2.3 Quadratic Functions


 Augustus Payne
 11 months ago
 Views:
Transcription
1 88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the quadratic functions. Definition.5. A quadratic function is a function of the form f) = a + b + c, where a, b and c are real numbers with a 0. The domain of a quadratic function is, ). The most basic quadratic function is f) =, whose graph appears below. Its shape should look familiar from Intermediate Algebra it is called a parabola. The point 0, 0) is called the verte of the parabola. In this case, the verte is a relative minimum and is also the where the absolute minimum value of f can be found., ), ), ), ) 0, 0) f) = Much like man of the absolute value functions in Section., knowing the graph of f) = enables us to graph an entire famil of quadratic functions using transformations. Eample... Graph the following functions starting with the graph of f) = and using transformations. Find the verte, state the range and find the  and intercepts, if an eist.. g) = + ). h) = ) + Solution.. Since g) = + ) = f + ), Theorem.7 instructs us to first subtract from each of the values of the points on = f). This shifts the graph of = f) to the left units and moves, ) to, ),, ) to, ), 0, 0) to, 0),, ) to, ) and, ) to 0, ). Net, we subtract from each of the values of these new points. This moves the graph down units and moves, ) to, ),, ) to, ),, 0) to, ),, ) to, ) and 0, ) to 0, ). We connect the dots in parabolic fashion to get
2 . Quadratic Functions 89, ), ), ) 0, ), ), ), ), ) 0, 0) f) =, ) g) = f + ) = + ) From the graph, we see that the verte has moved from 0, 0) on the graph of = f) to, ) on the graph of = g). This sets [, ) as the range of g. We see that the graph of = g) crosses the ais twice, so we epect two intercepts. To find these, we set = g) = 0 and solve. Doing so ields the equation + ) = 0, or + ) =. Etracting square roots gives + = ±, or = ±. Our intercepts are, 0).7, 0) and +, 0) 0.7, 0). The intercept of the graph, 0, ) was one of the points we originall plotted, so we are done.. Following Theorem.7 once more, to graph h) = ) + = f ) +, we first start b adding to each of the values of the points on the graph of = f). This effects a horizontal shift right units and moves, ) to, ),, ) to, ), 0, 0) to, 0),, ) to, ) and, ) to 5, ). Net, we multipl each of our values first b and then add to that result. Geometricall, this is a vertical stretch b a factor of, followed b a reflection about the ais, followed b a vertical shift up unit. This moves, ) to, 7),, ) to, ),, 0) to, ),, ) to, ) and 5, ) to 5, 7)., ) 5, ), ), ), ) 5 6, ), ), 7) 5, 7) 0, 0) f) = h) = f ) + = ) + The verte is, ) which makes the range of h, ]. From our graph, we know that there are two intercepts, so we set = h) = 0 and solve. We get ) + = 0
3 90 Linear and Quadratic Functions which gives ) =. Etracting square roots gives = ±, so that when we add to each side, we get = 6± ). Hence, our intercepts are 6, 0.9, 0) and ) 6+, 0.7, 0). Although our graph doesn t show it, there is a intercept which can be found b setting = 0. With h0) = 0 ) + = 7, we have that our intercept is 0, 7). A few remarks about Eample.. are in order. First note that neither the formula given for g) nor the one given for h) match the form given in Definition.5. We could, of course, convert both g) and h) into that form b epanding and collecting like terms. Doing so, we find g) = + ) = + + and h) = ) + = + 7. While these simplified formulas for g) and h) satisf Definition.5, the do not lend themselves to graphing easil. For that reason, the form of g and h presented in Eample.. is given a special name, which we list below, along with the form presented in Definition.5. Definition.6. Standard and General Form of Quadratic Functions: Suppose f is a quadratic function. The general form of the quadratic function f is f) = a + b + c, where a, b and c are real numbers with a 0. The standard form of the quadratic function f is f) = a h) + k, where a, h and k are real numbers with a 0. It is important to note at this stage that we have no guarantees that ever quadratic function can be written in standard form. This is actuall true, and we prove this later in the eposition, but for now we celebrate the advantages of the standard form, starting with the following theorem. Theorem.. Verte Formula for Quadratics in Standard Form: For the quadratic function f) = a h) + k, where a, h and k are real numbers with a 0, the verte of the graph of = f) is h, k). We can readil verif the formula given Theorem. with the two functions given in Eample... After a slight) rewrite, g) = + ) = )) + ), and we identif h = and k =. Sure enough, we found the verte of the graph of = g) to be, ). For h) = ) +, no rewrite is needed. We can directl identif h = and k = and, sure enough, we found the verte of the graph of = h) to be, ). To see wh the formula in Theorem. produces the verte, consider the graph of the equation = a h) +k. When we substitute = h, we get = k, so h, k) is on the graph. If h, then h 0 so h) is a positive number. If a > 0, then a h) is positive, thus = a h) +k is alwas a number larger than k. This means that when a > 0, h, k) is the lowest point on the graph and thus the parabola must open upwards, making h, k) the verte. A similar argument and rationalizing denominators! and get common denominators!
