Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.
|
|
- Christiana Woods
- 6 years ago
- Views:
Transcription
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 a Sstem of Linear Equations: - the intersecting point of two or more linear equations - on the Cartesian Coordinate Grid, the solution contains two parts: the -coordinate and the -coordinate Line Line (can be epressed as an Ordered Pair) Intersecting Point (, ) Eample : Find the solution of the following sstem of equations b graphing manuall using + Solution of the Sstem of Linear Equations We can find the solution of a sstem of linear equation graphing manuall or using a graphing calculator. a. the slope and -intercept form For Line : For Line : int (, ) Up + slope Right -int (, ) Down slope Right Line (, ) (, ) (, ) (, ) Line Intersecting Point (, ) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
2 Chapter : Linear and Quadratic Functions PreCalculus b. and -intercepts Line : For -int, let + () + (, ) For -int, let + + () (, ) Line : For -int, let () (, ) For -int, let () (, ) Line Intersecting Point (, ) (, ) (, ) (, ) (, ) Line Eample : Find the solution of the following sstem of equations b using the graphing calculator. + Step : Rearrange to Slope Step : Set WINDOWS to Standard Mode Step : Graph and intercept form Y ZOOM GRAPH Choose Option will set : [,, ] : [,, ] Step : Run INTERSECT nd CALC TRACE ENTER ENTER ENTER Step : Verif using TABLE nd TABLE GRAPH Page. Choose Option Coprighted b Gabriel Tang B.Ed., B.Sc.
3 Chapter : Linear and Quadratic Functions There are three tpes of solutions to a sstem of linear equations:. Intersecting Lines. Parallel Lines. Overlapping Lines One distinct Solution No Solution Man (Infinite) Solutions Different Slopes Identical Slopes Identical Slopes Different -Intercepts Different -Intercepts Identical -Intercepts Eample : Determine the number of solutions for the sstems of equations below. a. + b Line : m, -int Line : m, -int Line : m, -int Line : m, -int Identical slopes, but different - intercepts mean parallel lines. Therefore, this sstem has NO SOLUTION. Identical slopes and -intercepts mean overlapping lines. Therefore, this sstem has MANY SOLUTIONS. When using the substitution method to solve a sstem of linear equations:. Isolate a variable from one equation. (Alwas pick the variable with as a coefficient.). Substitute the resulting epression into that variable of the other equation.. Solve for the other variable.. Substitute the result from the last step into one of the original equation and solve for the remaining variable. Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
4 Chapter : Linear and Quadratic Functions PreCalculus Eample : Using the substitution method, algebraicall solve the sstem of equations below. ( ) ( ) ( ) ( + ) Epand each equation accordingl. Equation ( ) ( ) + Equation ( ) ( + ) Solve for both variables using the substitution method. Isolate from the first equation. + Substitute epression into in the second equation. ( ) Solve for the remaining variable. Pick the easier equation of the two Elimination b Multiplication: - most useful when neither equation has the same like terms. - b multipling different numbers (factors of their LCM) on each equation, we can change these equations into their equivalent form with the same like terms. + Eample : Solve the sstem of linear equations b elimination. First, rearrange the equations so and terms will line up. + + Eliminating Eliminate b addition. + LCM of and + ( + ) (Multipl Equation b to obtain ) (Multipl Equation b to obtain ) ( + ) + ( + ) + Net, substitute into one of the equations to solve for. + + () Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
5 Chapter : Linear and Quadratic Functions Review of Coordinate Geometr + + Midpoint M, Distance d ( ) + ( ) Eample : Find the distance (in eact value) between A (, ) and B (, ). A (, ) (, ) B (, ) (, ) d d AB AB ( ) + ( ) ( ) + ( ) + Eample : A circle has a diameter with endpoints (, ) and (, ). Find the eact length of the radius. Let A (, ) (, ) and B (, ) (, ) Length of Diameter AB Length of the Radius d d AB AB ( ) + ( ) + + Length of the Radius diameter AB Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
6 Chapter : Linear and Quadratic Functions PreCalculus Eample : Find the midpoints of the following line segments CD where C (, ) and D (, ) D (, ) + +,, Eample : Given that the midpoint of AB is M (, ). If one of the endpoint of AB is A (, ), find the coordinate of endpoint B. + + M AB, + + (, ), A (, ) M M (, ) C (, ) B + M CD M CD (, ) B (, ) Assignment: pg. # to (ever other odd) (Omit #ce, ) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
7 Chapter : Linear and Quadratic Functions - and -: Slopes of Lines and Finding Equations of Lines B (, ) A (, ) change in (run) ( ) Slope: - a measure on the steepness of the line segment. change in (rise) ( ) m Slope of a Line Segment rise run m change in change in Eample : Find the slope of the following line segments. a. CD where C (, ) and D (, ) b. PQ where P (, ) and Q (, ) C (, ) D (, ) - m CD ( ) ( ) rise ( ( )) run ( ()) - m CD P (, ) Q (, ) m PQ ( ) m PQ Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
8 Chapter : Linear and Quadratic Functions PreCalculus c. AB where A (, ) and B (, ) d. RS where R (, ) and S (, ) A (, ) B (, ) m AB ( ) m AB R (, ) S (, ) m RS ( ) m RS undefined In general, slopes can be classified as follows: Positive Slope (m > ) Line goes UP from left to right. Negative Slope (m < ) Line goes DOWN from left to right. Zero Slope (m ) Horizontal (Flat) Line [Rise ] Page. Undefined Slope Vertical Line [Run ] Coprighted b Gabriel Tang B.Ed., B.Sc.
