Exponential functions worksheet with answers pdf. Download for Desktop. Explore and practice Nagwa’s fre...

4.1 Exponential Functions and Their Graphs In this s

Exponential functions are functions of the form f(x) = b x for a fixed base b which could be any positive real number. The inverse of an exponential function is a logarithmic function. CALCULATIONS: Given exponential function is y = q x, also x = 4 and y = 1296. ∴ 1296 = q 4. ⇒ q = 6These math worksheets should be practiced regularly and are free to download in PDF formats. Solving Exponential Equations Worksheet - 1. Download PDF. Solving Exponential Equations Worksheet - 2. Download PDF. Solving Exponential Equations Worksheet - 3. Download PDF. Solving Exponential Equations Worksheet - 4. Download PDF.To get started, we'll try to guess C(a), for a few values of a, by plugging in some small values of h. Example 2.7.1 Estimates of C(a). Let a = 1 then C(1) = lim h → 0 1h − 1 h = 0. This is not surprising since 1x = 1 is constant, and so its derivative must be zero everywhere. Let a = 2 then C(2) = lim h → 0 2h − 1 h.30 sept. 2014 ... Record your answer and fill in the bubbles on your answer document. 10 The graph of the exponential function y = -400 5x is shown below. y x.Identify functions using differences or ratios EXAMPLE 2 Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function. ANSWER The table of values represents a quadratic function. x –2 –1 0 1 2 y –6 –6 –4 0 6Identify functions using differences or ratios EXAMPLE 2 Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function. ANSWER The table of values represents a quadratic function. x –2 –1 0 1 2 y –6 –6 –4 0 6 Apr 10, 2022 · 4. Why is \(b = 1\) excluded as a base in the definition of exponential functions? Explain. 5. Explain why an exponential function of the form \(y = b^{x}\) can never be negative. Answers to odd exercises: 1. Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original. 3. Writing Exponential Functions from Tables I can write a function from a table. I can write an exponential function from a table, using common ratios. E X P O N E N T I A L E Q U A T I O N Exponential function formula: y=a(B)x What do you plug in? a = B = Starting point Common ratio The amount it multiplies by. What is the first term? Where x=1. …The population is growing at a rate of 15% each year. When something grows at a percent, this is a clue to use exponential functions. Remember, the general form of an exponential function is y = a ( b) x, where a is the beginning value and b is the total growth rate. The beginning value is 20,000.316 cHAptER 5 Exponential Functions and Logarithmic Functions Finding Formulas for inverses Suppose that a function is described by a formula. If it has an inverse that is a func - tion, we proceed as follows to find a formula for f-1. Obtaining a Formula for an Inverse If a function f is one-to-one, a formula for its inverse can generally be found as follows: 1. …Interestingly, it turns out that a horizontal shift of an exponential function corresponds with a change in initial value of the function. Lastly, consider the effect of a vertical shift on an exponential function. Shifting down 4 units would give the equation f (x) 3(2)x 4. Graphing that, notice it is substantially differentlogarithms to solve exponential equations. They need to understand how exponential functions and logarithmic functions are related (they are the inverse of each.Programs such as Microsoft Excel, Apple Numbers and OpenOffice Calc allow users to create purposeful, adaptable spreadsheets. Spreadsheets are computer files that have the appearance of a paper worksheet with the added functionality of a co...EXPONENTIAL FUNCTION. For any real number x, an exponential function is a function with the form. f(x) = bx. where. b is any positive real number such that b ≠ 1. The domain of f is all real numbers. The range of f is all positive real numbers. The y -intercept is (0, 1), and the horizontal asymptote is y = 0.Exponential Functions Date________________ Simplify. Product and power. Keep the same base and include the exponent. Simplify quotients. Your answer should contain only positive exponents. Keep the same base and include the exponent. 