4 . Quadratic Functions 9 shows that if a < 0, h, k) is the highest point on the graph, so the parabola opens downwards, and h, k) is also the verte in this case. Alternativel, we can appl the machiner in Section.7. Since the verte of = is 0, 0), we can determine the verte of = a h) +k b determining the final destination of 0, 0) as it is moved through each transformation. To obtain the formula f) = a h) + k, we start with g) = and first define g ) = ag) = a. This is results in a vertical scaling and/or reflection. Since we multipl the output b a, we multipl the coordinates on the graph of g b a, so the point 0, 0) remains 0, 0) and remains the verte. Net, we define g ) = g h) = a h). This induces a horizontal shift right or left h units moves the verte, in either case, to h, 0). Finall, f) = g ) + k = a h) + k which effects a vertical shift up or down k units 5 resulting in the verte moving from h, 0) to h, k). In addition to verifing Theorem., the arguments in the two preceding paragraphs have also shown us the role of the number a in the graphs of quadratic functions. The graph of = a h) +k is a parabola opening upwards if a > 0, and opening downwards if a < 0. Moreover, the smmetr enjoed b the graph of = about the ais is translated to a smmetr about the vertical line = h which is the vertical line through the verte. 6 This line is called the ais of smmetr of the parabola and is dashed in the figures below. verte verte a > 0 a < 0 Graphs of = a h) + k. Without a doubt, the standard form of a quadratic function, coupled with the machiner in Section.7, allows us to list the attributes of the graphs of such functions quickl and elegantl. What remains to be shown, however, is the fact that ever quadratic function can be written in standard form. To convert a quadratic function given in general form into standard form, we emplo the ancient rite of Completing the Square. We remind the reader how this is done in our net eample. Eample... Convert the functions below from general form to standard form. Find the verte, ais of smmetr and an  or intercepts. Graph each function and determine its range.. f) = +.. g) = 6 Just a scaling if a > 0. If a < 0, there is a reflection involved. Right if h > 0, left if h < 0. 5 Up if k > 0, down if k < 0 6 You should use transformations to verif this!
5 9 Linear and Quadratic Functions Solution.. To convert from general form to standard form, we complete the square. 7 First, we verif that the coefficient of is. Net, we find the coefficient of, in this case, and take half of it to get ) =. This tells us that our target perfect square quantit is ). To get an epression equivalent to ), we need to add ) = to the to create a perfect square trinomial, but to keep the balance, we must also subtract it. We collect the terms which create the perfect square and gather the remaining constant terms. Putting it all together, we get f) = + Compute ) =.) = + ) + Add and subtract ) = to + ).) = + ) + Group the perfect square trinomial.) = ) Factor the perfect square trinomial.) Of course, we can alwas check our answer b multipling out f) = ) to see that it simplifies to f) =. In the form f) = ), we readil find the verte to be, ) which makes the ais of smmetr =. To find the intercepts, we set = f) = 0. We are spoiled for choice, since we have two formulas for f). Since we recognize f) = + to be easil factorable, 8 we proceed to solve + = 0. Factoring gives ) ) = 0 so that = or =. The intercepts are then, 0) and, 0). To find the intercept, we set = 0. Once again, the general form f) = + is easiest to work with here, and we find = f0) =. Hence, the intercept is 0, ). With the verte, ais of smmetr and the intercepts, we get a prett good graph without the need to plot additional points. We see that the range of f is [, ) and we are done.. To get started, we rewrite g) = 6 = + 6 and note that the coefficient of is, not. This means our first step is to factor out the ) from both the and terms. We then follow the completing the square recipe as above. g) = + 6 = ) + ) + 6 Factor the coefficient of from and.) ) = ) = ) + + ) + ) ) + 6 Group the perfect square trinomial.) = + ) If ou forget wh we do what we do to complete the square, start with a h) + k, multipl it out, step b step, and then reverse the process. 8 Eperience pas off, here!