9 Chapter : Linear and Quadratic Functions Eample : If the slope of a line is, and it passes through A (t, t ) and B (, ), find the value of t and point A. m t t t ( t ) ( t) ( t ) t t t + t + t t t A ((), () ) A (, ) Parallel Lines Perpendicular Lines slope of line slope of line slope of line negative reciprocal slope of line m l m l m l m l Eample : Given the slope of two lines below, determine whether the lines are parallel or perpendicular. a. m and m b. m and m c. m and m m and m (negative reciprocal slopes) m and m (same slopes) m and m (neither the same nor negative reciprocal) Perpendicular Lines Parallel Lines Neither Parallel nor Perpendicular Lines Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
10 Chapter : Linear and Quadratic Functions PreCalculus Eample : Find the slope of the line given and its perpendicular slope. a. + b. c. m ( means perpendicular) m (negative reciprocal) Eample : Find the equation in point-slope form and general form given the followings. a. (, ) and m b. (, ) and m + m m (negative reciprocal) (vertical line) m undefined m (negative reciprocal) m When given a slope (m) and a point (, ) on the line, we can find the equation of the line using the point-slope form: m (slope formula) m ( ) (Point-Slope form) If we rearrange the equations so that all terms are on one side, it will be in general form: A + B + C (General form) (A, the leading coefficient for the term must be positive) (, ) (, ) right For Slope-Point form: m ( ) ( ) ( ( ) ) + ( + ) up m up right For General form: + ( + ) ( + ) ( + ) (, ) down (, ) right m For Slope-Point form: m ( ) ( ( ) ) ( + ) down right For General form: ( + ) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
11 Chapter : Linear and Quadratic Functions When given a slope (m) and the -intercept (, b) of the line, we can find the equation of the line using the slope and -intercept form: m + b where m slope and b -intercept Eample : Given the -intercept and slope, write the equation of the line in slope and -intercept form, and standard form. Sketch a graph of the resulting equation. a. (, ) and m (, ) m right -intercept b Multipl both sides b. (, ) up up right b. b and m right (, ) down (, ) m -intercept b + Rearrange to the left side. + down right Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
12 Chapter : Linear and Quadratic Functions PreCalculus c. b and m d. -intercept and m undefined right up (, ) (, ) m -intercept b + up right (, ) right (, ) up intercept and undefined slope (Vertical Line). Rearrange to the left side. Rearrange to the left side. Vertical Line NO term. Horizontal Line NO term. Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
13 Chapter : Linear and Quadratic Functions Eample : Find the equation of a line parallel to + and passes through (, ). (, ) (, ) (, ) right Line : Eample : Find the equation of a line perpendicular to + and having the same - intercept as the line. Line Line up (, ) down right (, ) right (, ) up Line Line (, ) right (, ) up m + Line : m (parallel lines same slope as m ) Using (, ) as (, ) and the form m + b, we have: + + b + b () ( ) Line : b + + b b m Line : m (perpendicular lines negative reciprocal of m ) To find -intercept of, we let. () -int means (, ) Using (, ) as (, ) and the form m + b, we have: ( ) () + b b + b - Assignment: pg. # to (ever other odd) - Assignment: pg. # to (odd) (Omit #,,,,, ) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
14 Chapter : Linear and Quadratic Functions PreCalculus -: Linear Functions and Models Linear Function: - a set ordered pairs that ehibit a straight line when plotted on a graph. Mathematical Model: - one or more functions, graphs, tables, equations, or inequalities that describe a real-world situation. There are two tpes of data a. Discrete Data: - a graph with a series of separated ordered pairs or broken lines. Eample: Cost of Parking is $. ever hour with a Dail Maimum of $.. Time (hr) Cost ($) Cost ($) Parking Cost Tim e (hour) b. Continuous Data: - a graph with an unbroken line that connects a series of ordered pairs. Eample: Distance versus Time of a car with a constant speed of km/h. Time (hr) Distance (km) Distance (km) Distance versus Time Time (hour) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
15 Chapter : Linear and Quadratic Functions Direct Variation: - a variable that varies directl (b a rate of change) with another variable. ( is directl proportional to ) or k where k rate of change Eample : Gasoline at one time costs $. per Litre. a. Write out the cost of gasoline as a function of volume bought. b. What is the rate of change? c. Set up a table of values from V L to V L with a scale of L. d. Graph the function. a. C(V). V b. Rate of Change $./L (unit price of gasoline) c. d. V in L C(V) in $. ().. ().. ().. ().. ().. ().. (). Cost ($) Cost of Gasoline vs Volume Bought..... Volume (L) Eample : The amount of fuel used b a vehicle varies directl with the distance travelled. On a particular trip,. L of gasoline is used for a distance of. km. a. Calculate the rate of change. b. Find the function of volume of gasoline used in terms of distance travelled. c. What is the distance travelled if L of gasoline is used? d. How much would it cost to fuel up the car when the price of gasoline was $. / L if the entire trip was km? a. V(d) kd b. V(d) kd c. V L, d? d. d km, V?. k (.). d V. ().L k V(d).d d.km. V. L $. L. L k. L/km d. km $. Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
