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MATH 1314 College Algebra Matrices Matrices Section Notes: • A matrix is a __________ rectangular _____ array of numbers. • The numbers in a matrix are called ______ terms or _______. entries • An __________ augmented matrix is a matrix that represents a system of equations. Gaussian Method uses these matrices to solve a • The _________ system of linear equations.(Review in textbook.) row operations to eliminate any entry • This method relies on ____________ in the matrix. • Only 3 Row Operations are used on a matrix: Switch any two rows. ________ Multiply any row by a nonzero constant. ________ Add any two rows, …or add a non-zero constant ________ multiple of one row to another. Matrices • A matrix in Row Echelon Form has: main diagonal entries of 1 from upper left to lower right with 0 entries below each main diagonal entry. (The last entry on main diagonal may also be 0.) 1 0 0 1 1 0 1 2 3 7 1 0 Matrices • A matrix in Row Echelon Form can show that the system of linear equations has: only one solution infinitely many solutions or no solution. 1 0 0 𝒙 1 1 0 𝒚 1 1 −4 10 1 −2 𝒛 = 𝒄 1 −5 0 1 0 1 3 −7 0 0 0 0 1 0 0 0 1 0 1 1 −4 10 0 −2 P 1𝑧 = −2 𝑧 = −2 P 0𝑧 = 0 0=0 O 0𝑧 = −2 …here’s how you can tell… matrix is in Row Echelon Form.(read in textbook) last row shows one, many, or no solution exists. Matrices Matrix in Row Echelon Form. 1 – 0 0 1 1 0 1 2 3 7 1 5 – Does this matrix have: Pa) Only one solution b) Infinitely many solutions c) or No solution Matrices • Writing the solution set for Infinitely Many Solutions: 1 −5 0 1 → 𝑥 − 5𝑦 = 1 – Example: 0 1 3 −7 → 𝑦 + 3𝑧 = −7 0 0 0 0 – Solve Row2 equation for y. (Solutions will be in terms of 𝑧.) → 𝑦 + 3𝑧 = −7 𝑦 = −3𝑧 − 7 – Then solve Row1 Equation for x and simplify. You may need to substitute y. → 𝑥 − 5𝑦 = 1 𝑥 = 5𝑦 + 1 𝑥 = 5 −3𝑧 − 7 + 1 𝑥 = −15𝑧 − 35 + 1 𝑥 = −15𝑧 − 34 Matrices HW practice • Solve the system using Gaussian Elimination. Eqt1 2𝑥 + 𝑦 − 𝑧 = 3 Eqt2 −𝑥 + 2𝑦 + 4𝑧 = −3 Eqt3 𝑥 − 2𝑦 − 3𝑧 = 4 • First, rewrite as an augmented matrix. 𝑥 C1 R1 R2 R3 2 −1 1 𝑦 C2 𝑧 C3 1 2 −2 −1 4 −3 constants C4 3 −3 4 Matrices • Use Row Operations to solve. Need a 1 in R1C1. 2 1 −1 3 Try switching R1 and R3 −1 2 4 −3 1 −2 −3 4 1 −2 −3 4 • −1 2 4 −3 These two entries must be eliminated by using 2 1 −1 3 R1 and opposites. Matrices • • 1 −2 −3 −1 2 4 2 1 −1 1 0 2 −2 −3 0 1 1 −1 4 −3 3 4 1 3 Eliminate -1 with its opposite in R1. Use R2 + R1 to replace R2. R2 -1 2 4 -3 R1 + 1 -2 -3 4 0 0 1 1 New R2 Eliminate 2 with its opposite using R1. Use R3 + (-2)R1 to replace R3. R3 2 1 -1 3 (-2)R1 + -2 4 6 -8 0 5 5 -5 New R3 • 1 0 0 −2 −3 0 1 5 5 4 1 −5 C1 is complete, R1 no longer used. Matrices 1 −2 • 0 0 0 5 1 −2 • 0 0 0 1 1 −2 • 0 1 0 0 −3 1 5 −3 1 1 −3 1 1 4 1 −5 4 1 −1 4 −1 1 Need a 1 in R2C2. First try to multiply (1/5)R3. (1/5)R3 0 1 1 -1 New R3 Then switch R2 and R3. Matrix is in Row Echelon Form. Find solution for z in R3. Then use back-substitution to find solution for y and x. Variable coefficients • Matrices 1𝒙 −2𝒚 −3𝒛 = 4 → 𝑥 − 2𝑦 − 3𝑧 = 4 0 𝒙 1𝒚 1𝒛 = −1 → 𝑦 + 𝑧 = −1 0𝒙 0𝒚 1𝒛 = 1 → 