Math Review Packet #5 Algebra II (Part 2) Notes

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1 SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of a quadratic relation looks like this: A + B+ C + D+ E+ F 0 Each of these si constants (A, B, C, D, E, and F), has a particular effect on the graph of the relation, and there are four basic graphs of these relations: circles, ellipses, hperbolas, and parabolas. Circles Recall that the definition of a circle is a set of points in a plane which are the same distance from a fied point called the center. Let s take a look at what its graph looks like: (h, k) r (, ) Using this distance formula, we can set this up as r h + k Squaring this gives us the general equation of a circle: h + k r with the center of the circle at (h, k) and radius r. The characteristics of a circle s equation are that the coefficients of and are the equal, and the coefficient of the term is 0. Eample: The relation is the graph of a circle. Where is its center, and what is its radius? Solution: The graph of a circle is a sum of two binomial squares. To turn this relation into that form, we will use the completing the square method: ( + 6) + ( ) ( ) ( ) The center is at (, ) with radius 5

2 Eample: Write the equation of the circle with center at (5, ) containing the point (0, ). Solution: Because ever point on a circle is equidistant from the origin, we can use the center and the point to calculate r: r Therefore, the equation of the circle is ( ) ( ) Ellipses If and terms have different coefficients but the same sign, then the graph is called an ellipse. The longest distance between the points is called the major ais; the segment minor ais major ais perpendicular to the major ais is called the minor ais. If an ellipse has its major and minor aes parallel to the coordinate aes, then the -radius is the horizontal distance from the center to the ellipse and the -radius is the vertical distance from the center to the ellipse. The - and -radii are each half of either the major or minor ais (called the semi-major and semi-minor aes). The general equation for an ellipse is: h k +, r r with center at (h, k), -radius r and -radius r. Eample: Given the relation , find the center, and the lengths of the major and minor aes. Solution: To rewrite this relation into the above general form, we must complete the square. ( 5 00) + ( ) 8 5( 8 ) + 9( + ) 8 ( ) ( ) (the leading coeff. of each square must ) ( ) ( ) ( ) 9( + ) ( ) ( + ) center (h, k) Center at (, ); major ais 0, minor ais 6

3 The definition of an ellipse is the set of points in a plane in which the sum the distances, d+ d, from two fied points F and F is constant. The points F and F are each called a focus (plural foci), and the distance from the center to a focus is 5 called the focal radius. A simple wa to draw an ellipse is to put two pins in a piece of cardboard and tie a string to (, ) each pin. Use our pencil to pull the string taut as ou trace a path around the pins. This will create an ellipse. a d b -5 F c F 5-5 focal radius d If a is the length of the semi-major ais, b is the length of the minor ais, c is the focal radius, and d and d are the distances from a point (, ) on the ellipse to the two foci, then d + d a (length of the major ais) a b + c or c a b Eample: Find the foci of Solution: We have alread found the equation for this ellipse: +. 5 The denominators give us our semi-major and semi-minor aes, so a 5 and b : c c Since the major ais is in the -direction, the foci will be located verticall, ± units from the center or at (, 5) and (, ). Hperbolas If the coefficients of the and terms are unequal and have opposite signs, the graph is that of a hperbola. Hperbolas have two disconnected branches, each of which approaches diagonal asmptotes. This time, the -radius is the horizontal 8 Asmptotes distance between the center and a verte, while the - radius is the vertical distance from the verte to an asmptote. If the hperbola opens verticall, this is reversed. The general form for the hperbola is Foci -8 Vertices h k ± r r The hperbola opens in the -direction if the term containing is positive; otherwise it opens in the - direction. r The slope of the asmpototes is alwas ± r.

4 - - cfocal radius bsemi-minor ais asemi-major ais -5 5 c a +b focus If a is the length of the semi-major ais, b is the length of the semi-minor ais, c is the focal radius, and d and d are the distances from a point (, ) on the hperbola to the two foci, then d d a c a + b Eample: Find the center, lengths of the major and minor aes, coordinates of the vertices, and the focal radius of the following: Solution: As before, the first step is to complete the square: ( ) ( ) 97 9( + 0) ( 8) 97 ( + + ) ( + ) ( ) ( ) ( + 5) ( ) The hperbola opens in the -direction with a center at ( 5, ). The length of the major ais is 6, and the length of the minor ais is. The slope of the asmptotes is ±. The two vertices are at ( 5, ) and ( 5, 7), and the focal radius is +. Parabolas The previous parabolas we addressed all opened in the -direction. Parabolas can also open in the -direction, and their equations look like a + b+ c Everthing we talked about on parabolas applies here just switch the and around. The Term A non-zero coefficient of the term rotates the graph and affects its shape.