3 9) 33 3 10) 32 22 11) 2 22 12) 23 Variable bases. Simplify quotients.an exponential function that is defined as f(x)=ax. For example, f(x)=3x is an exponential function, and g(x)=(4 17) x is an exponential function. There is a big di↵erence between an exponential function and a polynomial. The function p(x)=x3 is a polynomial. Here the “variable”, x, is being raised to some constant power.©A Q2i0 D1K29 JK ku lt Pau lS Vo Lf gtyw Eatr 5ej VLALsCC.H 9 vA pl 0l x 6rli agchZtusm Tr2easheUrjv8e edF. 4 n SMgaSdLek Tw MiQtBh1 8I XnRffi 3n mi0t 4eQ RA7l 2g WepbUrKa1 X1N. g Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Properties of Exponents Date_____ Period____Vanier College Sec V Mathematics Department of Mathematics 201-015-50 Worksheet: Logarithmic Function 1. Find the value of y. (1) log 5 25 = y (2) log 3 1 = y (3) log 16 4 = y (4) log 2 1 8 = y (5) logdifferences follow a pattern similar to the y-values, the graph is exponential. See the examples below for clarity. Examples Based on each table, identify the shape of the graph. Example 1 The difference in y-values is always two, a constant. The graph is linear and is verified at right. Example 2 9 Exponential Word Problems: Growth & Decay Growth Formula: y = a (1 + r) t Decay Formula: y = a (1 – r) t where a = original number; r = rate (% in decimal form); t = time periods Write an exponential function to model each situation. Find each amount at the end of the specified time. Round your answers to the nearest whole number. 1.worksheet for a quadratic exponential worksheet answers and quadratic function that they can exit this? Assign quizizz to determine whether a linear ...1.1 Functions; 1.2 Inverse Functions; 1.3 Trig Functions; 1.4 Solving Trig Equations; 1.5 Trig Equations with Calculators, Part I; 1.6 Trig Equations with Calculators, Part II; 1.7 Exponential Functions; 1.8 Logarithm Functions; 1.9 Exponential and Logarithm Equations; 1.10 Common Graphs; 2. Limits. 2.1 Tangent Lines and Rates of …3. f x 1.25x x y - 3 - 2 - 1 0 1 2 3 1. Is the function increasing or decreasing? 2. Find the domain and range of the function.Exponential Functions 2 Exponential Functions: A basic exponential function has the form f x b y a b( ) or xx, where the base b is any positive real number other than 1 , x the exponent is any real number, and the constant a is any real number. Ex 3: Complete the table for the exponential function 3 2 x gx §· ¨¸ ©¹ and sketch its graph. Exponents Worksheets. Create an unlimited supply of worksheets for practicing exponents and powers. Students can solve simple expressions involving exponents, such as 3 3, (1/2) 4, (-5) 0, or 8 -2, or write multiplication expressions using an exponent. The worksheets can be made in html or PDF format (both are easy to print). End of Unit, Review Sheet. Exponential Growth (no answer key on this one, sorry) Compound Interest Worksheet #1 (no logs) Compound Interest Worksheet (logarithms required) Exponent Worksheets. Simplify Rational Exponents. Solve Equations with Rational Exponents.Exponential Functions: A basic exponential function has the form ( x ) b x or y a bx , where the base b is any positive real number other than 1 , x the exponent is any real number, and the constant any real number. Ex 3: −3 3 x and sketch its graph. 2 Complete the table for the exponential function g ( x ) y Ex 4:Answers to odd exercises: 1. Since the functions are inverses, their graphs are mirror images about the line \(y-x\). So for every point \((a,b)\) on the graph of a logarithmic function, there is a corresponding point \((b,a)\) on the graph of its inverse exponential function. 