𝑧 = 1 • First solution value: 𝑧 = 1 • Back-substitute z into R2 to find y-value: → 𝑦 + 𝑧 = −1 → 𝑦 + (1) = −1 → 𝑦 = −2 • Back-substitute z and y in R1 to find x-value: → 𝑥 − 2𝑦 − 3𝑧 = 4 → 𝑥 − 2(−2) − 3(1) = 4 → 𝑥+4−3=4 → 𝑥=3 Matrices HW practice So, the System of Linear Equations below 2𝑥 + 𝑦 − 𝑧 = 3 −𝑥 + 2𝑦 + 4𝑧 = −3 𝑥 − 2𝑦 − 3𝑧 = 4 has only One Solution, and the solution is: 𝑥 = 3, 𝑦 = −2, 𝑧 = 1 Substitute the solutions into each equation to verify they are correct. Matrices • Translating a written application problem to equations: 1. Read carefully.(read two or more times) 2. Identify and label variables and number of equations. variables: The last question gives you a hint. equations: How many quantity totals are stated? 3. Form system of equations from variables and quantity totals. Matrices • Section 8.2: Exercise #81-College Algebra e-text 1. Read. 2. Variables and Totals: Variables: Delta x, Beta y, Sigma z Totals(equations): painting, drying, polishing Matrices • Section 8.2: Exercise #81-College Algebra e-text 3. Form equations: • • • Painting hours Drying hours Polishing hours Delta 𝑥 Beta 𝑦 Sigma 𝑧 Total hours 10𝑥 + 16𝑦 + 8𝑧 = 240 3𝑥 + 5𝑦 + 2𝑧 = 69 2𝑥 + 3𝑦 + 𝑧 = 41 Now form augmented matrix to solve. MATH 1314 College Algebra Unit 1 Functions and their Graphs Section: 3.1 Functions Section :_____ Reading assignment-Terminology. Fill in the blank correctly. • A relation is: – a _____________ between two sets X and Y. correspondence ? or corollary ? • A function from set X into set Y is: – a relation in which each 𝑥-value is associated with ___________ one 𝑦-value. more than? or exactly ? – The notation for ordered pairs is ______. (𝑥, 𝑦)? or 𝑓(𝑥) ? • Fill in the blank with the correct term below. (range input dependent domain independent output) input domain independent – 𝑥-variable: ____________, ____________, _____________ dependent output range – 𝑦-variable: ____________, ____________, _____________ • Interval Notation for domain: – parentheses mean “not equal to”, brackets mean “equal to”. Functions Section :_____ • Relation or Function?(Refer to definitions.) 2) 1) Relation Function X Y X Y -1 5 8 3 1 -6 4 0 -9 7 -2 3 Each 𝑥-value associated with exactly one 𝑦-value. One 𝑥-value associated with two different 𝑦-values. 4) 3) Function Relation { (-4, 0), (1, 9), (0, -6), (-8, 3), (1, -2) } { (2, 1), (3, 0), (-4, 6), (1, -3), (0, 5) } One 𝑥-value associated with two different 𝑦-values. Each 𝑥-value associated with exactly one 𝑦-value. Functions Section :_____ • Is the equation below a function of 𝑥? 2) −𝑥 2 + 𝑥 + 6 + 𝑦 = 0 1) 𝑥 2 + 𝑦 2 = 9 Solve for 𝑦. Solve for 𝑦. 𝑦2 = 9 − 𝑥2 𝑦2 = ± 9 − 𝑥2 𝑦 = 𝑥2 − 𝑥 − 6 Only one equation for 𝑦. 𝑦 = ± 9 − 𝑥2 𝑦 = 9 − 𝑥 2 or 𝑦 = − 9 − 𝑥 2 Two different equations for 𝑦. 