16 Chapter : Linear and Quadratic Functions PreCalculus Partial Variation: - a variable that varies partiall (b a constant rate of change and a fied amount) with another variable. k + b where k rate of change and b fied amount (initial amount when ) Eample : The cost of a school dance organized b the student council $. per person and $ to hire the DJ. a. What is the rate of change and the fied amount? b. Epress the cost as a function of the number of people attending. c. Set up a table of values from n to n people with a scale of people. d. Graph the function. e. Eplain whether the graph should be discrete or continuous. f. How man people attended if the cost was $? a. Rate of Change $. / person (cost per person) b. C(n).n + Fied Amount $ (fied cost) c. d. n C(n) in $. () +. () +. () +. () +. () +. () + Cost ($) $, $ $ $ $ $ Cost of a School Dance n (people) e. The graph should be discrete because ou can not have a decimal number to describe the number of people attending. f. C(n) $, n? C(n).n +.n +.n.n n. n people Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
17 Chapter : Linear and Quadratic Functions Eample : A tai ride costs ou $. for km travelled, and $. if ou travelled km. a. Epress the equation as a function of Cost in terms of distance travelled. b. What is the rate of change and the fied amount? c. How much would it cost for a km ride to the airport? d. If the ride costs $., what is the distance travelled? a. There are two order pairs ( km, $.) and ( km, $.) m k $. $. km km k $./km C kd + b (Substitute an order pair into d and C) $. ($./km)( km) + b $. $. + b b $. Therefore, C(d).d + b. Rate of Change $./km (cost per km travelled) Fied Amount $. (Flat Rate) c. d km, C? d. C $., d? C(d).d + C(d).d + C(d).() +..d + C(d) $...d..d. d. d. km - Assignment: pg. # to (odd),, and (Omit #) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
18 Chapter : Linear and Quadratic Functions PreCalculus -: The Comple Numbers Comple Numbers: - non-real numbers that cannot be represented on a number line or a Cartesian plane. - also called imaginar number. Real Numbers (R) If i, then i (imaginar unit) Comple Numbers ( R) Q I W N Q Eample: or or i Rational Numbers (Q): - numbers that can be turned into a fraction b a, where a, b I, and b. - include all Terminating or Repeating Decimals. - include all Natural Numbers, Whole Numbers and Integers. - include an perfect roots (radicals). Irrational Numbers (Q ): - numbers that CANNOT be turned into a fraction b a, where a, b I, and b. - include all non-terminating, non-repeating decimals. - include an non-perfect roots (radicals). Real Numbers (R): - an numbers that can be put on a number line. - include all natural numbers, whole numbers, integers, rational and irrational numbers. Eample : Simplif the following. a. b. c. i i i d. ( ) e. ( ) Page. ( ) ( ) i Coprighted b Gabriel Tang B.Ed., B.Sc.
19 Chapter : Linear and Quadratic Functions Eample : Simplif the following. a. ( )( ) b. ( )( ) c. (i) i i i i i i i i () d. (i) e. i () Eample : Simplif the following. i i i i i i i i i f. i i i i i i i i i i i i i a. ( i) + ( i) b. ( i) ( i) i i i i i i i i (Pattern continues) i + i i i + i i Subtacting a Bracket, change signs in the barcket Solving Equations with Comple Numbers:. Rewrite the equation into two separate equations. One with Real Numbers, the other with imaginar numbers.. Solve the two equations separatel and combine the results.. Verif the answers b substituting each part into the original equation. Eample : Solve for and for ( + )i i. ( + )i i (Epand) i i i (Separate Real from Imaginar) i i i i + i i i i Verif: (() + )i i i i i ()i Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
20 Chapter : Linear and Quadratic Functions PreCalculus Multipling Comple Numbers: - multipl numerical coefficients and recall that i. Eample : Simplif the following. a. (i)(i) b. (i) c. ( + i)( i) d. ( i) i i Conjugate of Comple Number (a + bi) is a conjugate of (a bi) Eample : Simplif the following. + i + i i (Epand) + i () + i Product of Conjugate Comple Numbers (a + bi)(a bi) a bi a b () (a + bi)(a bi) a + b a. ( i)( + i) b. ( + i)( i) c. ( i) ( + i) + i i i () + i i i () ( i)( i) i i + i i + () i [( i)( + i)] [ + i i i ] [ ()] [] Rationalizing Comple Numbers: - similar to rationalizing radical binomial epression, we multipl b a fraction consists of the conjugate of the denominator over itself. Eample : Simplif the following. a. i ( i ( + i ) ) ( + i) ( + i) + i i i ( + i) ( ) ( + i) ( + i) + i b. i + i ( i ) ( i ) ( + i) ( i) i i + i i + i i i + ( ) ( ) i ( i) i + i c. i ( + i ) ( + i ) ( i) ( + i) i i + i i + i i i + ( ) ( ) i - Assignment: pg. # to (Omit #,, ) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
