Graphing Calculator Computations A) Write the graphing calculator notation and B) Evaluate each epression. 4 1) 15 43 8 e) 15 - -4 * 3^ + 8 ^ 4/ - 1) ) 5 ) 8 3 3) 3 4 1 8 3) 7 9 4) 1 3 5 4) 5) 5 5) 6) 5000 1 0.045 365 (4*365) 6) 4 6.5 7) 3 7) 3 8) 7 (Use our calculator!!!) 8) 9) 6 3.6 10 4.8 10 7 9) 10) 1 8 3 4 6 5 7 10) 3
Solving Quadratic Equations Outline I Factoring Method II Calculator A. Solve f ( ) g( ) Y1 f( ) Y g( ) Intersection of two graphs B. Solve f ( ) g( ) 0 Y1 f( ) g( ) Y 0 Intersection of two graphs C. Solve f ( ) g( ) 0 Y1 f( ) g( ) Find the Zero(s) of the function III Square Root Method IV Complete the Square V Quadratic Formula
Steps to Factor a Polnomial Prep Arrange in descending order of powers and e) combine like terms. ) Determine our signs:, or ** See Below ** 10 3 + 5 = 3 + 15 I Factor Out the Greatest Common Factor (GCF), e) 3 + 15 = 3( 5) Get a positive leading coefficient. II If the Polnomial has 4 terms or more, e) 3 + + + Factor b Grouping = ( + 1 ) + ( + 1 ) III Factoring Trinomials ( 3 terms ) A. Trial and Error e) ( 1)( ) = + + 1 1) Write down pairs of parentheses ( )( ) ++ + ( + )( ) 3) Factor the front term ( + )( ) 4) Tr different factors of the last term ( + 3) ( 4) until the binomials FOIL to the trinomial. B. Perfect Square Trinomial e) IV Factoring Binomials ( terms ) V + 6 + 9 1 1 1 6 3 4 ( 3)( 3) ( 3) = + + = + A. Difference of Two Squares e) 9 = ( + 3 )( 3 ) B. Sum of Two Squares - Does Not Factor e) + 5 Does Not Factor C. Difference of Two Cubes e) 3 3 = ( )( + + ) D. Sum of Two Cubes e) 3 + 3 = ( + )( + ) Check for Complete Factorization: VI The Polnomial Does Not Factor e) ( )( ) 4 e) 16 = + 4 4 ( 4)( )( ) = + + + 5 + 1 Does Not Factor Determine the signs of the factors: + +, or + + 6 + 5 = ( + 1) ( + 5) SIGNS ARE THE SAME, SIGNS BOTH + 6 + 5 = ( 1 ) ( 5 ) SIGNS ARE THE SAME, SIGNS BOTH ( ) ( ) ( + ) ( ) + 3 = + 3 1 3 = 1 3 SIGNS ARE OPPOSITES: +
Factoring Polnomials 1 Factor the Following Polnomials Completel 1) + 6 + 8 19) + 5 + 6 ) 7 + 10 0) 13 + 3) + 4 5 1) 4 5 4) + 8 + 15 ) 10 + 1 5) + 13 + 1 3) 1 1 1 6) + 8 + 1 4) 8 + 1 7) + 7 + 1 5) 13 + 1 8) + 4 1 6) 1 9) + 1 7) 7 + 1 10) 4 1 8) + 11 1 11) + 3 + 1 9) 10 + 9 1) + 7 + 6 30) + 1 + 7 13) 3 4 31) 6 14) 3 10 3) 3 + 10 15) + 3 + 10 33) + 3 10 16) 8 9 34) + + 1 17) + 6 + 9 35) 8 + 16 18) + 8 16 36) 14 + 48
Solving Equations with our Graphing Calculator Solve the following equations GRAPHICALLY. Round our answer(s) to decimal places. 1) 5 7 = 3 1) ) 0.10 = 3.95 + 0.04 ) 3) 3 7 5 = 0 3) 4) (7 )(5 ) = 0 4) 5) ( ) 3 ( ) ( ) 1 + 3 = 0 5) 6) + 7 = + 1 6) 7) 4 = + 3 7) 8) 1 + 3 = 1 8) 9).5 0.1 = e 9) 10) 000 = 1000 1 + 0.06 1 1 10) 11) log( ) + log( + 3) = 3 11) 1) 3 + 3 = + + 1 1)
The Quadratic Formula If a + b + c = 0 for a 0, then = 4 b ± b ac a E) Solve 3 + 5 + 1 = 0 3) Solve 3 4 = 0 a = 3 b = 5 c = 1 = = ( 5) ( 5) 4( 3)( 1) ( 3) ± 5 ± 13 6 1) Solve + 3 1 = 0 a = b = c = = ( ) ( ) 4 ( )( ) ( ) ± 4) Solve 6 + = 0 ) Solve 6 9 + = 0 a = b = c = 5) Solve 8 10 + 1 = 0 = ( ) ( ) 4 ( )( ) ( ) ±
