Introduction to Functions - CCS
Exponential FunctionsCommon Core Algebra IRecall:Linear functions – constant rate of change through addition or subtraction.Exponential growth and decay – growth through multiplication.Exponential functions have the variable as the exponent.Exponential functionsConsider the pattern found in the last set of notes. New = starting amount ? factor of changenumber of changes This is the format for the formula for exponential functions :12965423456Exponential Functions A general exponential function has the form: y = abx where a is the initial value, b is the growth factor, and x is the number of changes.When graphing, the initial value, a, becomes the starting point, b, in y = mx + b.00Exponential Functions A general exponential function has the form: y = abx where a is the initial value, b is the growth factor, and x is the number of changes.When graphing, the initial value, a, becomes the starting point, b, in y = mx + b.Exercise #1: Consider the exponential function . Answer the following.Evaluate each of the following and indicate what point must lie on the graph of based on each:(i) (ii) (iii) Exercise #1 (con’t) Consider the exponential function . Answer the following.142875278765(b)Calculate the average rate of change of f over the interval -1 ≤ x ≤ 000(b)Calculate the average rate of change of f over the interval -1 ≤ x ≤ 0Calculate the average rate of change of f over the interval 0 ≤ x ≤ 2 .95250167640(d)What does comparing answers from (b) and (c) tell you about this function? Explain.00(d)What does comparing answers from (b) and (c) tell you about this function? Explain.94615360680(e)Using your calculator, draw a sketch of this function on the axes below using the window indicated.00(e)Using your calculator, draw a sketch of this function on the axes below using the window indicated.203334610634350Exercise #2: Consider the function .-1397060325(a)Evaluate . What point does this indicate on the graph of g?00(a)Evaluate . What point does this indicate on the graph of g?-6556375938530(c)Using your graphing calculator, sketch a graph of this function using the window and . Mark the y-intercept.00(c)Using your graphing calculator, sketch a graph of this function using the window and . Mark the y-intercept.-6554470311150(b)Without the use of your calculator, determine the values of .00(b)Without the use of your calculator, determine the values of .607325137453700217805106045(d)Why is this exponential function always decreasing while the one in Exercise #1 is always increasing?00(d)Why is this exponential function always decreasing while the one in Exercise #1 is always increasing?1201003118385Increasing Vs. Decreasing Exponentials will increase if ___________________________ will decrease if ___________________________00Increasing Vs. Decreasing Exponentials will increase if ___________________________ will decrease if ___________________________Exercise #3: For each of the following exponential functions, give its y-intercept and tell whether it is increasing or decreasing.(a) (b) (c) Exercise #4: Find the equation of the exponential function, in form, for the function given in the table below. Show or explain your thinking.x01234y103090270810Introduction to Exponential FunctionsCommon Core Algebra I HomeworkFluency1.Consider the exponential function .67310931545(b)Is this an increasing or decreasing exponential function? How can you tell based on its equation?00(b)Is this an increasing or decreasing exponential function? How can you tell based on its equation?6731043815(a)Find the value of . What point does this represent on the graph of ?00(a)Find the value of . What point does this represent on the graph of ?6794582550(c)Is this function’s average rate of change over the interval greater or less than that of the linear function ? Justify.00(c)Is this function’s average rate of change over the interval greater or less than that of the linear function ? Justify.1624084729473017018040005(d)Using your calculator, sketch a graph of this function on the axes shown below. Use the window indicated. Mark the y-intercept.00(d)Using your calculator, sketch a graph of this function on the axes shown below. Use the window indicated. Mark the y-intercept.2.Which of the following is a decreasing exponential function whose y-intercept is 20?(1) (3) (2) (4) 3.Which of the following functions would best describe the data in the table?313690044450x01234y21050250125000x01234y210502501250(1) (3) (2) (4) 4.Graphing a basic exponential can be challenging because of how quickly they grow (or decay). In this exercise, we will graph one of the most basic.f(x) = 2xEvaluate each of the following and state the coordinate point that occurs on the graph of based on the calculation.(b)Evaluate each of the following. Remember your facts about negative exponents and give the point on the graph of . (c)Using the points you found in (a) and (b), graph this function for the domain interval .187656781876yx00yx5.Classify each of the following exponential functions as either increasing or decreasing and give the value of their y-intercepts.(a) (b) (c) Reasoning4563745169545y1020x00y1020x6.Which of the following could be the equation of the exponential function shown graphed below? Explain your choice.(1) (3) (2) (4) Explanation ................
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