6 . Quadratic Functions 9 From g) = + ) + 5, we get the verte to be, 5 ) and the ais of smmetr to be =. To get the intercepts, we opt to set the given formula g) = 6 = 0. Solving, we get = and =, so the intercepts are, 0) and, 0). Setting = 0, we find g0) = 6, so the intercept is 0, 6). Plotting these points gives us the graph below. We see that the range of g is, 5 ] , 5 ) 6 5 0, 6) 5 = 0, ), 0), 0) 5, ) f) = +, 0) =, 0) g) = 6 With Eample.. fresh in our minds, we are now in a position to show that ever quadratic function can be written in standard form. We begin with f) = a + b + c, assume a 0, and complete the square in complete generalit. f) = a + b + c = a + ba ) + c Factor out coefficient of from and.) = a ba b + + = a ba b + + a = a + b ) + a a b a ) a ac b a ) + c b a ) + c Group the perfect square trinomial.) Factor and get a common denominator.) Comparing this last epression with the standard form, we identif h) with + a) b so that h = b ac b a. Instead of memorizing the value k = a, we see that f b ) a = ac b a. As such, we have derived a verte formula for the general form. We summarize both verte formulas in the bo at the top of the net page.
7 9 Linear and Quadratic Functions Equation.. Verte Formulas for Quadratic Functions: Suppose a, b, c, h and k are real numbers with a 0. If f) = a h) + k, the verte of the graph of = f) is the point h, k). If f) = a + b + c, the verte of the graph of = f) is the point b a, f b )). a There are two more results which can be gleaned from the completedsquare form of the general form of a quadratic function, f) = a + b + c = a + b ) ac b + a a We have seen that the number a in the standard form of a quadratic function determines whether the parabola opens upwards if a > 0) or downwards if a < 0). We see here that this number a is none other than the coefficient of in the general form of the quadratic function. In other words, it is the coefficient of alone which determines this behavior a result that is generalized in Section.. The second treasure is a rediscover of the quadratic formula. Equation.5. The Quadratic Formula: If a, b and c are real numbers with a 0, then the solutions to a + b + c = 0 are = b ± b ac. a Assuming the conditions of Equation.5, the solutions to a + b + c = 0 are precisel the zeros of f) = a + b + c. Since f) = a + b + c = a + b ) ac b + a a the equation a + b + c = 0 is equivalent to a + b ) ac b + a a = 0. Solving gives
8 . Quadratic Functions 95 a + b ) ac b + a a a + b a [ a + b a a = 0 ) ac b = a ) ] = a b ) ac a + b ) = b ac a a + b a = ± b ac a etract square roots + b a = ± = b a ± b ac a b ac a = b ± b ac a In our discussions of domain, we were warned against having negative numbers underneath the square root. Given that b ac is part of the Quadratic Formula, we will need to pa special attention to the radicand b ac. It turns out that the quantit b ac plas a critical role in determining the nature of the solutions to a quadratic equation. It is given a special name. Definition.7. If a, b and c are real numbers with a 0, then the discriminant of the quadratic equation a + b + c = 0 is the quantit b ac. The discriminant discriminates between the kinds of solutions we get from a quadratic equation. These cases, and their relation to the discriminant, are summarized below. Theorem.. Discriminant Trichotom: Let a, b and c be real numbers with a 0. If b ac < 0, the equation a + b + c = 0 has no real solutions. If b ac = 0, the equation a + b + c = 0 has eactl one real solution. If b ac > 0, the equation a + b + c = 0 has eactl two real solutions. The proof of Theorem. stems from the position of the discriminant in the quadratic equation, and is left as a good mental eercise for the reader. The net eample eploits the fruits of all of our labor in this section thus far.
9 96 Linear and Quadratic Functions Eample... Recall that the profit defined on page 8) for a product is defined b the equation Profit = Revenue Cost, or P ) = R) C). In Eample..7 the weekl revenue, in dollars, made b selling PortaBo Game Sstems was found to be R) = with the restriction carried over from the pricedemand function) that The cost, in dollars, to produce PortaBo Game Sstems is given in Eample..5 as C) = for 0.. Determine the weekl profit function P ).. Graph = P ). Include the  and intercepts as well as the verte and ais of smmetr.. Interpret the zeros of P.. Interpret the verte of the graph of = P ). 5. Recall that the weekl pricedemand equation for PortaBos is p) = , where p) is the price per PortaBo, in dollars, and is the weekl sales. What should the price per sstem be in order to maimize profit? Solution.. To find the profit function P ), we subtract P ) = R) C) = ) ) = Since the revenue function is valid when 0 66, P is also restricted to these values.. To find the intercepts, we set P ) = 0 and solve = 0. The mere thought of tring to factor the left hand side of this equation could do serious pschological damage, so we resort to the quadratic formula, Equation.5. Identifing a =.5, b = 70, and c = 50, we obtain = b ± b ac a = 70 ± 70.5) 50).5) = 70 ± 8000 = 70 ± 0 70 ) ) and To find the intercept, we set We get two intercepts:, 0, 0 = 0 and find = P 0) = 50 for a intercept of 0, 50). To find the verte, we use the fact that P ) = is in the general form of a quadratic function and appeal to Equation.. Substituting a =.5 and b = 70, we get = 70.5) = 70.