3. Shifting the function right or left and reflecting the function …ANSWER KEY 1. A. Exponential Growth B. 1200 C. 0.3 or 30% 2. A. Exponential Decay B. 55 C. 0.02 or 2% 3. A. Exponential Growth B. 100 C. 0.25 or 25% 4. A. Exponential Decay B. 5575 C. 0.35 or 35% 5. A. Exponential Growth B. 2000 C. 0.05 or 5% 6. A. Exponential Decay B. 14000 C. 0.08 or 8% 7. A. Exponential Decay B. 2250 …Save as PDF Page ID ... Steps for Solving an Equation involving Exponential Functions. Isolate the exponential function. ... It is a fair question which we answer unfairly: it’s our book. This page titled 6.3: Exponential Equations and Inequalities is shared under a CC BY-NC-SA 3.0 license and was authored, ...278 Chapter 6 Exponential Functions and Sequences 6.1 Lesson WWhat You Will Learnhat You Will Learn Use zero and negative exponents. Use the properties of exponents. Solve real-life problems involving exponents. Using Zero and Negative Exponents Using Zero and Negative Exponents Evaluate each expression. a. 6.70 b. (−2)−4 SOLUTION a. Selected Answers Index Acknowledgments Brief Contents Also Available Algebra 2 Foundations Series. Chapter 1: Expressions, Equations, and Inequalities Get Ready! 1 ... 7-2 Properties of Exponential Functions 442 7-3 Logarithmic Functions as Inverses 451 Concept Byte: TECHNOLOGY Fitting Curves to Data 459 Mid-Chapter Quiz 461Determine if the following functions are linear, exponential, or quadratic, then describe all of the transformations that occur. 4. x0 (1 transformation) 5. fx)x (1 transformation) 6. 1 2 2 f x x (1 transformation) 7. x7 (2 transformatons) 8. f …EXPONENTIAL FUNCTION. For any real number x, an exponential function is a function with the form. f(x) = bx. where. b is any positive real number such that b ≠ 1. The domain of f is all real numbers. The range of f is all positive real numbers. The y -intercept is (0, 1), and the horizontal asymptote is y = 0.Download for Desktop. Explore and practice Nagwa’s free online educational courses and lessons for math and physics across different grades available in English for Egypt. Watch videos and use Nagwa’s tools and apps to help students achieve their full potential.Finding angles of triangles. Finding side lengths of triangles. Statistics. Visualizing data. Center and spread of data. Scatter plots. Using statistical models. Free Algebra 1 worksheets created with Infinite Algebra 1. Printable in convenient PDF format.L 1 lMYaEdje P awWiztGhE MIHnyfYiCn7iPtxe v tA SlZg ieWbDr4ai K2r. m Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Exponential Equations Not Requiring Logarithms Date_____ Period____ Solve each equation. 1) 42 x + 3 = 1 2) 53 − 2x = 5−x 3) 31 − 2x = 243 4) 32a = 3−a ©9 U2e041 q3Y 1KQuUtka 5 TSXobfXt2w ca qr YeS XLoL oCg.e 3 ZARlslr grri Tg9hwtbsl ir9e RsbeurYvde PdN.0 9 yMZaud aeC 5wvi Ptshx cIsn 0fYi3n kiWt0e0 ZCbail 9c ju clFuAsg.R Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Name_____ Differentiation - Logs and Exponentials Date_____ Period____4.1 Exponential Functions and Their Graphs In this section you will learn to: • evaluate exponential functions • graph exponential functions • use transformations to graph exponential functions • use compound interest formulas An exponential function fwith base bis defined by f ( or x) =bx y=bx, where b> 0, b ≠ 1, and xis any real number.Download for Desktop. Explore and practice Nagwa’s free online educational courses and lessons for math and physics across different grades available in English for Egypt. Watch videos and use Nagwa’s tools and apps to help students achieve their full potential.Exponential functions change at a vastly increased or decreased rate than your everyday function. As a result of this, you should expect graphed curved to move quickly up or down. These worksheets explain how to solve exponential functions, to include finding domains, ranges, and values. Your students will use these worksheets in order to ...24/7 Homework Help. Stuck on a homework question? Our verified tutors can answer all questions, from basic math to advanced rocket science! Post question.Vanier College Sec V Mathematics Department of Mathematics 201-015-50 Worksheet: Logarithmic Function 1. Find the value of y. (1) log 5 25 = y (2) log 3 1 = y (3) log 16 4 = y (4) log 2 1 8 = y (5) logANSWER KEY 1. A. Exponential Growth B. 1200 C. 0.3 or 30% 2. A. Exponential Decay B. 55 C. 0.02 or 2% 3. A. Exponential Growth B. 100 C. 0.25 or 25% 4. A. Exponential Decay B. 5575 C. 0.35 or 35% 5. A. Exponential Growth B. 2000 C. 0.05 or 5% 6. A. Exponential Decay B. 14000 C. 0.08 or 8% 7. A. Exponential