Solving Quadratic Equations b Completing the Square I Move the Constant Term to the RIGHT side of the equation, Move the and terms to the LEFT side of the equation. II Get a leading coefficient of 1 (i.e. 1 +...). If necessar, divide both sides of the equation b the coefficient of. III Complete the Square b Taking 1 of the coefficient of, Square this result, Then add the squared result to Both Sides of the equation. IV Factor the Perfect Square Trinomial on the left side of the equation. V Use the Square Root Method (take the Square Root of Both SIDES of the equation) to solve the resulting equation. Solve the following Quadratic Equation b Completing the Square + 1+ 4 = 0 + 6 + 9 = 7 + 1+ 4 = 0 4 4 + 1 = 4 ( + 3)( + 3) = 7 ( + 3) = 7 1 4 + = ( + 3) = 7 + 3 = ± 7 + 6 = 1 6 = 3, (3) = 9 = 3 ± 7 + 6 + 9 = + 9 + 6 + 9 = 7
Solve: 1 3 Solve: 3 5
Solving Equations & Inequalities Review Worksheet Solve the following equations b the indicated method. Show our work on another piece of paper. 1) Factoring Method: 8 0 ) Complete the Square Method: 6 10 0 3) Quadratic Formula: 3 5 1 0 4) Radical Equations: 5 5 5) Rational Equations: 1 1 1 3 5 6) Absolute Value Equations: 3 1 7 Solve the following absolute value inequalities, Graph our solution on a number line, and Give the interval notation (i.e. (a,b) or [a,b] ) for our solution. 7) 5 7 8) 7 4
MAC1105 Sample Test 1 (017) Graphing Calculator Name 1) Use our calculator and evaluate the following epressions. Round our answers to 3 decimal places. A) 1, 000 1 0.03 1 4 B) 5 7 = Solve the following equations with our calculator. Round our answers to decimal places. 0. 1 ( 1) 5 ) 3 4 3) 3 8 Comple Numbers 4) Perform the indicated operations. Simplif our answer. A) ( 8 3 i) ( 8 7 i) B) ( 7 i) ( 3 i) C) i( 3 6 i) D) ( 5 4 i)( 3 i) E) 5 4i F) 5 3 i Solve the Following Equations b Hand - EXACT Values. Simplif. Show Your Work. Don t forget: You can check our answers using our graphing calculator 5) 5 4 6 10 6) 15 1 7) 4
π MAC1105 College Algebra Sample Test 1 page of 3 8) 3 7 9) 3 7 0 10) 5 3 1 0 11) 3 7 1) 1 6 13) 3 4 5
π MAC1105 College Algebra Sample Test 1 page 3 of 3 14) Solve b Completing the Square Method onl. Show our work. 1 0 A) Solve the inequalities, B) Graph the solution & C) Epress the Solution using Interval Notation 15) 3 9 Interval Notation 16) 3 7 Interval Notation 17) 1 5 Interval Notation 18) Suppose that a watermelon is dropped from the top of a 400 foot tall building. The formula showing the height of the watermelon at time t is given b: h( t) 16t 400. A) Find the height of the watermelon 3 seconds after dropping it (i.e at t = 3). B) When did the watermelon hit the ground? Show our work. Round our answer to decimal places.
Finding an Equation of a Circle : ( ) ( ) h + k = r 1) ) 3) 4) 5) 6)
Analze the Graph 1) Viewing Rectangle Xmin: Xma: Xscl: ) -intercept(s): 3) -intercept: 4) Function? 5) Domain: 6) Range: 7) Where does f ( ) = 0? List the -values. Ymin: Yma: Yscl: 8) Where is f ( ) < 0? State the -values, interval notation. 9) Where is f ( ) 0? State the -values, interval notation. 10) Find f (). 11) Find f ( 5). 1) How man times does the line = intersect the graph? 13) Where does f ( ) = 4? List the -values 14) Where does f( ) = 5? List the -values 15) Find f( 1) f(). 16) Find 3 f (1). 17) Absolute Maimum value: 18) Absolute Minimum value: 19) Relative Maimum value: 0) Relative Minimum value: 1) Where is the graph increasing? State the -values, interval notation. ) Where is the graph decreasing? State the -values, interval notation. 3) Is the Graph a One-to-One Function?
