Objectives
1. We will determine if a sequence is geometric. MAFS.912.F-LE.1.2
2. We will analyze real-world examples of arithmetic and geometric sequences. MAFS.912.F-IF.1.3, MAFS.912.F-LE.1.2
3. We will determine if a function is exponential. MAFS.912.F-LE.1.3, MAFS.912.AREI.4.10 , MAFS.912.F-LE.1.1, MAFS.912.A-CED.1.1
4. We will determine key features of an exponential graph and graph the function. MAFS.912.A-REI.4.10, MAFS.912.F-LE.1.3, MAFS.912.F-IF.3.7.e
5. We will graph an exponential function and its transformations. MAFS.912.F-BF.2.3, MAFS.912.F-LE.1.3
6.We will determine whether an exponential function represents growth or decay. MAFS.912.F-LE.1.1
Standards
MAFS.912.A-REI.4.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
MAFS.912.F-LE.1.1DEF Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
MAFS.912.F-LE.1.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
MAFS.912.F-LE.1.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
MAFS.912.F-LE.2.5 Interpret the parameters in a linear or exponential function in terms of a context.
MAFS.912.F-IF.1.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
MAFS.912.F-IF.2.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
MAFS.912.F-IF.2.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
MAFS.912.F-IF.3.7.e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift.
MAFS.912.F-IF.3.8.b Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
b. Use the properties of exponents to interpret expressions for exponential functions.
MAFS.912.F-IF.3.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
MAFS.912.A-SSE.2.3.c Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
c. Use the properties of exponents to transform expressions for exponential functions
1. We will determine if a sequence is geometric. MAFS.912.F-LE.1.2
2. We will analyze real-world examples of arithmetic and geometric sequences. MAFS.912.F-IF.1.3, MAFS.912.F-LE.1.2
3. We will determine if a function is exponential. MAFS.912.F-LE.1.3, MAFS.912.AREI.4.10 , MAFS.912.F-LE.1.1, MAFS.912.A-CED.1.1
4. We will determine key features of an exponential graph and graph the function. MAFS.912.A-REI.4.10, MAFS.912.F-LE.1.3, MAFS.912.F-IF.3.7.e
5. We will graph an exponential function and its transformations. MAFS.912.F-BF.2.3, MAFS.912.F-LE.1.3
6.We will determine whether an exponential function represents growth or decay. MAFS.912.F-LE.1.1
Standards
MAFS.912.A-REI.4.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
MAFS.912.F-LE.1.1DEF Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
MAFS.912.F-LE.1.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
MAFS.912.F-LE.1.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
MAFS.912.F-LE.2.5 Interpret the parameters in a linear or exponential function in terms of a context.
MAFS.912.F-IF.1.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
MAFS.912.F-IF.2.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
MAFS.912.F-IF.2.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
MAFS.912.F-IF.3.7.e Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude and using phase shift.
MAFS.912.F-IF.3.8.b Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
b. Use the properties of exponents to interpret expressions for exponential functions.
MAFS.912.F-IF.3.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
MAFS.912.A-SSE.2.3.c Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
c. Use the properties of exponents to transform expressions for exponential functions
Geometric SequencesConsider the sequence 3, 6, 12, 24, … . What pattern do you notice in the sequence?
We are multiplying by 𝟐 each time. This is an example of a geometric sequence. Each term in the sequence is the product of the previous term and some real number 𝑟. A geometric sequence represents an exponential pattern because the numbers in the sequence rapidly increases or decreases. In order to find the next term in your sequence, all you have to do is take the previous term and multiply it by the common ratio. In order to find a future term, you plug in the number of the term into the exponent and simplify by using order of operations. Key Features of an Exponential GraphIn order to sketch an exponential function, you need to know how to evaluate exponents. Once you know how to solve using exponent rules/properties, you can organize your ordered pairs into a t-chart. Always plug in 0 as a value (y-intercept) and some numbers greater than 0 and some numbers less than 0. This will help you determine if your graph is increasing or decreasing based on the x values.
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Comparing Arithmetic and Geometric SequencesGeometric sequence follows an exponential
pattern. Arithmetic sequences follow a linear pattern. When determining if a situation is geometric or arithmetic, you need to see how the pattern is increasing or decreasing. If the pattern is changing by a multiple, then it's geometric. If the pattern is changing by a common difference (a constant), then it's arithmetic. Exponential TransfomationsHorizontal shifts- occur on the x axis.
To recognize horizontal shifts, there will be a change on the variable which is typically found in the exponent. Vertical shifts- occur on the function (y axis) To recognize vertical shifts, there will be a change to f(x). If there is a negative in the exponent, that signifies a reflection over the y axis. If there is a negative on the function, that signifies a reflection over the x-axis. |
Writing Exponential FunctionsWhen writing an exponential function from a graph or table, determine two features: common ratio and the y-intercept. The y intercept is your a value and the common ratio is your b value.
Follow these steps: 1. Pick two points. It’s helpful to use the 𝑦-intercept and the coordinate where 𝑥 = 1. 2. Substitute the coordinates into the exponential equation 𝑦 = 𝑎𝑏^x. Solve for 𝑎 and 𝑏. 3. Substitute 𝑎 and 𝑏 into the equation 𝑦 = 𝑎𝑏^x Growth and DecayConsider an exponential function in the form 𝑓(𝑥) = 𝑎 ⋅ 𝑏^x.
Assume that 𝑎 (the 𝒚-intercept) is positive. If 𝑏 (the common ratio) is greater than 1, the function is growing. If 𝑏 is between 0 and 1, the function is decaying. You will see the rate of growth/decay expressed as a decimal or a percent. To find the decay rate, you must subtract 𝑏 from 1. To find the growth rate, you subtract 1 from 𝑏. |