10 . Quadratic Functions 97 To find the coordinate of the verte, we compute P ) 70 = 000 and find that our verte is 70, 000 ). The ais of smmetr is the vertical line passing through the verte so it is the line = 70. To sketch a reasonable graph, we approimate the intercepts, 0.89, 0) and., 0), and the verte, 56.67, ). Note that in order to get the intercepts and the verte to show up in the same picture, we had to scale the ais differentl than the ais. This results in the lefthand intercept and the intercept being uncomfortabl close to each other and to the origin in the picture.) The zeros of P are the solutions to P ) = 0, which we have found to be approimatel 0.89 and.. As we saw in Eample.5., these are the breakeven points of the profit function, where enough product is sold to recover the cost spent to make the product. More importantl, we see from the graph that as long as is between 0.89 and., the graph = P ) is above the ais, meaning = P ) > 0 there. This means that for these values of, a profit is being made. Since represents the weekl sales of PortaBo Game Sstems, we round the zeros to positive integers and have that as long as, but no more than game sstems are sold weekl, the retailer will make a profit.. From the graph, we see that the maimum value of P occurs at the verte, which is approimatel 56.67, ). As above, represents the weekl sales of PortaBo sstems, so we can t sell game sstems. Comparing P 56) = 666 and P 57) = 666.5, we conclude that we will make a maimum profit of $ if we sell 57 game sstems. 5. In the previous part, we found that we need to sell 57 PortaBos per week to maimize profit. To find the price per PortaBo, we substitute = 57 into the pricedemand function to get p57) =.557) + 50 = 6.5. The price should be set at $6.50. Our net eample is another classic application of quadratic functions. Eample... Much to Donnie s surprise and delight, he inherits a large parcel of land in Ashtabula Count from one of his e)stranged) relatives. The time is finall right for him to pursue his dream of farming alpaca. He wishes to build a rectangular pasture, and estimates that he has enough mone for 00 linear feet of fencing material. If he makes the pasture adjacent to a stream so no fencing is required on that side), what are the dimensions of the pasture which maimize the area? What is the maimum area? If an average alpaca needs 5 square feet of grazing area, how man alpaca can Donnie keep in his pasture?
11 98 Linear and Quadratic Functions Solution. It is alwas helpful to sketch the problem situation, so we do so below. river w pasture l w We are tasked to find the dimensions of the pasture which would give a maimum area. We let w denote the width of the pasture and we let l denote the length of the pasture. Since the units given to us in the statement of the problem are feet, we assume w and l are measured in feet. The area of the pasture, which we ll call A, is related to w and l b the equation A = wl. Since w and l are both measured in feet, A has units of feet, or square feet. We are given the total amount of fencing available is 00 feet, which means w + l + w = 00, or, l + w = 00. We now have two equations, A = wl and l + w = 00. In order to use the tools given to us in this section to maimize A, we need to use the information given to write A as a function of just one variable, either w or l. This is where we use the equation l + w = 00. Solving for l, we find l = 00 w, and we substitute this into our equation for A. We get A = wl = w00 w) = 00w w. We now have A as a function of w, Aw) = 00w w = w + 00w. Before we go an further, we need to find the applied domain of A so that we know what values of w make sense in this problem situation. 9 Since w represents the width of the pasture, w > 0. Likewise, l represents the length of the pasture, so l = 00 w > 0. Solving this latter inequalit, we find w < 00. Hence, the function we wish to maimize is Aw) = w +00w for 0 < w < 00. Since A is a quadratic function of w), we know that the graph of = Aw) is a parabola. Since the coefficient of w is, we know that this parabola opens downwards. This means that there is a maimum value to be found, and we know it occurs at the verte. Using the verte formula, we find w = 00 ) = 50, and A50) = 50) ) = Since w = 50 lies in the applied domain, 0 < w < 00, we have that the area of the pasture is maimized when the width is 50 feet. To find the length, we use l = 00 w and find l = 00 50) = 00, so the length of the pasture is 00 feet. The maimum area is A50) = 5000, or 5000 square feet. If an average alpaca requires 5 square feet of pasture, Donnie can raise = 00 average alpaca. We conclude this section with the graph of a more complicated absolute value function. Eample..5. Graph f) = 6. Solution. Using the definition of absolute value, Definition., we have { f) = 6 ), if 6 < 0 6, if 6 0 The trouble is that we have et to develop an analtic techniques to solve nonlinear inequalities such as 6 < 0. You won t have to wait long; this is one of the main topics of Section.. 9 Donnie would be ver upset if, for eample, we told him the width of the pasture needs to be 50 feet.