5 Eponential Functions An eponential function is one whose general equation is and b is positive. This is read as varies eponentiall with. ab, where a and b are constants, Eponentiation Rules 0 (for 0) Adding/Subtracting You can onl add and subtract coefficients of bases with the same eponents. The eponent does not change. Multipling When multipling terms whose bases are the same, add the eponents and leave the base the same. Dividing When dividing terms whose bases are the same, subtract the eponents and leave the base the same. As a general rule, invert negative eponents. Power to a power When raising a power to a power, multipl each eponent b the power. Now, let s look at fractional eponents. Consider the following: This means that is a number that when squared gives for the answer which we also know as the square root. (In general, an nth root of is a number that when raised to the n power give for the answer.) Therefore, we can also use eponents to represent roots: n n If the eponent s numerator is a number other than, rules of eponents still appl: or more generall, a b ( ) b a, where a and b are integers (b 0) This process also works with decimal eponents since decimals are rational numbers. Logarithms We know that 0 00, but what about an equation that looks like this: 0 or what is the power of 0 that will produce? For this tpe of operation, we use logarithms. The definition of the base-0 logarithm is log if and onl if 0 The base-0 logarithm and 0 are inverse functions, so it is also true that log 0

6 Most scientific and graphing calculators have both a log button and a 0 button. Eample: Solve and check 0 57 Solution: 0 57 log 0 log (store this number to check) Check: 0 57 (use the 0 button) A ke concept to remember when working with logs is that the are eponents. So far, we have focused on base-0 logarithms, but a log can be in an base. Therefore, our earlier definition becomes log b if and onl if b where > 0, b > 0, and b This is called a base-b logarithm. If the log has no subscript, it is assumed to be base-0. Eample: Solve log for. Solution: Remember that the log is the eponent of the base. This means we can rewrite this as: 8 Eample: Find if Solution: 8 log The properties of logarithms are thus ver similar to the properties of eponents that we looked at earlier. Properties of Logarithms The logarithm of a product is the sum of the logs of the two factors. The logarithm of a quotient is the difference of the logs of the numerator and denominator. The logarithm of a power is the product of the power and the log of the base. Eample: Epress log7 + log7 5 as a single logarithm of a single argument. Solution: log7 + log7 5 log + log 5 Converse of log of a power 7 7 log 75 Converse of log of a product 7

7 Eample: Solve 7 8 b taking the log of each member. Solution: 7 8 log 7 log 8 log 7 log 8 log 8 log This is reasonable, as 7 9 and 7. If we want to change from one base to another, we use the change-of-base propert: log a logb log b The natural log (ln) is the logarithm whose base is the transcendental number called e, which is approimatel.788. (A transcendental number is a real, irrational, number that is not a solution of an integer polnomial.) All of the properties and procedures for using logarithms appl to natural logs. Composition of Functions Sometimes one function just cannot do everthing we need. A composite function, a f g, uses the output of g as the input of f. It is onl defined for values in the domain of g whose g( ) values are in the domain of f. To evaluate a composition of functions, ou evaluate the inner function first, and then use that as our input value for the outer function. Eample: Let f and g. Find h f g and j g f Solution: h g ( ) j f. f g. Composite functions are sometimes written as f g, which is the same as Inverses of Functions Consider a particular function f that has a domain D and range R. We could write this as f D R g R D. We would describe. Now suppose there is another functions g such that f and g as inverses of each other. If a function has an inverse, it is said to be invertible.

8 Eample: The cricket function, which gives temperature, T, in terms of chirp rate, R, is T f ( R) R+ 0 R f T. Find a formula for the inverse function, Solution: The inverse function gives the chirp rate in terms of the temperature, so we solve the following equation for R: T R+ 0 T 0 R R T 0 f T Here is a more formal definition: Q f t is a function with the propert that each value of Q determines Suppose eactl one value of t. Then f has an inverse function, f and if and onl if Q f ( t) f Q t The definition of logarithms has the same form as the definition of f, which means that log and 0 are inverse functions. The trig and inverse trig functions also work the same wa ( sin and sin, etc.). Man tetbooks eplain that to find the inverse of a function, interchange the variables and solve for. While this method works, the process loses one of the essential parts of the definition that the function and the inverse cancel each other out. A more mathematicallcorrect wa would be to take the composition of the function and its inverse and solve for the inverse: Eample: Find f, if f +. Solution: f + f f f + f f f + Check: ( ) + f ( f )