21 Chapter : Linear and Quadratic Functions -: Solving Quadratic Equations Roots: - solution to an algebraic equation; commonl called zeros or -intercepts. There are several was of solving quadratic equations Solve Quadratic Equations b Factoring:. Epand, Simplif, Cross-Multipl, and /or Rearrange the quadratic equation into the general form a + b + c (with the preference for a > ).. FACTOR the resulting general quadratic equation.. EQUATE each factor to and solve for. Eample : Solve the following quadratic equations. a. b. + A B means A and / or B m m + m (Rearrange Terms) ( + )( + ) ( + )( + ) m(m + ) (m + ) m + m m + m m (m + )(m ) m m Sometimes, when the quadratic equation is alread in standard form, ou ma find that it cannot be factored b conventional approach. In those cases, we might need to solve b completing the square. Epand and Simplif a ( h) + k a + b + c Standard Form General Form Complete the Square To Complete the Squares from General Form to Standard Form. Factor out the leading coefficient if a is NOT out of the a and b terms.. Group the first two terms in a bracket.. Complete the Square in the bracket b adding another constant. Be sure to subtract this constant at the end of the equation. Also be aware that sometimes we have to subtract the PRODUCT of this constant and a (when a is NOT ).. Factor the perfect trinomial and combine the like terms at the end of the equation.. Solve for. Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
22 Chapter : Linear and Quadratic Functions PreCalculus Eample : Solve b completing the square. All solutions must be in the simplest radical form. a. b. + ( ) ( + ) ( ) ( ) ( ) ( + ) ( ) ( )( + ) + + ( )( + ) ( + ) ( + + ) ( + ) ( ) ± ± ( + ) ( + ) ± + ± When we need to find the solution to a non-factorable quadratic equation algebraicall, we need to complete the square. As we have seen before, the algebra can get quite long and tedious. In order to speed up this process, we need to find a better wa to solve for non-factorable quadratic equations. To determine EXACT SOLUTIONS of a NON-Factorable Quadratic Equation, use the b ± b ac Quadratic Formula: (MEMORIZE!!) a Eample : Use the quadratic formula to find the solutions of the following quadratic equations. a. + b. + + a b c () b ± b ac ( ) ± ( ) ( )( ) a () + a b c ± ± ± b ± b ac ± a Page. ± ( ) ( ) ( )( ) ( ) No Real Solutions Coprighted b Gabriel Tang B.Ed., B.Sc.
23 Chapter : Linear and Quadratic Functions Discriminant: - the part of the quadratic formula that determine the tpe of solution(s) of the equation b ac > (Discriminant is Positive) Two Distinct Real Solutions Sometimes, due to various algebraic steps involved in solving an equation, a root can be omitted or an etra root ma appeared. Gaining a Root: - when both sides of the equations are squared. - all roots must be verified against the original equation and all etraneous roots must be rejected. Eample : Solve Quadratic Formula: b ± ( )( ) ( )( + ) ( )( ) ( ) ( ( ) ( ) ) + + ( )( + ) b ac a Determinant b ac For : b ac (Discriminant ) Equal Real Solutions Coprighted b Gabriel Tang B.Ed., B.Sc. Page. - b ac < (Discriminant is Negative) Imaginar Solutions (No Real Solutions) Verif b substitution into the original equation. For : undefined undefined is an Etraneous Root
24 Chapter : Linear and Quadratic Functions PreCalculus Losing a Root: - commonl occur when dividing both sides of the equation b a common factor - when the same variable common factor appears on both sides of the equation, one must epand each sides before solving. - the onl eception to the above rule is if the common factor was numerical onl (without variable). Eample : Solve ( + )( ) ( ) The Correct Method: ( + )( ) ( ) (Epand both sides) + (Quadratic Equation) + (Collect all terms on one side) ( )( ) (Factor resulting equation) Verif Both Roots: For, (() + ) (() ) () (() ) ()() ()() The Incorrect Method: For, + ( ) ( ) ( + )( ) ( ) (Cancel Variable Common Factor on both sides WRONG!) + (Missing a Root!) - Assignment: pg. # to (odd) and (Omit #) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
25 Chapter : Linear and Quadratic Functions -: Quadratic Functions and their Graphs Verte at (p, q) g() is opening DOWN (a < ) (Verte is a Maimum) Maimum at q g () a ( p) + q Equation for Ais of Smmetr p Domain: R Range: g() q Eample : Using a graphing calculator, find the coordinate of the verte of. Determine the direction of the opening. State the maimum or minimum value of the parabola and the range. Indicate the equation of the ais of smmetr. Find the - and -intercepts. Using the graphing calculator to find the verte Graph the Equation. Run Minimum function (if a <, then run the Maimum function). GRAPH nd CALC a. Take cursor left b. Take cursor right of the ENTER of the verte. verte. ENTER TRACE f () a ( p) + q Verte at (p, q) f() is opening UP (a > ) (Verte is a Minimum) Minimum at q Equation for Ais of Smmetr p Domain: R Range: g() q The process of completing the square from the general form, a + b + c, to the standard form, a ( p) + q, has the following generalization. a ( b a ) + ac b a p b a and q ac b a Be sure the verte is on the screen. ENTER c. Press again ENTER d. Read the Result. V(,.) Range:. Minimum at. opens up (a > ) Ais of Smmetr: Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