12 . Quadratic Functions 99 Nevertheless, we can attack this problem graphicall. To that end, we graph = g) = 6 using the intercepts and the verte. To find the intercepts, we solve 6 = 0. Factoring gives ) + ) = 0 so = or =. Hence,, 0) and, 0) are intercepts. The intercept 0, 6) is found b setting = 0. To plot the verte, we find = b a = ) =, and = ) ) 6 = 5 = 6.5. Plotting, we get the parabola seen below on the left. To obtain points on the graph of = f) = 6, we can take points on the graph of g) = 6 and appl the absolute value to each of the values on the parabola. We see from the graph of g that for or, the values on the parabola are greater than or equal to zero since the graph is on or above the ais), so the absolute value leaves these portions of the graph alone. For between and, however, the values on the parabola are negative. For eample, the point 0, 6) on = 6 would result in the point 0, 6 ) = 0, 6)) = 0, 6) on the graph of f) = 6. Proceeding in this manner for all points with coordinates between and results in the graph seen below on the right = g) = 6 = f) = 6 If we take a step back and look at the graphs of g and f in the last eample, we notice that to obtain the graph of f from the graph of g, we reflect a portion of the graph of g about the ais. We can see this analticall b substituting g) = 6 into the formula for f) and calling to mind Theorem. from Section.7. { g), if g) < 0 f) = g), if g) 0 The function f is defined so that when g) is negative i.e., when its graph is below the ais), the graph of f is its refection across the ais. This is a general template to graph functions of the form f) = g). From this perspective, the graph of f) = can be obtained b reflection the portion of the line g) = which is below the ais back above the ais creating the characteristic shape.
a. plotting points in Cartesian coordinates (Grade 9 and 10), b. using a graphing calculator such as the TI83 Graphing Calculator,
GRADE PRECALCULUS 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
More information3.1 Graph Quadratic Functions
3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your
More informationLESSON #11  FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON #  FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More information3.1Quadratic Functions & Inequalities
3.1Quadratic Functions & Inequalities Quadratic Functions: Quadratic functions are polnomial functions of the form also be written in the form f ( ) a( h) k. f ( ) a b c. A quadratic function ma Verte
More informationSECTION 3.1: Quadratic Functions
SECTION 3.: Quadratic Functions Objectives Graph and Analyze Quadratic Functions in Standard and Verte Form Identify the Verte, Ais of Symmetry, and Intercepts of a Quadratic Function Find the Maimum or
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More informationLESSON #12  FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON #  FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More informationUnit 10  Graphing Quadratic Functions
Unit  Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
More informationMath 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.
Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the
More informationUnit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
More informationChapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs
Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The
More informationReview Topics for MATH 1400 Elements of Calculus Table of Contents
Math 1400  Mano Table of Contents  Review  page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical
More informationMA123, Chapter 1: Equations, functions and graphs (pp. 115)
MA123, Chapter 1: Equations, functions and graphs (pp. 115) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand
More information8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.
8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,
More informationSection 5.3: Other Algebraic Functions, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a
Section 5.: Other Algebraic Functions, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons AttributionNonCommercialShareAlike.0 license.
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE.1 The Rectangular Coordinate Systems and Graphs. Linear Equations in One Variable.3 Models and Applications. Comple Numbers.5 Quadratic Equations.6 Other
More informationPRINCIPLES OF MATHEMATICS 11 Chapter 2 Quadratic Functions Lesson 1 Graphs of Quadratic Functions (2.1) where a, b, and c are constants and a 0
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 1 Graphs of Quadratic Functions (.1) Date A. QUADRATIC FUNCTIONS A quadratic function is an equation that can be written in the following
More informationGraphing and Optimization
BARNMC_33886.QXD //7 :7 Page 74 Graphing and Optimization CHAPTER  First Derivative and Graphs  Second Derivative and Graphs 3 L Hôpital s Rule 4 CurveSketching Techniques  Absolute Maima and Minima
More informationNonlinear Systems. No solution One solution Two solutions. Solve the system by graphing. Check your answer.
810 Nonlinear Sstems CC.91.A.REI.7 Solve a simple sstem consisting of a linear equation and a quadratic equation in two variables algebraicall and graphicall. Objective Solve sstems of equations in two
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More informationCHAPTER 3 Graphs and Functions
CHAPTER Graphs and Functions Section. The Rectangular Coordinate Sstem............ Section. Graphs of Equations..................... 7 Section. Slope and Graphs of Linear Equations........... 7 Section.
More informationAlgebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample  Find the degree of 5 7 yz Degree of a polynomial:
More informationLESSON #48  INTEGER EXPONENTS COMMON CORE ALGEBRA II
LESSON #8  INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will
More information4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4
SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward
More informationf ( x ) = x Determine the implied domain of the given function. Express your answer in interval notation.
Test Review Section.. Given the following function: f ( ) = + 5  Determine the implied domain of the given function. Epress your answer in interval notation.. Find the verte of the following quadratic
More informationLinear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?