𝑥 2 + 𝑦 2 = 9 is not a function of 𝑥. −𝑥 2 + 𝑥 + 6 + 𝑦 = 0 is a function of 𝑥 . Functions Section :_____ • Function Notation 𝑦 = 𝑓(𝑥): “the value of 𝑓 at 𝑥”. – To find the value of 𝑓(𝑥), substitute 𝑥-value into 𝑓(𝑥) and simplify. Example: Find the following for 𝑓 𝑥 = 𝑥 2 + 2𝑥. a) 𝑓(3) b) 𝑓 𝑥 + 𝑟 𝑓 𝑥 = 𝑥 2 + 2𝑥 𝑓 𝑥 = 𝑥 2 + 2𝑥 𝑓 𝑥 + 𝑟 = 𝑥 + 𝑟 2 + 2(𝑥 + 𝑟) 𝑓 3 = 3 2 + 2(3) = 𝑥 + 𝑟 𝑥 + 𝑟 + 2(𝑥 + 𝑟) = 9 + 6 = 15 = 15 = (𝑥 2 + 𝑥𝑟 + 𝑥𝑟 + 𝑟 2 ) + 2𝑥 + 2𝑟 = 𝑥 2 + 2𝑥𝑟 + 𝑟 2 + 2𝑥 + 2𝑟 Refer to page 210 for other examples. P • Domain of 𝑓(𝑥) is the set of all real numbers 𝑥 that define 𝑓, except 𝑥-values that: – cause division by zero – cause even roots of negative numbers. P Functions Section :_____ 𝑥+3 • Find the domain of 𝑓 𝑥 = . (Write the domain using 𝑥−5 Denominator may need to Interval Notation.) 1). Find restricted 𝑥-value(s) of function. Cannot divide by zero, so 𝑥−5≠0 Restricted value is 𝑥≠5 2). Cross out restricted value(s) −∞ on number line. 3). Write domain interval(s) from left to right. be factored in order to solve for restricted values in other related problems, producing two or more restricted values. ∞ 0 5 −∞ ∞ 0 5 −∞, 5 ∪ (5, ∞) P Functions Section :_____ • Find the domain of 𝑓 𝑥 = 6𝑥 + 18 . (Write the domain using Interval Notation.) 1). Find restricted 𝑥-values of function. 6𝑥 + 18 is not real if 6𝑥 + 18 is negative, so 6𝑥 + 18 ≥ 0 Solve for x: 6𝑥 ≥ −18 𝑥 ≥ −3 All x-values are restricted except 𝑥 ≥ −3. 2). Cross out restricted value(s) on number line. −∞ 3). Write domain interval(s) from left to right. −∞ ∞ -3 -3 0 ∞ 0 [−3, ∞) P Functions Section :_____ • Use the previous two Practice Exercises to find the 𝑥 domain. 𝑓 𝑥 = 𝑥−2 1). Find restricted 𝑥-values of function. First, numerator 𝑥 requires that 𝑥 ≥ 0. Second, denominator requires that 𝑥 − 2 ≠ 0, so 𝑥 ≠ 2. Together, this means 𝑥 ≥ 0, but 𝑥 ≠ 2. 2). Cross out restricted value(s) on number line. −∞ 3). Write domain interval(s) from left to right. −∞ ∞ 0 2 ∞ 0 2 [0,2) ∪ (2, ∞) P MATH 1314 College Algebra Functions and their Graphs Section: 3.2 Graphs of Functions Section :____ • A function can be identified from its graph by Vertical Line the ____________Test. • The Vertical Line Test states: if a vertical line intersects a graph _____________, more than once the graph is ___________________. not a graph of a function • Is this the graph of a function? A vertical line intersects the graph only once. YES! A vertical line intersects the graph more than once. NO! Graphs of Functions Section :____ • Information on a function from its graph: 𝑦 = 𝑓(𝑥), so (𝑥, 𝑦) can also be (𝑥, 𝑓(𝑥)) To find 𝑓(𝑐) on graph, look for 𝑦 on graph when 𝑥 = 𝑐. The 𝑦-value is 𝑓(𝑐). (The value 𝑐 is a real number.) To find 𝑥 when 𝑓 𝑥 = 𝑏, look for 𝑥 on graph when 𝑦 = 𝑏. (The value 𝑏 is a real number.) Domain of graph: domain is set of all 𝑥-values across graph. (Always read domain from graph left to right.) Range of graph: range is set of all 𝑦-values across graph. (Always read range from graph bottom to top.) 𝑥-intercepts: (𝑥, 0) Points where graph intersects 𝑥-axis. 𝑦-intercepts: (0, 𝑦) Points where graph intersects 𝑦-axis. Graphs of Functions Section :___ • Information from the graph of a function. – a) Find 𝑓(−21). What−6 is 𝑦 when 𝑥 is −21? Answer: – b) For what numbers 𝑥 is 𝑓(𝑥) = 0? Answer: −18, −3, 12 – c) For what numbers 𝑥 is 𝑓(𝑥) = 6? Answer: −6, 18 – d) What is the domain of 𝑓? Answer: [−21,18] – e) What is the range? Answer: [−6,9] Graphs of Functions Section :___ • Information from a function about its graph. If 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1: Equation Editor a) Is point (2,5) on graph of 𝑓(𝑥)? Yes Only if 𝑦 = 5, when 𝑥 = 2. 