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Warm Up Thursday – have each group do one of the following 1. Make a problem using the zero exponent rule that simplifies to 12x6 2. Make a problem using the power of a quotient that simplifies to 12x6 3. Make a problem using the power of a product rule that simplifies to 12x6 4. Make a problem using the power of a power rule that simplifies to 12x6 5. Make a problem using negative exponents that simplifies to 12x6 Solving Simple Exponential Equations • When we “solve” an equation, we are really setting two equations equal to each other and finding the value of x that satisfies both equations. We could then substitute the solution (the x) into either equation and find the “y” of that point. • Make up two equation, set them equal to each other, and show what “equal” means geometrically on geo sketch pad • Solve as an equality, then “guess and check” to see which side of the equality is the solution. Solving Simple Exponential Equalities One-to-One Property: • For a > 0 and a ≠ 1, ax = ay iff x = y • Example: if 2x = 29, then x = 9 NOTE: we must have the same base. What if we do not have the same base? Solve for x: 3 x 2 9 2 x 9 Practice - Thursday • Packet page 6 Warm Up for Friday 2/10 • Use each power rule (seven of them) to write an expression that reduces to 16x6 and identify the power rule used. • Explain why 00 ≠ 1 • Explain what the domain restrictions to x-1 are. Solving Simple Exponential Inequalities • Go back and review what solving an algebraic equation really means geometrically. • When we are solving an inequality, we want to know the values of “x” that makes the inequality true. We need to know the values of “x” that makes “y” value of one equation greater or less than the other equation. Solving Simple Exponential Inequalities • To solve an inequality: 1. Solve the equation as an equality, 2. then “guess and check” to see which side of the equality satisfies the original equation. Practice - Friday • Packet pages 8, make all the evens less than (<) • Make all the odds greater than (>) Warm Up Monday 2/13 • Create an expression that reduces to 716 by using each exponential property (there are seven of them) Reduce & Solve Exponent Review • Coach, lesson 11, page 77 • American Choice, lesson 2, page 5 & 6 Review of Exponent Rules • Am Ch Lesson 1 • Coach Lesson 4 Properties of Exponents, pg 27 - 30 • Coach Lesson 11 Solving Exponential Equations, pg 73 - 77 • Coach Lesson 12 Solving Exponential Inequalities, pg 78 - 82 Warm up Wednesday 2/15 Simplify: 2x x y 3 3 2 4 x 2 7 0 Standards: MM2A2. Students will explore exponential functions. (Book 4.4, 4.5, 4.6)(Coach 9 – 13)(Am Ch 1 – 7) b. Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rates of change, and end behavior. c. Graph functions as transformations of f x b x • Basic exponential functions (good) • http://www.onlinemathlearning.com/expone ntial-functions.html Graphing exponential functions (Khanacademy.com)(six minutes to make one graph) • http://www.khanacademy.org/video/graphing -exponential-functions?topic=developmentalmath-3 Exponential Functions – Just Changing the Base (Wednesday) • Investigate various exponential functions using the web page: • http://www.mathopenref.com/graphfunctions .html • Use 3 functions: f(x) = a^x, g(x) = b^x, h(x) = c^x, y from -1 to 9, and x from -10 to 10 • Emphasize we are taking a positive number to an exponent. We are NOT playing with stretch, reflection, or translations. We will do those later. Exponential Functions – Just Changing the Base • Look at the graph as “b” increases and decreases • What are the similarities? – The basic shape remains the same – Y-intercept is always 1 • Explain why that would be so – All curves have an asymptote of y = 0 – Domain: All real numbers – Range: y > 0 Exponential Functions – Just Changing the Base • What are the differences? – The curve reacts faster, i.e., gets steeper when “b” gets farther from 1 • Explain why – The curve raises to the right when b > 1 • As x increases, y increases • Explain why – The curve raises to the left when 0 < b < 1. • As x increases, y decreases • Explain what is happening – The graph is a straight line of y = 1 when b = 1 • Why? Make a “T” chart, graph on the same sheet, and explain what happens as the base changes • • • • • • f(x) = 3x g(x) = 2x h(x) = 1.4x i(x) = 0.7x j(x) = 0.5x k(x) = 0.3x Describe the transformation from: f(x) = x2 • g(x) = -3(x – 4)2 + 7 • The 4 moves (translates) the graph 4 units to the right • The 3 stretches the graph vertically by a factor (multiplies) by 3 • The negative in front of the 3 can be thought of as “times -1”, and it reflects the graph over the x-axis. • The 7 moves (translates) the graph up 7 units. Exponential Functions – Transformations • REMEMBER VERTEX FORM OF QUADRATIC: 2 f ( x) a( x h) k • a – vertical stretch • If there is a negative in front of the a, reflects across the x-axis • h – horizontal transformation (shift) • k – vertical transformation (shift) Exponential Functions – Transformations • Investigate various exponential functions using the web page: • http://www.mathopenref.com/graphfunctions .html • Use: f(x) = a*b^(x-c)+d, y from -10 to 10, and x from -10 to 10 • We are now using the function: xh f ( x) a b k Exponential Functions – Transformations • Look at the graph as a, h, and k are varied • What happens? – All constants perform the same transformation as in a quadratic – The “a” stretches the function – The “h” is “inside the house” so it translates the function horizontally – The “k” is “outside the house” so it translates the function vertically, shifting the asymptote from y=0 Exponential Functions – Transformations • Look at the graph as a, h, and k are varied • What happens? – If there is a negative in front of the a, the graph is reflected across the x-axis – If there is a negative in front of the x, the graph is reflected across the y-axis (this is new for you) Exponential Functions – Transformations • How does that change the domain and range? • Domain: is always all real numbers • Range: – y > asymptote if a > 0 – y < asymptote if a < 0 Describe the transformation from f(x) = 3x and give the domain, range, y-intercept end conditions • • • • g(x) = 3*2(x – 2) h(x) = 2-x i(x) = -2x + 3 j(x) = -3*2(-x + 4) - 7 Make a “T” chart, graph f(x) or f’(x) on each graph, and explain what happened from f(x) or f’(x) as a result of changing a, b, h, and/or k, or making a or x negative f(x) = a*b(x – h) + k • • • • • • f(x) = 1.3x g(x) = 1.3(-x) h(x) = 2*1.3x i(x) = 1.3(x + 3) j(x) = 1.3x - 4 k(x) = -1*1.3x • • • • • • f’(x) = 0.7x g’(x) = 0.7(-x) h’(x) = 0.5*0.7x i’(x) = 0.7(x - 3) j’(x) = 0.7x + 4 k’(x) = -1*0.7x