )
38. An orchard has trees per hectare. The average ield is 3 oranges per tree. It is estimated that for each additional tree added per hectare, that the average ield per tree will be reduced b oranges. a. What is the optimum number of trees per hectare to ield the most oranges. 8 b. What is a good mathematical equation to model the ield vs the number of trees? 38. You make the best Bannock in town. You have 4 customers weekl who pa $3. per loaf. But ou want to optimize our weekl income: make it a maimum. You know that for ever quarter ($.) ou increase the price of our bannock ou will get two fewer customers. Price $ $3. $8. # Customers 4 Income $ a. at what price do ou have no customers? b. from the table what is the optimum price to sell our bannock at so that ou make maimum income? c. what will be our weekl income at the optimum price? d. what is a good formula that models our income vs our price?
Glossar - GRADE PRE-CALCULUS UNIT C QUADRATIC FUNCTIONS GLOSSARY Binomial A polnomial is an epression consisting of two terms, ( ), ( + 3), (3 + ) are binomials. Absolute value Ais of Smmetr Coefficient Complete the square Domain Equation Eponential equation Factoring Polnomials Function the distance between an real number and on a number line without regard to being left or right of zero; for eample, 3 = 3, 3 = 3. The Absolute Value strips off negative values. It doesn t care if a number has a negative sense. To sa that < 3 means that is between 3 and 3. A line about which a curve looks mirror image. The Line of smmetr (or ais of smmetr it is sometimes called) for a quadratic is simpl the - value of the verte. If the verte of the parabola is at ( 4, 8) the line of smmetr is the vertical line = 4. the numerical factor of a term; for eample, in the terms 3 and 3, the coefficient is 3. In a polnomial the coefficient of the highest power term is called the leading coefficient. Eample: 4 + +7, the leading coefficient is a 4. add or subtract constants to reform a quadratic epression from the general form to the standard form of the equation a + b + c as a( h) + k. The h can be calculated as : b/a. the set of -values (or valid input numbers) represented b the graph or the equation of a function. For quadratics all values of are valid inputs, so the domain of a quadratic will alwas be (- < < ) for a quadratic a mathematical statement that two epressions are equal An equation that has the variable as an eponent. Eg: = 3*. Eponential equations are great models for population growth, mone growth, bacteria growth, radioactive deca, temperature cooling. to factor a polnomial with integer coefficients means to write it as a product of polnomials with integer coefficients. Eg: + 4 + 3 can be written as ( + )*( + 3) a rule that gives a single output number for ever valid input number. A relationship between sets of numbers that assigns one value to each number of a set. On da 3 ou have zero mone, our press the gas pedal cm ou go 4 km/h, when ou are our sister is 3, etc.
Glossar - Standard Form (of a quadratic) Intercept(s) An equation in the form = a( h) + k Note some references talk about = a( + h) + k, no big deal if h is a negative h. Where a curve crosses an ais. A parabola, the graph of a quadratic function, will alwas have a -intercept. A parabola ma have,, or -intercepts Linear equation a function whose defining equation can be written in the form = m + b, where m and b are constants. An increase in gives the same proportional increase in at all points. It has a constant slope of m. On-Line Grapher parabola Perfect square trinomial (a = ) Quadratic function Range (of a function) Root of an equation Translate -intercept -intercept zero or root of a function There are lots of good on-line graphers and apps if ou do not have a graphing calculator hand: the name given to the shape of the graph of a quadratic function. In this simple contet a quadratic epression of the form + b + c which can be readil factored to two binomials ( + [b/]). Eample: the epression + 6 + 9 can be readil factored to ( + 3). a function with defining equation f() =a + b + c where a, b, c are constants and a cannot be zero. It can also be written in the standard or verte form: = a( h) + k. Where a and h and k are constants. A quadratic can also be epressed in a factored form: = a( r )( r ); where r and r are the -intercepts. the set of -values (or output numbers) represented b the graph or the equation of a function. A quadratic will alwas have a limited range since it will alwas have a maimum or a minimum. a value of the variable that satisfies the equation To shift. If in an equation is replaced b ( h), the curve looks eactl the same ecept it translates right b an amount h. Curves can have vertical translation also. the -coordinate where the graph of a line or a function intersects the -ais the -coordinate where the graph of a line or a function intersects the -ais a value of the variable for which the function has value of zero. EG: the quadratic function f()= 4 has a zero or root of =. An quadratic can have either,, or zeros though. See: intercepts