𝑓 𝑥 = 𝑦 = 2𝑥 2 − 𝑥 − 1 𝑦 = 2(2)2 − 2 − 1 𝑦=5 P TI-83/84 : 1) Press Y=. Enter 𝑓(𝑥) in Equation Editor. 2) Press 2nd GRAPH for TABLE. 3) Look for 𝑦 when 𝑥 = 2. X-variable Exponent Graphs of Functions Section :___ • Information from a function about its graph. If 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1: b) If 𝑥 = −3, what is 𝑓(𝑥)? 20 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1 𝑓(−3) = 2(−3)2 − −3 − 1 = 20 P With this information, list a point on graph of 𝑓. (−3,20) Graphs of Functions Section :___ • 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1: c) If 𝑓(𝑥) = −1, what is 𝑥? What point(s) are on graph of 𝑓? 𝑥 = 0, 1 2 1 ( 0, −1 ), 2 1 ( , −1 ) 2 Set 𝑓 𝑥 = −1, solve for 𝑥 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1 = −1 When solving Quadratic equations, always write in Standard Form first. 2 2 Standard Form: 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 2𝑥 − 𝑥 = 0 𝑥 2𝑥 − 1 = 0 𝑥 = 0, 2𝑥 − 1 = 0 𝑥 = 0, 𝑥 = 1 2 P Graphs of Functions Section :___ • 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1: d) What is the domain of 𝑓(𝑥)? −∞, ∞ 1) Does 𝑓 𝑥 show possible division by zero, or possible square roots of negative values? No, so 𝑓(𝑥) can use all real numbers. P Graphs of Functions Section :___ • If 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1: 1 − ,1 2 e) What x-intercept(s), if any, are on graph of 𝑓? x-intercept is 𝑥, 0 , so… Factoring is needed to solve 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1 = 0 the Quadratic equation. (2𝑥 + 1 )( 𝑥 − 1 ) = 0 2𝑥 + 1 = 0, 𝑥−1=0 𝑥=− 1 2 𝑥=1 f) What y-intercept, if any, is on graph of 𝑓? P −1 y-intercept is 0, 𝑦 , so… 𝑓 0 = 2(0)2 −(0) − 1 = MATH 1314 College Algebra Unit 1 Library of Functions Section: 3.4 Library of Functions Section :____ • This section covers several basic functions to begin the main study of this course. • Study and learn all properties and graphs for the functions listed in this section that you were assigned. • The next section will cover a collection of techniques called Transformations to graph functions similar to these basic functions. • After completing the work in this section, your goal is to successfully: – recognize and sketch the graph of each function, – identify the Domain and Range of each function. • If you do not meet this goal, you will need to review this section until you can in order to prepare for Section 3.5. Library of Functions Section :____ • Quick Check! ∞ 3 From your Notes assignment: State the basic function for the given graph shape. Square Function _____________________ −∞ (−∞, ∞) State it’s Domain: _________ [0, ∞) State it’s Range: ________ −3 3 −3 −∞ ∞ Library of Functions Section :____ • Quick Check! ∞ 3 From your Notes assignment: State the basic function for the given graph shape. Reciprocal Function _____________________ −∞ −3 3 −3 −∞ (−∞, 0) ∪ (0, ∞) State it’s Domain: _________________ (−∞, 0) ∪ (0, ∞) State it’s