9 Eample: Find the inverse of +. Solution: Define f ( ) ( f ) ( f ) ( ) f f f f for 0, because the range of 0 f Now, let s go back to our eponential function, ab. This formula is often used to calculate eponential growth ( ) and eponential deca ( 0< < ). The number a is the initial quantit (because 0), and b is the rate of growth or deca (for eample, b.05 represents a growth rate of 5%, while b 0.95 represents a deca rate of 5%). Eample: Carbon- has a half-life of 578 ears. That means for a given initial quantit of carbon-, half of it will have decaed after 578 ears. What is its annual deca rate? Solution: Let Q be an amount of carbon-, and let t be the time in ears. Let the initial amount of carbon- be denoted as Q 0, so we have Q0 0 ab, which means that a Q0. The half-life of 578 gives us Q0 578 Q 0 b Q0 578 b Q log 0 b 578 log log log log b log b 578 ( b ) ( b) 578 log b Therefore, the deca rate is %

10 k Another tpe of eponential function uses e and the natural log. If we let b e, then we can see that t ab a e ae t k kt If b>, then k is positive; if 0< b<, then k is negative. Therefore, we would interpret k 0. to be a growth rate of 0% and k 0.5 to be a deca rate of 5%. To solve a problem using continuous growth/deca, use the natural log to undo the e and work as before. The function t ab is usuall used for quantities that grow at a constant rate; the function kt ae is usuall used for quantities that grow at a continuous rate. In word problems, look at the description to decide which formula to use. Eample: The voltage V across a charged capacitor is given b V( t) Rational Functions seconds. (a) What is the voltage after seconds? 0.( ) 0.9 V 5e 5e.0 volts (b) When will the voltage be? 0.t 5e e 0.t 0.t ( e ) 0. ln ln 0. 0.t ln 0. ln 0. t 5.65 seconds 0. 0.t 5e where t is in P A rational function is a function with the general equation f, where P( ) and Q( ) Q stand for polnomials. The most interesting thing about rational functions is what happens when the value of makes the denominator equal zero. Consider the function f rewrite the function:. To simplif this, let s factor the denominator and 6 f + ( + )( ) We can cancel the + factor, but we can t ignore it. Our function now becomes f, for

11 If we look at the graph of this function, we notice two things: The closer gets to, the larger the absolute value of f. Since can never equal, there is a vertical Eample: Simplif the function g asmptote at. Values of close to produce less dramatic results; as gets closer to, f( ) get closer to. Since can never 5 equal, there is simpl a point missing from the graph, leaving a hole at,. This hole is called a 5 discontinuit. + and note an discontinuities and/or 6 asmptotes. Solution: The denominator can be factored as a difference of squares, which produces: + g ( + )( ) g, where. This means that we have a discontinuit at and a vertical asmptote at. A trick to remember when ou are factoring is that b a ( a b) need to rewrite a factor to be able to cancel it. 9 Eample: Multipl and simplif +. 9 Solution: + 9 i + ( + )( ) ( + )( ) i + ( + )( ). This can be helpful if ou + Since this is not an equation or function, we don t have to note an discontinuities.

12 Comple Fractions 6 At first glance, the epression + 5 looks like something that would be wa too 7 + complicated to simplif. It turns out, however, that simplifing it is a fairl straightforward process. We want to multipl b epressions that will allow us to cancel out the problematic denominators. If we were to multipl the main numerator b + 5, that would cancel out the + 5 denominator; likewise, multipling the main denominator b would cancel out the denominator. Since we have to multipl b a form of to keep from changing the value of the fraction, we just multipl top and bottom b both factors: ( ) 5 ( 5)( ) i 7 ( 5)( ) ( + 5) ( + 5 6)( ) ( + 9)( ) ( + )( ) Sums and Differences of Rational Epressions Since we cannot add or subtract fractions without finding a common denominator, we have to do the same thing for rational epressions. The common denominator will be the least common multiple (LCM) of the two denominators. 7 6 Eample: Simplif. 7 6 Solution: ( )( + ) ( )( + ) 7 6 i i ( )( + )( ) ( )( + )( ) ( )( + )( ) ( )( + )( ) ( + )( + ) + + ( )( + )( ) ( )( ) 5+ 6 The LCM would be ( )( + )( )

Chapter 8 Notes SN AA U2C8

Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of