3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationName Class Date. Quadratic Functions and Transformations. 4 6 x
 Quadratic Functions and Transformations For Eercises, choose the correct letter.. What is the verte of the function 53()? D (, ) (, ) (, ) (, ). Which is the graph of the function f ()5(3) 5? F 6 6 O
More informationName: Period: SM Starter on Reading Quadratic Graph. This graph and equation represent the path of an object being thrown.
SM Name: Period: 7.5 Starter on Reading Quadratic Graph This graph and equation represent the path of an object being thrown. 1. What is the ais measuring?. What is the yais measuring? 3. What are the
More informationAlgebraic Functions, Equations and Inequalities
Algebraic Functions, Equations and Inequalities Assessment statements.1 Odd and even functions (also see Chapter 7)..4 The rational function a c + b and its graph. + d.5 Polynomial functions. The factor
More informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.
Chapter : Linear and Quadratic Functions Chapter : Linear and Quadratic Functions : Points and Lines Sstem of Linear Equations:  two or more linear equations on the same coordinate grid. Solution of
More informationMATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)
MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the
More information4Cubic. polynomials UNCORRECTED PAGE PROOFS
4Cubic polnomials 4.1 Kick off with CAS 4. Polnomials 4.3 The remainder and factor theorems 4.4 Graphs of cubic polnomials 4.5 Equations of cubic polnomials 4.6 Cubic models and applications 4.7 Review
More informationThe speed the speed of light is 30,000,000,000 m/s. Write this number in scientific notation.
Chapter 1 Section 1.1 Scientific Notation Powers of Ten 1 1 1.1.1.1.1 Standard Scientific Notation N n where 1 N and n is an integers Eamples of numbers in scientific notation. 8.17 11 Using Scientific
More informationLESSON #17  FACTORING COMMON CORE ALGEBRA II FACTOR TWO IMPORTANT MEANINGS
1 LESSON #17  FACTORING COMMON CORE ALGEBRA II In the study of algebra there are certain skills that are called gateway skills because without them a student simply cannot enter into many more comple
More informationAlgebra Notes Quadratic Functions and Equations Unit 08
Note: This Unit contains concepts that are separated for teacher use, but which must be integrated by the completion of the unit so students can make sense of choosing appropriate methods for solving quadratic
More informationQuadratic equationsgraphs
All reasonable efforts have been made to make sure the notes are accurate. The author cannot be held responsible for an damages arising from the use of these notes in an fashion. Quadratic equationsgraphs
More informationProperties of the Graph of a Quadratic Function. has a vertex with an xcoordinate of 2 b } 2a
0.2 Graph 5 a 2 b c Before You graphed simple quadratic functions. Now You will graph general quadratic functions. Wh? So ou can investigate a cable s height, as in Eample 4. Ke Vocabular minimum value
More informationQuadratic Function. Parabola. Parent quadratic function. Vertex. Axis of Symmetry
Name: Chapter 10: Quadratic Equations and Functions Section 10.1: Graph = a + c Quadratic Function Parabola Parent quadratic function Verte Ais of Smmetr Parent Function =  1 0 1 1 Eample 1: Make a table,
More informationSection 1.2: Relations, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons
Section.: Relations, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons AttributionNonCommercialShareAlike.0 license. 0, Carl Stitz.
More informationSolve Quadratic Equations
Skill: solve quadratic equations by factoring. Solve Quadratic Equations A.SSE.A. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. For eample, see
More informationPolynomial Functions of Higher Degree
SAMPLE CHAPTER. NOT FOR DISTRIBUTION. 4 Polynomial Functions of Higher Degree Polynomial functions of degree greater than 2 can be used to model data such as the annual temperature fluctuations in Daytona
More informationNorthwest High School s Algebra 2/Honors Algebra 2
Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar
More informationChapter 5: Systems of Equations
Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.
More informationUsing Intercept Form
8.5 Using Intercept Form Essential Question What are some of the characteristics of the graph of f () = a( p)( q)? Using Zeros to Write Functions Work with a partner. Each graph represents a function of
More informationSection 7.3: Parabolas, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons
Section 7.: Parabolas, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons AttributionNonCommercialShareAlike.0 license. 0, Carl Stitz.
More informationAlgebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.
Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and
More information2.1 Evaluate and Graph Polynomial
2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of
More informationMAT 1033C  MartinGay Intermediate Algebra Chapter 8 (8.1, 8.2, 8.5, 8.6) Practice for the Exam
MAT 33C  MartinGa Intermediate Algebra Chapter 8 (8.1 8. 8. 8.6) Practice for the Eam Name Date Da/Time: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
More informationMathematics 2201 Midterm Exam Review
Mathematics 0 Midterm Eam Review Chapter : Radicals Chapter : Quadratic Functions Chapter 7: Quadratic Equations. Evaluate: 8 (A) (B) (C) (D). Epress as an entire radical. (A) (B) (C) (D). What is the
More informationMth 95 Module 4 Chapter 8 Spring Review  Solving quadratic equations using the quadratic formula
Mth 95 Module 4 Chapter 8 Spring 04 Review  Solving quadratic equations using the quadratic formula Write the quadratic formula. The NUMBER of REAL and COMPLEX SOLUTIONS to a quadratic equation ( a b
More informationFinal Exam Review Part 2 #4
Final Eam Review Part # Intermediate Algebra / MAT 135 Fall 01 Master (Prof. Fleischner) Student Name/ID: 1. Solve for, where is a real number. + = 8. Solve for, where is a real number. 9 1 = 3. Solve
More informationUnit 11  Solving Quadratic Functions PART TWO
Unit 11  Solving Quadratic Functions PART TWO PREREQUISITE SKILLS: students should be able to add, subtract and multiply polynomials students should be able to factor polynomials students should be able
More informationIf C(x) is the total cost (in dollars) of producing x items of a product, then
Supplemental Review Problems for Unit Test : 1 Marginal Analysis (Sec 7) Be prepared to calculate total revenue given the price  demand function; to calculate total profit given total revenue and total
More informationMathematics 2201 Midterm Exam Review
Mathematics 0 Midterm Eam Review Chapter : Radicals Chapter 6: Quadratic Functions Chapter 7: Quadratic Equations. Evaluate: 6 8 (A) (B) (C) (D). Epress as an entire radical. (A) (B) (C) (D). What is the
More informationFair Game Review. Chapter 2. and y = 5. Evaluate the expression when x = xy 2. 4x. Evaluate the expression when a = 9 and b = 4.
Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five
More information9 (0, 3) and solve equations to earn full credit.
Math 0 Intermediate Algebra II Final Eam Review Page of Instructions: (6, ) Use our own paper for the review questions. For the final eam, show all work on the eam. (6, ) This is an algebra class do not
More informationLaw of Sines, Law of Cosines, Heron s Formula:
PreAP Math Analsis nd Semester Review Law of Sines, Law of Cosines, Heron s Formula:. Determine how man solutions the triangle has and eplain our reasoning. (FIND YOUR FLOW CHART) a. A = 4, a = 4 ards,
More informationIntermediate Algebra. Gregg Waterman Oregon Institute of Technology
Intermediate Algebra Gregg Waterman Oregon Institute of Technolog c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license
More informationPRACTICE FINAL EXAM. 3. Solve: 3x 8 < 7. Write your answer using interval notation. Graph your solution on the number line.
MAC 1105 PRACTICE FINAL EXAM College Algebra *Note: this eam is provided as practice onl. It was based on a book previousl used for this course. You should not onl stud these problems in preparing for
More informationOne of the most common applications of Calculus involves determining maximum or minimum values.
8 LESSON 5 MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..
More information4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?
3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSASSE.A. HSAREI.B.b HSFIF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the
More informationChapter 1. Functions, Graphs, and Limits
Chapter 1 Functions, Graphs, and Limits MA1103 Business Mathematics I Semester I Year 016/017 SBM International Class Lecturer: Dr. Rinovia Simanjuntak 1.1 Functions Function A function is a rule that
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More information[ ] 4. Math 70 Intermediate Algebra II Final Exam Review Page 1 of 19. Instructions:
Math Intermediate Algebra II Final Eam Review Page of 9 Instructions: Use our own paper for the review questions. For the final eam, show all work on the eam. This is an algebra class do not guess! Write
More information5. 2. The solution set is 7 6 i, 7 x. Since b = 20, add
Chapter : Quadratic Equations and Functions Chapter Review Eercises... 5 8 6 8 The solution set is 8, 8. 5 5 5 5 5 5 The solution set is 5,5. Rationalize the denominator. 6 The solution set is. 8 8 9 6
More informationLesson 10.1 Solving Quadratic Equations
Lesson 10.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with each set of conditions. a. One intercept and all nonnegative yvalues b. The verte in the third quadrant and no
More information5. Determine the discriminant for each and describe the nature of the roots.
4. Quadratic Equations Notes Day 1 1. Solve by factoring: a. 3 16 1 b. 3 c. 8 0 d. 9 18 0. Quadratic Formula: The roots of a quadratic equation of the form A + B + C = 0 with a 0 are given by the following
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More informationExponential, Logistic, and Logarithmic Functions
CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic
More informationHonors Algebra 2 ~ Spring 2014 Name 1 Unit 3: Quadratic Functions and Equations
Honors Algebra ~ Spring Name Unit : Quadratic Functions and Equations NC Objectives Covered:. Define and compute with comple numbers. Operate with algebraic epressions (polnomial, rational, comple fractions)
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationSection 3.1 Power Functions & Polynomial Functions
Chapter : Polynomial and Rational Functions Section. Power Functions & Polynomial Functions... 59 Section. Quadratic Functions... 67 Section. Graphs of Polynomial Functions... 8 Section.4 Factor Theorem
More informationNEXTGENERATION Advanced Algebra and Functions
NEXTGENERATIN Advanced Algebra and Functions Sample Questions The College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunit.