Range: _________________ ∞ Library of Functions Section :____ • Quick Check! ∞ 3 From your Notes assignment: State the basic function for the given graph shape. Cube Root Function _____________________ −∞ (−∞, ∞) State it’s Domain: _________ (−∞, ∞) State it’s Range: ________ −3 3 −3 −∞ ∞ Library of Functions Section :____ • Quick Check! ∞ 3 From your Notes assignment: State the basic function for the given graph shape. Identity Function _____________________ −∞ (−∞, ∞) State it’s Domain: _________ (−∞, ∞) State it’s Range: ________ −3 3 −3 −∞ ∞ Library of Functions Section :____ • Piecewise-defined Function: is a function defined using different equations on different parts of its domain. 𝑥2 Example: 𝑓 𝑥 = 𝑥−3 𝑓 𝑥 = 𝑥 2 is defined only over this interval. −∞ -5 𝑖𝑓 − 2 ≤ 𝑥 < 0 𝑖𝑓 𝑥 ≥ 0 𝑓 𝑥 = 𝑥 − 3 is defined only over this interval. ∞ 5 [ )[ 5 -3 −∞ ) ∞ P Library of Functions Section :____ • Piecewise-defined Function. Practice: 𝑓 𝑥 = 𝑥2 𝑥−3 𝑖𝑓 − 2 ≤ 𝑥 < 0 𝑖𝑓 𝑥 ≥ 0 a) Find 𝑓(−1). 𝑥 = −1 lies in interval −2 ≤ 𝑥 < 0. Find 𝑓(−1) using 𝑓 𝑥 = 𝑥 2 . 𝑓(−1) = (−1)2 = 1 P 𝑥 = 7 lies in interval 𝑥 ≥ 0. b) Find 𝑓 7 . Find 𝑓(7) using 𝑓 𝑥 = 𝑥 − 3. 𝑓 7 = 7 −3= 4 P Library of Functions Section :____ Practice: Read domain interval for each section of piecewise-defined graph given. y 7 Section I x -5 5 -3 Domain intervals: Section I Section II [−5,2] [2,4) Section II Library of Functions Section :____ Practice: Write a definition for the piecewise-defined graph given. 4 𝑓(𝑥) = − 3 𝑥 if −3 ≤ 𝑥 ≤ 0 4 4 𝑚𝑥 + 𝑏 → − 𝑥 + 0 → − 𝑥 3 3 7 (−3, 4) (5,2) Domain for Left line segment is: Left line segment is 𝑦 = 𝑚𝑥 + 𝑏. Find slope 𝑚 and 𝑦-intercept. y −4 -5 P (0,0) 3 5 -3 Complete the second section in the same way. 𝑓(𝑥) = 2 𝑥 if 5 0 ≤𝑥≤ 5 P 2 5 x Review and Complete HW 3.4-Library of Functions MATH 1314 College Algebra Unit 1 Transformations Section: 3.5 Transformations Section :____ • As stated in the textbook for today’s Lecture(Section 3.5), techniques used to graph functions similar to basic functions Transformations are called _______________. three • How many different categories were outlined? ______ • These main categories of transformations were: Reflections ______________ 𝑥 if 𝑦 = −𝑓(𝑥), reflect 𝑓(𝑥) about the ___-axis 𝑦 -axis if 𝑦 = 𝑓(−𝑥), reflect 𝑓(𝑥) about the ___ Stretch/Compress _____________________ vertically if 𝑦 = 𝑎𝑓 𝑥 , 𝑎 > 0, stretch/compress 𝑓(𝑥) __________ if 𝑦 = 𝑓 𝑎𝑥 , 𝑎 > 0, stretch/compress 𝑓(𝑥) ____________ horizontally Shifts ________ horizontally if 𝑦 = 𝑓 𝑥 ± ℎ , ℎ > 0, shift 𝑓(𝑥) ____________ vertically if 𝑦 = 𝑓(𝑥) ± 𝑘, 𝑘 > 0, shift 𝑓(𝑥) __________ Review symmetry of graphs as needed(p165). Transformations Section :____ ∞ Prepare your mind for Transformations: In the following graph: A (3,2) B 1 −∞ -4 -1 ∞ 3 if point A moves to point B, is the x- or y-coordinate value x-coordinate transformed? _____________ −∞ horizontally why? …because point A shifted __________. what operation shifts point A to point B? subtract 7 from x-coordinate __________________________ (3,2) find the new coordinate for point A after (3 − 7,2) it moves to point B. (−4,2) P Library of Functions