A RELATION A FUNCTION RELATIONS AND FUNCTIONS - is a correspondence between two variables. - is a set of points. - is a relation such that for each -value in the Domain, there is eactl 1 corresponding -value in the Range. Vertical Line Test for a Function A vertical line can intersect the graph of a function in at most one point, or using an alternate form, If an vertical line intersects the graph at more than one point, then the graph is NOT the graph of a function. ( 5) + ( 3) = 4 Line Segment = 3 Function? Function? Function? Domain: Domain: Domain: Range: Range: Range: = ( ) 5 ( ) = + 1 = 3 + Function? Function? Function? Domain: Domain: Domain: Range: Range: Range:
Finding Equations of Lines Worksheet I Directions: Find an equation of each line (b inspection) and put in the form: = m + b 1) ) 3) 4) 5) 6) 7) 8) 9)
6 1 3 4 5 1.. 3. 4. 5. 6.
I Transformations of Functions Translations A) Move Up B) Move Down C) Move Left D) Move Right E) Combinations of A D II Reflections A) Reflect about the -ais (Upside-Down) III Distortions A) Vertical Stretch (Steeper) B) Vertical Compression (Flatter) C) Horizontal Stretch (Wider) D) Horizontal Compression (Narrower)
Equation of a Parabola = a - h + k Reflect about the ais Translate Right or Left Translate Up or Down Vertical Stretch (steeper): a > 1 or a < 1 Vertical Compression (wider): 1 < a < 1
Transformations of = f() = Vertical Shift Up Vertical Shift Down 4 Horizontal Shift Right 3 Horizontal Shift Left = + = 4 = ( 3) = ( + ) = f() + = f() 4 = f( 3) = f( + ) Vertical Stretch Vertical Compression Reflection about the -ais Upside Down Steeper / Narrower Flatter / Wider Upside Down and Down = 3 3f() = 1 = 1 = = = f( ) = f( ) = f() Steeper, Opens Up, Right, Down 4 Wider, Opens Down, Left 3, Up 1 = 3( ) 4 = ( + 3) +1 = 3f( ) 4 = 1 f( + 3) +1 1
Transform Parabolas Worksheet 1.. 3. 4. 5. 6. 7. 8. 9.
Transformation Worksheet II Find an equation of each parabola and put in the form: = a( h) + k 1) ) 3) 4) 5) 6) 7) 8) 9)
PARENT FUNCTIONS f() = a f() = f() = f() = int( ) = [ ] Constant Linear Absolute Value Greatest Integer 3 f() = f() = f() = f() = Quadratic Cubic Square Root Cube Root 3 f() = a f() = a log 1 f() = f() = ( + 1 )( ) ( + 1)( ) Eponential Logarithmic Reciprocal Rational f() = sin f() = cos f() = tan Trigonometric Functions
TRANSFORM PARENT FUNCTIONS 6 5 4 3 1-6 -5-4 -3 - -1 1 3 4 5 6-1 - -3-4 -5-6 8 7 6 5 4 3 1-8 -7-6 -5-4 -3 - -1 1 3 4 5 6 7 8-1 - -3-4 -5-6 -7-8 1.. 8 7 6 5 4 3 1-8 -7-6 -5-4 -3 - -1 1 3 4 5 6 7 8-1 - -3-4 -5-6 -7-8 8 7 6 5 4 3 1-8 -7-6 -5-4 -3 - -1 1 3 4 5 6 7 8-1 - -3-4 -5-6 -7-8 3. 4. 8 7 6 5 4 3 1-8 -7-6 -5-4 -3 - -1 1 3 4 5 6 7 8-1 - -3-4 -5-6 -7-8 8 7 6 5 4 3 1-8 -7-6 -5-4 -3 - -1 1 3 4 5 6 7 8-1 - -3-4 -5-6 -7-8 5. 6.