More informationCore Connections Algebra 2 Checkpoint Materials
Core Connections Algebra 2 Note to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactly the same way at the same time. At some point you will
More informationSection 3.3 Graphs of Polynomial Functions
3.3 Graphs of Polynomial Functions 179 Section 3.3 Graphs of Polynomial Functions In the previous section we eplored the short run behavior of quadratics, a special case of polynomials. In this section
More informationChapter 11 Quadratic Functions
Chapter 11 Quadratic Functions Mathematical Overview The relationship among parabolas, quadratic functions, and quadratic equations is investigated through activities that eplore both the geometric and
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Spring 0 Math 08 Eam Preparation Ch Dressler Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the quadratic equation b the square root propert.
More informationAlgebra I Quadratics Practice Questions
1. Which is equivalent to 64 100? 10 50 8 10 8 100. Which is equivalent to 6 8? 4 8 1 4. Which is equivalent to 7 6? 4 4 4. Which is equivalent to 4? 8 6 From CCSD CSE S Page 1 of 6 1 5. Which is equivalent
More informationSkills Practice Skills Practice for Lesson 1.1
Skills Practice Skills Practice for Lesson. Name Date Lots and Projectiles Introduction to Quadratic Functions Vocabular Give an eample of each term.. quadratic function 9 0. vertical motion equation s
More informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions Section 3.1 Power Functions & Polynomial Functions... 155 Section 3. Quadratic Functions... 163 Section 3.3 Graphs of Polynomial Functions... 176 Section 3.4
More informationTable of Contents. Module 1
Table of Contents Module 1 11 Order of Operations 16 Signed Numbers 1 Factorization of Integers 17 Further Signed Numbers 13 Fractions 18 Power Laws 14 Fractions and Decimals 19 Introduction to Algebra
More informationMATH 021 UNIT 1 HOMEWORK ASSIGNMENTS
MATH 01 UNIT 1 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usuall, the part (A) questions ask for eplanations,
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More informationObjectives To solve quadratic equations using the quadratic formula To find the number of solutions of a quadratic equation
96 The Quadratic Formula and the Discriminant Content Standards A.REI..a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( p) 5 q... Derive the
More informationCHAPTER 1 Functions and Their Graphs
PART I CHAPTER Functions and Their Graphs Section. Lines in the Plane....................... Section. Functions........................... Section. Graphs of Functions..................... Section. Shifting,
More informationM122 College Algebra Review for Final Exam
M1 College Algebra Review for Final Eam Revised Fall 017 for College Algebra  Beecher All answers should include our work (this could be a written eplanation of the result, a graph with the relevant feature
More informationDerivatives of Multivariable Functions
Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit
More informationUnit 2. Quadratic Functions and Modeling. 24 Jordan School District
Unit Quadratic Functions and Modeling 4 Unit Cluster (F.F.4, F.F.5, F.F.6) Unit Cluster (F.F.7, F.F.9) Interpret functions that arise in applications in terms of a contet Analyzing functions using different
More informationMethods for Advanced Mathematics (C3) Coursework Numerical Methods
Woodhouse College 0 Page Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on
More informationPure Core 1. Revision Notes
Pure Core Revision Notes Ma 06 Pure Core Algebra... Indices... Rules of indices... Surds... 4 Simplifing surds... 4 Rationalising the denominator... 4 Quadratic functions... 5 Completing the square....
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS
Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic
More informationMATH College Algebra Review for Test 2
MATH 4  College Algebra Review for Test Sections. and.. For f (x) = x + 4x + 5, give (a) the xintercept(s), (b) the intercept, (c) both coordinates of the vertex, and (d) the equation of the axis of
More informationVertex. March 23, Ch 9 Guided Notes.notebook
March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function
More informationSolutions to the Exercises of Chapter 4
Solutions to the Eercises of Chapter 4 4A. Basic Analtic Geometr. The distance between (, ) and (4, 5) is ( 4) +( 5) = 9+6 = 5 and that from (, 6) to (, ) is ( ( )) +( 6 ( )) = ( + )=.. i. AB = (6 ) +(
More information2 variables. is the same value as the solution of. 1 variable. You can use similar reasoning to solve quadratic equations. Work with a partner.
9. b Graphing Essential Question How can ou use a graph to solve a quadratic equation in one variable? Based on what ou learned about the intercepts of a graph in Section., it follows that the intercept
More information