Decay B. 2250 …Functions - Exponential Functions Objective: Solve exponential equations by finding a common base. As our study of algebra gets more advanced we begin to study more involved functions. One pair of inverse functions we will look at are exponential functions and logarithmic functions. Here we will look at exponential functions and then we326 Chapter 6 Exponential Functions and Sequences 6.5 Lesson Property of Equality for Exponential Equations Words Two powers with the same positive base b, where b ≠ 1, are equal if and only if their exponents are equal. Numbers 2 If x= 25, then x= 5.If =5, then 2 = 25. Algebra If b > 0 and ≠ 1, then x = by if and only if x = y. WWhat You Will Learnhat …Sketch each of the following. Sketch the graph of f (x) = e−x f ( x) = e − x. Solution. Sketch the graph of f (x) = ex−3 +6 f ( x) = e x − 3 + 6. Solution. Here is a set of practice problems to accompany the Exponential Functions section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at ...Exponents and exponential functions worksheet answers. The questions in these worksheets include direct application of formulas conversion of exponents to logarithms and vice versa as well as work problems. ... Graph Exponential Functions Add to my workbooks 7 Download file pdf Embed in my website or blog Add to Google …Free worksheet(pdf) and answer key on Compound interest. 20 scaffolded questions that start relatively easy and end with some real challenges. Plus model problems explained step by stepExponential Functions: A basic exponential function has the form ( x ) b x or y a bx , where the base b is any positive real number other than 1 , x the exponent is any real number, and the constant any real number. Ex 3: −3 3 x and sketch its graph. 2 Complete the table for the exponential function g ( x ) y Ex 4:The question: can you convert a PDF to a Microsoft Word doc file? The answer: absolutely. This conversion can be accomplished by a few different methods, but here’s one easy — and high-quality — method.EXPONENTIAL FUNCTIONS Determine if the relationship is exponential. If so, determine a function relating the variable. > Is it exponential? no What is the starting point (a)? What is the common ratio (B)? Write the equation in y=a(B)x form: x y 1 26 2 24 3 22 4 20 5 18 > > 22÷24 = .917 20÷22 = .909 > 18÷20=.9 24÷26 = .923 T q zAGl8lY Or7iMgvhMtMsv 3rEefsneSrFvne Ed0. t I 5M Za bdbe M Zw ai jt mhI wIsn Nfmi2n qiktre 1 xAElugUeKbvr7ad D29.H Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Graphing Exponential Functions Date_____ Period____ Sketch the graph of each function. 1)As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers x > 0, S > 0, T > 0 and any positive real number b, where b ≠ 1, log b S = log b T if and only if S = T. For example, If log 2 ( x − 1) = log 2 ( 8), then x ...Divide both sides by the coefficient, 80, to isolate the exponential expression. 2500 80 = 80 80e0.12t 31.25 = e0.12t. Rewrite the equation in logarithmic form. 0.12t = ln(31.25) Divide both sides by 0.04 to isolate t; then use your calculator and its natural log function to evaluate the expression and solve for t.Finding angles of triangles. Finding side lengths of triangles. Statistics. Visualizing data. Center and spread of data. Scatter plots. Using statistical models. Free Algebra 1 worksheets created with Infinite Algebra 1. Printable in convenient PDF format.Lesson 8: Determining an Exponential Function from a Table or Graph Date LESSON Day #1 Ok, so we spent a lot of time focusing on exponential growth and decay problems and how to write a function to model each situation. We used those functions to then determine future values for each situation. But what happens if you don’t know the function? transformations; growth and decay word problems; simple and compound interest application problems. 13 pages + all answer keys included. Terms of Use: This ...Chapter 2 Hyperbolic Functions 38 Activity 6 Find the values of x for which cosh x = 13 5 expressing your answers as natural logarithms. Example Solve the equation 2cosh2x +10sinh2x =5 giving your answer in terms of a natural logarithm. Solution cosh2x = 1 2 ()e2x +e−2x; sinh2x =1 2 ()e2x −e−2x So e 2x +e−2x +5e2x −5e−2x =5 6e 2x ...Practice Worksheet 9.1. Algebra 2. Solving Exponential Equations and. Graphing Exponential Function. Solve each exponential equation given below. 