Section :____ • If a basic function graph changes from its initial direction, shape, or position, then its function has had one or more Transformations _____________ applied. For the given graph shape: ∞ 3 Identify its initial basic function. Square Root Function ____________________ ∞ −∞ −3 3 Explain how the initial basic function graph has −3 been Transformed? −∞ Shifted right two units and shifted up one unit ________________________________________ State the Domain and Range of the [2, ∞) Range______ [1, ∞) Transformed function: Domain ______ Transformations Section :____ Prepare your mind: • If 𝑓 𝑥 = − 𝑥 is a transformed function, 𝑥 Its basic function is: _______. The basic function graph is: Graph of 𝑓 𝑥 = − 𝑥 looks like: (0,0) (1,1)(4,2) (0,0) (1, −1) (4, −2) What effect does leading − sign have reflects graph about 𝑥-axis on basic graph? _________________________ multiply 𝑦-coordinates by −1 on its basic coordinates?________________________ Transformations Section :____ Prepare your mind: • If 𝑓 𝑥 = (𝑥 + 3)2 −2 is a transformed function, 𝑥2 Its basic function is: _______. The basic function graph is: The graph of 𝑓(𝑥) looks like: What effect does + 3 have on basic graph? shifts graph left 3 units ________________________________ subtract 3 from 𝑥-coordinates The coordinate rule is: ____________________________. What effect does − 2 have on basic graph? shifts graph down 2 units ________________________________ subtract 2 from 𝑦-coordinates The coordinate rule is: ____________________________. Transformations Section :____ Local maximum or minimum on graph of 𝑓(𝑥): • A point on section of graph that is higher or lower than any other points around it. Transformations Section :____ Determining increasing, decreasing Intervals 𝑓(𝑥) increases over an open interval of 𝑥-values if graph only climbs from left-right over the interval. 𝑓(𝑥) decreases over an open interval of 𝑥-values if graph only falls from left-right over the interval. Open intervals only use ( ). ∞ Increasing Decreasing Increasing 5 Local −1) maximum (−1,3) (−∞, (3, ∞) (−1, 𝑦) −∞ ∞ -5 5 -5 −∞ 3, 𝑦 Local minimum Transformations Section :____ To find local maximum on TI-83/84: 𝑓 𝑥 = 𝑥 3 − 4𝑥 • Enter equation in Y1 . • Press 2nd TRACE Select command :maximum • Left Bound?: move cursor to left side of maximum, press ENTER. • Right Bound?: move cursor to right side of maximum, press ENTER. • Guess?: press ENTER. For local minimum use :minimum Transformations Section :____ Practice: If 𝑓 𝑥 = 𝑥 3 − 16𝑥, for −7 < 𝑥 < 7 find the following: a) Choose the correct graph of 𝑓(𝑥) below. PA B Before graphing, change WINDOW settings Xmin: −7 Xmax: 7. (Adjust Ymin Ymax as needed.) C Transformations Section :____ Practice: If 𝑓 𝑥 = 𝑥 3 − 16𝑥, for −7 < 𝑥 < 7 find the following: b) 𝑥-intercepts of 𝑓 𝑥 Solve for 𝑥. (or use TI-83) c) 𝑥-intercepts of 𝑓 𝑥 + 2 * 𝑓 𝑥 + 2 → shift 𝑓(𝑥) left 2 units → subtract 2 from 𝑥-values 𝑥 3 − 16𝑥 = 0 𝑥(𝑥 + 4)(𝑥 − 4) = 0 𝑥 = 0, 𝑥 + 4 = 0, 𝑥 − 4 = 0 𝑥 = 0, 𝑥 = −4, 𝑥=4 −4, 0, 4 P −4 − 2 0−2 −6, −2, 2 4−2 P End - Unit 1 Complete all Ch3 assignments in time to take Ch3 Exam by Due Date.