26 Chapter : Linear and Quadratic Functions Using the graphing calculator to find -intercepts.. Graph the Equation. Run Zero (-intercept) function GRAPH nd CALC a. Take cursor left of the desired -intercept. TRACE PreCalculus b. Take cursor right of the desired -intercept. Be sure all -intercepts are on the screen. c. Press ENTER again ENTER d. Read the Result. ENTER. Redo the same steps for the other -intercept. ENTER ver near zero -intercept (let ) () () -int Eample : Turn the general equation,, into the standard form, a( p) + q. Find the coordinate of the verte. Determine the direction of the opening. State the maimum or minimum value of the parabola and the range. Indicate the equation of the ais of smmetr. Find the - and -intercepts. ( + ) ( + + ) + ( + ) + V, opens down (a < ) Maimum at Range: Ais of Smmetr: -intercept (let ) () () -intercepts (let ) b ± b ac a -int V, ( ) ± ( ) ( )( ) ( ) ± ± ints ± ± Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
27 Chapter : Linear and Quadratic Functions Eample : Find the intersecting points between + and + using a. an algebraic method. b. a graphing calculator. a. Using the substituting method b equating (since both equations have alread isolated ). + and ( + ) ( + )( ) For : () + For : () + and Intersecting Points: (, ) and (, ) b. Using a graphing calculator,. Enter the equations. Graph Y GRAPH. Run INTERSECT nd CALC TRACE a. Bring Cursor near the left intersecting point b. Press three times ENTER. Run INTERSECT nd CALC TRACE a. Bring Cursor near the right of intersecting point b. Press three times ENTER - Assignment: pg. #a & b, to (odd) (Omit #) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
28 Chapter : Linear and Quadratic Functions PreCalculus -: Quadratic Models Eample : The dimensions of a garden are m b m. It is to be surrounded b a brick path that has the same width on all sides of the garden. The area of the brick path itself has to be the same as the area of the garden. Determine the width of the garden. Let the width of the brick path Garden Area m m m Brick Path Area ( + )( + ) m m ( + ) Garden Area Brick Path Area ( + )( + ) ( ) + (Solve for factor) ( + ) ( )( + ) (not possible negative width). m The width of the brick path is. m ( + ) Eample : A marketing firm for the Santa Clara Transit has determined that there will be, less people riding the light rail for ever five cents increased on the fare. There are currentl, people ride the light rail at $. on a dail basis. At what price should the Santa Clara Transit charge per fare to ield the maimum profit? Let the number of cents increase on the fare. (If, the new fare is $.; if, the new fare is $., and so on.) Revenue Fare Number of Riders Ma Revenue (. +.)(,,) Ma Revenue,, +, Ma Revenue + +, (Ma/Min Problem Complete the Square) Ma Revenue ( + ) +, Ma Revenue ( + ) +, + Ma Revenue ( ) +, V(, $,) Since, the amount of fare increase is ($.) $. Hence, the new fare is $. + $. $. Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
29 Chapter : Linear and Quadratic Functions Eample : The height of an object thrown upward in metres can be epressed as a function of time in seconds b h(t) t + v i t + h i. The initial velocit is represented b v i in m/s and h i is the initial height in metres. A cannon ball was fired with a height of m and a initial velocit of m/s. a. State the function of height in terms of time and sketch the graph representing the function. b. Find the maimum height of the cannon ball and the time it happened. c. Determine the time at which the object landed on the ground. a. h(t) t + v i t + h i v i m/s h i m h(t) t + t + b. Maimum Height need to Complete the Square. h(t) t + t + h(t) (t t ) + h(t) (t t + ) + + h(t) (t ) + V( sec, m) The maimum height is at m in seconds. Time (s) Height (m) - Height (m) Cannon Ball Height vs. Time Time (s) Maimum Height V( sec, m) Landing Time (. sec, m) c. Time takes to land need to find t-intercept (at m, t?) h(t) t + t + (t t ) t t t b ± b ac a ( ) ± ( ) ( )( ) () Coprighted b Gabriel Tang B.Ed., B.Sc. Page. ± ± t. seconds and. seconds (time cannot be negative) The time to land is. seconds. ±
30 Chapter : Linear and Quadratic Functions PreCalculus Eample : Three-tiers wedding cakes are sold in sizes of inches, inches and inches in diameter. Their prices are $, $ and $ respectivel. Determine the quadratic function that models the prices of the wedding cakes in terms of their diameters b a. an algebraic approach. b. using the graphing calculator. a. using an algebraic approach. P(d) ad + bd + c (, ), (, ), (, ) a() + b() + c a + b + c a() + b() + c a + b + c a() + b() + c a + b + c Combining the first and second equations: Combining the second and third equations: a + b + c a + b + c ( a + b + c) ( a + b + c) a + b a + b a + b ( a + b) a a Substitute a : a + b () + b b b P(d) d + d + Substitute a and b : a + b + c () + () + c + + c c b. Using a Graphing Calculator,. To Enter a Table of values, use the STAT menu. STAT Choose Option ENTER Enter Coordinates When finish entering, nd QUIT MODE. Obtaining Equation with Correlation of Determination: a. Setting DiagnosticOn for Correlation of Determination nd ENTER CATALOG Again and Move Down using ENTER Select DiagnosticOn Note: After DiagnosticOn is selected; it will remain ON even when the calculator is turned Off. However, resetting the calculator will turn the Diagnostic Off (factor setting). Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