Transformation Worksheet 4 Find an equation of each of the following transformed relations. 1) ) 3) 4) 5) 6) 7) 8) 9)
TRANSFORM PARENT FUNCTIONS 5 4 3 1-5 -4-3 - -1 1 3 4 5-1 - -3-4 -5 7. for f( ) = for for 7 6 5 4 3 1-7 -6-5 -4-3 - -1 1 3 4 5 6 7-1 - -3-4 -5-6 -7 8. for f( ) = for for
Find an equation for each part of the face. State restrictions, if necessar. 1) ) for 3) for 4) 5) 6) for 7) for 8) for 9) for 10) for
3 for 5 < f() = + 3 for < 1 3 1 for 1 3 Piecewise-Defined Functions f() = for for for ( ) Find f( 4) Find the -intercept(s) f( ) f(0) f(1) Find the -intercept(s) Domain: Range: Maimum: Minimum Find the Interval(s) on where the function is: Increasing: Decreasing: Constant:
Smmetr about the ais / Even Function ais ais (, ) (, ) ( 3,5) (3, 5) (,0) (, 0) ( 1, 3) (1, 3) f ( ) f( ) 4 Smmetr about the Origin / Odd Function (, ) (1,1) (8, ) (, ) ( 8, ) ( 1, 1) f ( ) f( ) 3 Smmetr about the ais / Not a Function (, ) (, ) ais (3,1) (3, 1) (6, ) (6, ) (11, 3) (11, 3)
Smmetr Is the given relation smmetric about the; an Odd fcn Even fcn origin -ais -ais 4 3 3 4 9
Eamine the Graph 1) Where is f ( ) > 0? ) Where is f ( ) 0? Interval Notation Interval Notation 3) Where is f ( ) increasing? 4) Where is f ( ) decreasing? Interval Notation Interval Notation
Combinations of Functions Let f( ) = 4 and g( ) = 3+ 1 Sum: f( ) + g ( ) = ( f + g)( ) ( ) ( ) 4 + 3 + 1 = + 3 3 f( ) g ( ) = ( f g)( ) Difference: ( ) ( ) 4 3 + 1 = 3 5 Product: f( ) g ( ) = ( fg)( ) ( ) ( ) 3 4 3 + 1 = 3 + 1 4 Quotient: f( ) 4 f = = ( ) for g ( ) 0 g ( ) 3 + 1 g The Domain of the Combination of two functions is the Intersection of the Domains of those two functions. Domain = Domain Domain f ± g f g Domain = Domain Domain f g f g Domain = Domain Domain { g ( ) 0} f/g f g
16 15 14 13 1 11 10 9 8 7 6 5 4 3 1-16 -15-14- 13-1- 11-10 -9-8 -7-6 -5-4 -3 - -1-1 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 - -3-4 -5-6 -7-8 -9-10 -11-1 -13-14 -15-16 1) ) 3) 4) 5) 6) 7) 8) 9) 10)
Analze Graphs # 1) Equation of Graph 3 4 ) Calculator Notation 3) What Tpe of Graph is this? 4) Function? es/no 5) One-to-One Function? es/no 6) State an Smmetr: 7) Domain: 14) Where is f ( ) increasing? State the -values using interval notation 8) Range: 9) -intercept(s): 15) Where is f ( ) decreasing? State the -values using interval notation 10) -intercept: 16) What is the Absolute Maimum value? 11) Where is f( ) 0? State the -values using interval notation 17) What is the Absolute Minimum value? 1) Where is f( ) 0? State the -values using interval notation 18) Are there an Asmptotes? If es, then give the equation(s) 13) Where does f( ) 0? List the -value(s)
The Difference Quotient f( + h) ( + h, f( + h) ) f( ) (, f ( ) ) h + h m sec f( + h) f( ) f ( + h) f( ) ( + h) h 1 = = = 1 The Difference Quotient: f( + h) f( ) h
MAC1105 College Algebra Sample Test Name 1) Use the graph to the right to answer questions A S. A) Is the graph a function? Yes / No B) Domain: C) Range: D) -intercept(s): E) -intercept: F) Absolute Maimum value: G) Absolute Minimum value: For items H L, State the corresponding -values using Interval Notation. H) Where is the graph increasing? I) Where is the graph decreasing? J) Where is the graph constant? K) Where is f( ) 0? L) Where is f( ) 0? M) How man times does the line 1 intersect the graph? N) Find f (3) O) Find f ( 4) P) Find f () Q) Where does f( ) 0? List the corresponding value(s) of. R) Where does f( ) 4? List the corresponding value(s) of. S) Find a piecewise-defined function for the function graphed above: for f( ) for for
π MAC1105 College Algebra Sample Test page of 4 Find an equation of each of the following Lines ) 3) Find an equation of each of the following transformed graphs 4) 5) 6) 7) 8) 9)
π MAC1105 College Algebra Sample Test page 3 of 4 10) Refer to the graph above. 11) Refer to the graph above. 1) Graph 3 5 A) Is it a function? A) Is it a function? on the grid above B) State the Domain B) State the Domain C) State the Range C) State the Range D) Find f ( 3) E) Find f () 13) Graph ( ) ( 3) 16 on the grid to the right 14) Graph 3 for 4 1 f ( ) 1 for 1 3 4 for 3 5
π MAC1105 College Algebra Sample Test page 4 of 4 Smmetr 15) Is B) Is C) Is D) Is E) Is 6 (Yes/No) f ( ) 3 smmetric about the -ais? A) f ( ) 3 smmetric about the origin? B) f ( ) 1 smmetric about the -ais? C) f ( ) ( 3) an even function? D) 3 f ( ) an odd function? E) 16) A) Find an equation of the line that passes through the points (8, ) and (11, 4) Write our answer in slope-intercept form: m b. 17) Let f ( ) 3 5 and g( ) 3. Find the following: A) f ( 4) B) f(1) g(4) C) f g () D) g f (0) E) f g ( ) 18) Find the Difference Quotient of f ( ) 5 3. Diff. Quotient: f ( h) f ( ) h