1. 5x²-2x=125.Unit 23: PDF and CDF Lecture 23.1. In probability theory one considers functions too: De nition: A non-negative piece-wise continuous function f(x) which has the property that R 1 1 f(x) dx= 1 is called a probability density function. For every interval A= [a;b], the number P[A] = Z b a f(x) dx is the probability of the event. 23.2.1. Exponential functions Consider a function of the form f(x) = ax, where a > 0. Such a function is called an exponential function. We can take three different cases, where a = 1, 0 < a < 1 and a > 1. If a = 1 then f(x) = 1x = 1. So this just gives us the constant function f(x) = 1. What happens if a > 1? To examine this case, take a numerical ...Exponential Equations Worksheet #1 Name_____ Solve the Exponential Equation. Author ...MGSE9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. • MGSE9-12.F.LE.1a Show that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. (This can be shown by algebraic proof, with a Linear Functions Exponential Functions General Equation Function Notation y ax b f(x) ax b General Equation Function Notation y abx f(x) abx (recall: variable is the exponent for an exponential function) a = b = a = b = x = Exponential Function are able to have both a_____ or _____ rate of change278 Chapter 6 Exponential Functions and Sequences 6.1 Lesson WWhat You Will Learnhat You Will Learn Use zero and negative exponents. Use the properties of exponents. Solve real-life problems involving exponents. Using Zero and Negative Exponents Using Zero and Negative Exponents Evaluate each expression. a. 6.70 b. (−2)−4 SOLUTION a. Now, with expert-verified solutions from enVision Algebra 2 1st Edition, you’ll learn how to solve your toughest homework problems. Our resource for enVision Algebra 2 includes answers to chapter exercises, as well as detailed information to walk you through the process step by step. With Expert Solutions for thousands of practice problems ...LT1 3.1 Exponential Functions and their Graphs Sketch graphs of exponential functions with transformations. LT2 3.1 Exponential Functions and their Graphs Apply compound interest formulas to solve real-world problems. LT3 3.2 Logarithmic Functions and their Graphs Evaluate logarithmic expressions of different bases without a calculator.Repeat Exploration 1 for the exponential function 1 16 . 2 x y = Do you think the statement below is true for any exponential function? Justify your answer. “As the independent variable x changes by a constant amount, the dependent variable y is multiplied by a constant factor.” 1 EXPLORATION: Exploring an Exponential FunctionAlgebra 2 Chapter 10 Worksheet 1—Exponential Functions Author: Smithers403 Last modified by: Smithers403 Created Date: 4/22/2013 3:20:00 AM Company: WSFCS Other titles: Algebra 2 Chapter 10 Worksheet 1—Exponential Functionsan exponential function that is defined as f(x)=ax. For example, f(x)=3x is an exponential function, and g(x)=(4 17) x is an exponential function. There is a big di↵erence between an exponential function and a polynomial. The function p(x)=x3 is a polynomial. Here the “variable”, x, is being raised to some constant power.When working with exponential functions, re-member to use the various rules of exponentia-tion. Here, the rules to use are ea+ b= eae and e ab = e=eb. Answer. 13. Hint. Z 7 p ... Just more practice with trig and exponential functions. Answer. 17. Hint. Z 3 p 7vdv You can write 3 p 7vas 3 p 7 3 p v. And remember you can write 3 p vas v1=3 ...-2- Worksheet by Kuta Software LLC 5) y x x y 6) y ( ) x x y Write an equation for each graph. 