31 Chapter : Linear and Quadratic Functions b. Use Quadratic Regression to obtain equation STAT Select CALC, use Choose Option QuadReg ENTER ENTER Again The quadratic equation is + +. The function that models the price of a wedding cake in terms of its diameter is P(d) d + d + Correlation of Determination (r ) is, which means the equation is an ecellent fit for the coordinates. (The closer it is to, the better the fit.) - Assignment: pg. # to (odd) (Omit #) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Unit 3: Relations and Functions
Unit 3: Relations and Functions 5-1: Binar Relations Binar Relation: - a set ordered pairs (coordinates) that include two variables (elements). (, ) = horizontal = vertical Domain: - all the -values (first
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 informationUNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
More informationAlgebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.
Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the U-shaped graph
More informationCHAPTER 2 Polynomial and Rational Functions
CHAPTER Polnomial and Rational Functions Section. Quadratic Functions..................... 9 Section. Polnomial Functions of Higher Degree.......... Section. Real Zeros of Polnomial Functions............
More informationf(x) = 2x 2 + 2x - 4
4-1 Graphing Quadratic Functions What You ll Learn Scan the tet under the Now heading. List two things ou will learn about in the lesson. 1. Active Vocabular 2. New Vocabular Label each bo with the terms
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
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 informationCh 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.
Ch Alg L Note Sheet Ke Do Activit 1 on our Ch Activit Sheet. Chapter : Quadratic Equations and Functions.1 Modeling Data With Quadratic Functions You had three forms for linear equations, ou will have
More informationModule 2, Section 2 Solving Equations
Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying
More informationUnit 2: Rational Expressions
Rational Epressions Pure Math 0 Notes Unit : Rational Epressions -: Simplifing Rational Epressions Rational Epressions: - fractions with polnomials as numerator and / or denominator. To Simplif (Reduce)
More informationReady To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions
Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte
More informationStudy Guide and Intervention
6- NAME DATE PERID Stud Guide and Intervention Graphing Quadratic Functions Graph Quadratic Functions Quadratic Function A function defined b an equation of the form f () a b c, where a 0 b Graph of a
More informationPreCalculus. Ocean Township High School Mathematics Department
PreCalculus Summer Assignment Name Period Date Ocean Township High School Mathematics Department These are important topics from previous courses that ou must be comfortable doing before ou can be successful
More informationAnswers. Chapter Warm Up. Sample answer: The graph of h is a translation. 3 units right of the parent linear function.
Chapter. Start Thinking As the string V gets wider, the points on the string move closer to the -ais. This activit mimics a vertical shrink of a parabola... Warm Up.. Sample answer: The graph of f is a
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 informationReteaching (continued)
Quadratic Functions and Transformations If a, the graph is a stretch or compression of the parent function b a factor of 0 a 0. 0 0 0 0 0 a a 7 The graph is a vertical The graph is a vertical compression
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 informationKeira Godwin. Time Allotment: 13 days. Unit Objectives: Upon completion of this unit, students will be able to:
Keira Godwin Time Allotment: 3 das Unit Objectives: Upon completion of this unit, students will be able to: o Simplif comple rational fractions. o Solve comple rational fractional equations. o Solve quadratic
More information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
More informationMini-Lecture 8.1 Solving Quadratic Equations by Completing the Square
Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.
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 informationLearning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1
College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,
More informationAlgebra 2 Honors Summer Packet 2018
Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear
More informationMath 030 Review for Final Exam Revised Fall 2010 RH/ DM 1
Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab
More informationREVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES
Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.
More informationEssential Question How can you use a quadratic function to model a real-life situation?
3. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..E A..A A..B A..C Modeling with Quadratic Functions Essential Question How can ou use a quadratic function to model a real-life situation? Work with a partner.
More informationThe standard form of the equation of a circle is based on the distance formula. The distance formula, in turn, is based on the Pythagorean Theorem.
Unit, Lesson. Deriving the Equation of a Circle The graph of an equation in and is the set of all points (, ) in a coordinate plane that satisf the equation. Some equations have graphs with precise geometric
More informationCoordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
More informationAlgebra 2 Semester Exam Review
Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation
More informationAdditional Factoring Examples:
Honors Algebra -3 Solving Quadratic Equations by Graphing and Factoring Learning Targets 1. I can solve quadratic equations by graphing. I can solve quadratic equations by factoring 3. I can write a quadratic
More informationa [A] +Algebra 2/Trig Final Exam Review Fall Semester x [E] None of these [C] 512 [A] [B] 1) Simplify: [D] x z [E] None of these 2) Simplify: [A]
) Simplif: z z z 6 6 z 6 z 6 ) Simplif: 9 9 0 ) Simplif: a a a 0 a a ) Simplif: 0 0 ) Simplif: 9 9 6) Evaluate: / 6 6 6 ) Rationalize: ) Rationalize: 6 6 0 6 9) Which of the following are polnomials? None
More information2.1 The Rectangular Coordinate System
. The Rectangular Coordinate Sstem In this section ou will learn to: plot points in a rectangular coordinate sstem understand basic functions of the graphing calculator graph equations b generating a table
More informationUnit 4 Practice Problem ANSWERS
Unit Practice Problem ANSWERS SECTION.1A 1) Parabola ) a. Root, Zeros b. Ais of smmetr c. Substitute = 0 into the equation to find the value of. -int 6) 7 6 1 - - - - -1-1 1 - - - - -6-7 - ) ) Maimum )
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 informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polnomial Degree and Finite Differences 1. Identif the degree of each polnomial. a. 1 b. 0. 1. 3. 3 c. 0 16 0. Determine which of the epressions are polnomials. For each polnomial, state its
More informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationGlossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
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 informationLinear Equation Theory - 2
Algebra Module A46 Linear Equation Theor - Copright This publication The Northern Alberta Institute of Technolog 00. All Rights Reserved. LAST REVISED June., 009 Linear Equation Theor - Statement of Prerequisite
More informationPrecalculus Notes: Unit P Prerequisite Skills
Syllabus Objective Note: Because this unit contains all prerequisite skills that were taught in courses prior to precalculus, there will not be any syllabus objectives listed. Teaching this unit within
More informationAlgebra 2 CPA Summer Assignment 2018
Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou
More information4. exponential decay; 20% 9.1 Practice A found square root instead of cube root 16 =
9.. eponential deca; 0% 9. Practice A.. 7. 7.. 6. 9. 0 7.. 9. 0. found square root instead of cube root 6 = = = 9. = 7, 9. =,.. 7n 7n. 96. =, 97. =, 9. linear function: = + 0 99. quadratic function: =
More informationReview for Intermediate Algebra (MATD 0390) Final Exam Oct 2009
Review for Intermediate Algebra (MATD 090) Final Eam Oct 009 Students are epected to know all relevant formulas, including: All special factoring formulas Equation of a circle All formulas for linear equations
More informationMath Intermediate Algebra
Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and
More informationMATH 60 Review Problems for Final Exam
MATH 60 Review Problems for Final Eam Scientific Calculators Onl - Graphing Calculators Not Allowed NO CLASS NOTES PERMITTED Evaluate the epression for the given values. m 1) m + 3 for m = 3 2) m 2 - n2
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 informationINTRODUCTION GOOD LUCK!
INTRODUCTION The Summer Skills Assignment for has been developed to provide all learners of our St. Mar s Count Public Schools communit an opportunit to shore up their prerequisite mathematical skills
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 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 informationUnit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents
Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE
More informationCourse 15 Numbers and Their Properties
Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.
More informationCOUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
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 informationAPPLIED ALGEBRA II SEMESTER 1 EXAM ITEM SPECIFICATION SHEET & KEY
APPLIED ALGEBRA II SEMESTER 1 EXAM ITEM SPECIFICATION SHEET & KEY Constructed Response # Objective Sllabus Objective NV State Standard 1 Graph a polnomial function. 1.1.7.1 Analze graphs of polnomial functions
More informationLESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
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 informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationChapter 13. Overview. The Quadratic Formula. Overview. The Quadratic Formula. The Quadratic Formula. Lewinter & Widulski 1. The Quadratic Formula
Chapter 13 Overview Some More Math Before You Go The Quadratic Formula The iscriminant Multiplication of Binomials F.O.I.L. Factoring Zero factor propert Graphing Parabolas The Ais of Smmetr, Verte and
More informationUNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Instruction
Prerequisite Skills This lesson requires the use of the following skills: knowing the standard form of quadratic functions using graphing technolog to model quadratic functions Introduction The tourism
More informationH.Algebra 2 Summer Review Packet
H.Algebra Summer Review Packet 1 Correlation of Algebra Summer Packet with Algebra 1 Objectives A. Simplifing Polnomial Epressions Objectives: The student will be able to: Use the commutative, associative,
More informationCHAPTER 3 Polynomial Functions
CHAPTER Polnomial Functions Section. Quadratic Functions and Models............. 7 Section. Polnomial Functions of Higher Degree......... 7 Section. Polnomial and Snthetic Division............ Section.