7) x y y x 8) x y y x Create your own worksheets like this one with Infinite Precalculus. Free trial available at KutaSoftware.comGraphs of Exponential Functions Example 1) Graph the given function. State the domain and range. List any asymptotes. Is it growth, decay, or neither? a. …How can you solve an exponential equation graphically? Solving an Exponential Equation Graphically Work with a partner. Use a graphing calculator to solve the exponential equation 2.5x 3 6.25 graphically. Describe your process and explain how you determined = the solution. The Number of Solutions of an Exponential Equation Work with a partner. a.Worksheet by Kuta Software LLC Secondary Math 1 Writing Exponential Functions From a Graph Name_____ Date_____ Period____ ©O P2w0X1x8a EK]u`tIaV ASUoKf`tMwRamraeS PLhLdCd.w h AAslElS IryibgjhMthsN ErqeYsVeWrBvOe^dK.-1-Write an equation for each graph. 1) x y-6-4-2246 2 4 6 8 10 12 14 16 18 20 2) x y-6-4-2246 2 4 6 8 10 12 14 16 18 15 jan. 2019 ... Graphing Exponential Functions Math 1 Mr. Trutna Clovis High School Worksheet #2 Solutions to #4.Evaluating Functions Worksheets. Bank on our printable evaluating function worksheets to equip high school students with a sound knowledge and practice in evaluating a variety of functions beginning with linear, moving to quadratic, polynomial, rational, exponential, trigonometry, and piecewise functions. A bunch of revision pdfs with a mix of ...Download for Desktop. Explore and practice Nagwa’s free online educational courses and lessons for math and physics across different grades available in English for Egypt. Watch videos and use Nagwa’s tools and apps to help students achieve their full potential.3. f x 1.25x x y - 3 - 2 - 1 0 1 2 3 1. Is the function increasing or decreasing? 2. Find the domain and range of the function. Worksheet. The following problems will help you in your study about Exponential and Logarithmic Functions and their Applications. This is an extra source for revising the ... Answer: 0.993968 . b) ln 2 3(+) Answer: 1.316958 . c) 1 ln 2 4020. k = Answer: -0.0001724 . Answer: 0.0001394 . d) ln2 5025 k = 14. Use the definition and properties of ...exponential function. Below is some advice that will help you decide. Linear Function Exponential Function f(x) = mx + b or f(x) = m (x t x1) + y1 f(x) = a · bx b is the starting value , m is the rate or the slope . m is positive for growth, negative for decay. a is the starting value , b is the base or the multiplier .Answer 1. Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve. Answer 3Definition of the derivative. Instantaneous rates of change. Power rule for differentiation. Motion along a line. Approximating area under a curve. Area under a curve by limit of sums. Indefinite integrals. Free Precalculus worksheets created with Infinite Precalculus. Printable in convenient PDF format.A function that models exponential growth grows by a rate proportional to the amount present. For any real number x and any positive real numbers a and b such that b ≠ 1, an exponential growth function has the form. f ( x) = a b x. where. a. a is the initial or starting value of the function. b.Printable Worksheets @ www.mathworksheets4kids.com Name: Answer key A) Evaluate each function at the speci!ed value. B) Evaluate each function. =(±5) (±3) = ±54 ± ...As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers x > 0, S > 0, T > 0 and any positive real number b, where b ≠ 1, log b S = log b T if and only if S = T. For example, If log 2 ( x − 1) = log 2 ( 8), then x .... Linear vs Exponential Functions Practice Name _____ Pd____T q zAGl8lY Or7iMgvhMtMsv 3rEefsneSrFvne Ed0. t 76 Exponential and Logarithmic Functions 5.2 Exponential Functions An exponential function is one of form f(x) = ax, where is a positive constant, called the base of the exponential function. For example f(x)=2x and f(x)=3x are exponential functions, as is f(x)= 1 2 x. If we let a=1 in f(x) =ax we get f(x) 1x =1, which is, in fact, a linear function. For this reason we agree that the base of ... Application solutions are available for purchase! Worksheet 2:7 Logarithms and Exponentials Section 1 Logarithms The mathematics of logarithms and exponentials occurs naturally in many branches of science. It is very important in solving problems related to growth and decay. The growth and decay may be that of a plant or a population, a crystalline structure or money in the bank. Therefore The general form of an exponential functio...

Continue Reading