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition
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 informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationx Radical Sign: Radicand: the number beneath the radical sign
Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing.
More informationQUADRATIC FUNCTION REVIEW
Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important
More informationAlgebra 2 Unit 2 Practice
Algebra Unit Practice LESSON 7-1 1. Consider a rectangle that has a perimeter of 80 cm. a. Write a function A(l) that represents the area of the rectangle with length l.. A rectangle has a perimeter of
More information5.2 Solving Linear-Quadratic Systems
Name Class Date 5. Solving Linear-Quadratic Sstems Essential Question: How can ou solve a sstem composed of a linear equation in two variables and a quadratic equation in two variables? Resource Locker
More informationSummer Packet Honors PreCalculus
Summer Packet Honors PreCalculus Honors Pre-Calculus is a demanding course that relies heavily upon a student s algebra, geometry, and trigonometry skills. You are epected to know these topics before entering
More informationMaths A Level Summer Assignment & Transition Work
Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first
More informationSpeed (km/h) How can you determine the inverse of a function?
.7 Inverse of a Function Engineers have been able to determine the relationship between the speed of a car and its stopping distance. A tpical function describing this relationship is D.v, where D is the
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 informationCHAPTER 8 Quadratic Equations, Functions, and Inequalities
CHAPTER Quadratic Equations, Functions, and Inequalities Section. Solving Quadratic Equations: Factoring and Special Forms..................... 7 Section. Completing the Square................... 9 Section.
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 informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 111.
Algera Chapter : Polnomial and Rational Functions Chapter : Polnomial and Rational Functions - Polnomial Functions and Their Graphs Polnomial Functions: - a function that consists of a polnomial epression
More informationLaurie s Notes. Overview of Section 3.5
Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.
More informationGraph the linear system and estimate the solution. Then check the solution algebraically.
(Chapters and ) A. Linear Sstems (pp. 6 0). Solve a Sstem b Graphing Vocabular Solution For a sstem of linear equations in two variables, an ordered pair (x, ) that satisfies each equation. Consistent
More informationc) domain {x R, x 3}, range {y R}
Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..
More informationAn equation is a statement that states that two expressions are equal. For example:
Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the
More informationUNIT 6 MODELING GEOMETRY Lesson 1: Deriving Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: appling the Pthagorean Theorem representing horizontal and vertical distances in a coordinate plane simplifing square roots writing
More information3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
More informationMth Quadratic functions and quadratic equations
Mth 0 - Quadratic functions and quadratic equations Name Find the product. 1) 8a3(2a3 + 2 + 12a) 2) ( + 4)( + 6) 3) (3p - 1)(9p2 + 3p + 1) 4) (32 + 4-4)(2-3 + 3) ) (4a - 7)2 Factor completel. 6) 92-4 7)
More informationChapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
More informationALGEBRA 1 CP FINAL EXAM REVIEW
ALGEBRA CP FINAL EXAM REVIEW Alg CP Sem Eam Review 0 () Page of 8 Chapter 8: Eponents. Write in rational eponent notation. 7. Write in radical notation. Simplif the epression.. 00.. 6 6. 7 7. 6 6 8. 8
More information8.1 Exponents and Roots
Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents
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 informationa 2 x y 1 x 1 y SOL AII.1a
SOL AII.a The student, given rational, radical, or polnomial epressions, will a) add, subtract, multipl, divide, and simplif rational algebraic epressions; Hints and Notes Rules for fractions: ) Alwas
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 informationQUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta
QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape
More informationmath0320 FALL interactmath sections developmental mathematics sullivan 1e
Eam final eam review 180 plus 234 TSI questions for intermediate algebra m032000 013014 NEW Name www.alvarezmathhelp.com math0320 FALL 201 1400 interactmath sections developmental mathematics sullivan
More informationMath Analysis Chapter 2 Notes: Polynomial and Rational Functions
Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do
More informationChapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100
Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 5 1 Quadratics 5 The Discriminant 54 Completing the Square 55 4 Sketching Parabolas 57 5 Determining the Equation
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 information3.1 Graphing Quadratic Functions. Quadratic functions are of the form.
3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations.
More informationG r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Midterm Practice Exam Answer Key
G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Midterm Practice Eam Answer Key G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s Midterm Practice Eam Answer Key Name:
More informationy = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is
Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,
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 informationPractice Problems for Test II
Math 117 Practice Problems for Test II 1. Let f() = 1/( + 1) 2, and let g() = 1 + 4 3. (a) Calculate (b) Calculate f ( h) f ( ) h g ( z k) g( z) k. Simplify your answer as much as possible. Simplify your
More information