#### Exponential Functions

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###### Exercises
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Exercises 1 Recall that the y-intercept is the y-coordinate of the point whose x-coordinate is x=0. Therefore, the condition for our function is that y=2 when x=0. Let's review the general form of an exponential function. y=abx​ In this form a, and b are real numbers such that a​=0, b>0, and b​=1. Now, let's apply the conditions we mentioned above to a function with this form. y=abxx=0, y=22=ab0 Solve for a a0=12=a⋅1a⋅1=a2=aRearrange equation a=2 As we can see, y(0)=2 if we choose a=2. This happens because if b​=0, b0=1. We can choose any positive value for b, as long as b​=1, to obtain an exponential function that satisfies the exercise's conditions. Therefore, there are infinitely many solutions satisfying the given requirements. For example, y=2(3)x.
Exercises 2 Recall that the y-intercept is the y-coordinate of the point whose x-coordinate is x=0. Therefore, to find it we just need to evaluate the exponential function y=abx at x=0. y=abxx=0y=ab0a0=1y=a⋅1a⋅1=ay=a As we can see, y(0)=a. This happens because b0 is always 1, since for an exponential function b​=0. For this reason, the y-intercept of any exponential function is a.
Exercises 3 We are given the exponential functions shown below for us to compare. We will label them in order to tell them apart. y1​=2(5)x and y2​=5x​ Notice that the first of them is of the form y=abx, while the second one is of the form y=bx. yy1​​=abx=2(5)x   ​​y =bxy2​=5x​ Since a=2>1, we know that y1​=2(5)x is a vertical stretch of y2​=5x by a factor of 2. We can also find their y-intercepts by evaluating the functions at x=0. Recall that for any nonzero number a, a0=1. ​y1​(0)=2(5)0y1​(0)=2(1)y1​(0)=2​​y2​(0)=50y2​(0)=1​ We can also visualize these differences by graphing both functions together.From the graph, we can confirm that the function y1​=2(5)x is a vertical stretch of y2​=5x by a factor of 2. We also see that the y-intercept of y1​ is 1, while the y-intercept of y2​ is 2.
Exercises 4 We are given the equations shown below and are asked to find the one that is different. We will label each of them in order to tell them apart. ​A) y=3x​C) f(x)=(-3)x​​​B) f(x)=2(4)x​D) y=5(3)x​​ All of these equations have a similar format, since they all have a power with the variable x as the exponent. Therefore, we can check if they are exponential functions. Recall that an exponential function has the form shown below. y=abx​ Here, a, and b are real numbers such that a​=0, b>0, and b​=1. We can compare this form to the equations given. ​A) y=1(3)x​C) f(x)=1(-3)x​​​y=abxB) f(x)=2(4)x​D) y=5(3)x​​ As we can see, all the equations represent exponential functions except for C, f(x)=(-3)x, since it has a negative b-value, b=-3. Therefore, this is the odd one out.
Exercises 5 We want to determine whether the given rule represents an exponential function. Notice that the independent variable x is an exponent. Therefore, the rule represents an exponential function. General form​y=a⋅bx⇔y=a(b)xGiven function​y=4(7)x​
Exercises 6 We want to determine whether the given rule represents an exponential function. Notice that the independent variable x is not an exponent. Therefore, the rule doesn't represent an exponential function. y=-6x​
Exercises 7 We want to determine whether the given rule represents an exponential function. Notice that the independent variable x is not an exponent. Therefore, the rule doesn't represent an exponential function. y=2x3​
Exercises 8 We want to determine whether the given rule represents an exponential function. Notice that the independent variable x is an exponent. Therefore, the rule represents an exponential function. General form:Given function:​ y=a⋅bx y=-1⋅3x​
Exercises 9 We want to determine whether the given rule represents an exponential function. Notice that even though the independent variable x is an exponent, b cannot be negative. Therefore, the rule doesn't represent an exponential function. General form:Given function:​ y=a⋅bx y=9(-5)x​
Exercises 10 We want to determine whether the given rule represents an exponential function. Notice that even though the independent variable x is an exponent, b is equal to 1. Therefore, the rule doesn't represent an exponential function. y=21​(1)x​
Exercises 11 We want to check whether the given table represents a linear or an exponential function. If the ratios of consecutive y-values are equal, then the data represents an exponential function. If the difference of consecutive y-values is constant, then the data represents a linear function. Consider the given table.xy 1-2 20 32 44 Notice that the difference between each x-value is 1. Let's calculate the difference between consecutive y-values. 0−(-2)=2,2−0=2,4−2=2​ Each difference is equal to 2. Since the x-values are at regular intervals and the y-values differ by a constant, the table represents a linear function.
Exercises 12 Let's determine whether the table represents a linear or an exponential function. If the ratios of consecutive y-values are equal, then the table represents an exponential function. If the difference of consecutive y-values is constant, then the table represents a linear function. Consider the given table.xy 16 212 324 448 Let's calculate the difference between consecutive y-values. 12−6=6,  24−12=12,  48−24=24​ We can see that the differences are not constant, so the table does not represent a linear function. Let's determine the ratios of the consecutive y-values. 612​=2,1224​=2,2448​=2​ Each ratio is equal to 2, so the table represents an exponential function.
Exercises 13 We want to check whether the given table represents a linear or an exponential function. If the ratios of consecutive y-values are equal, then the data represents an exponential function. If the difference of consecutive y-values is constant, then the data represents a linear function. Consider the given table.x-10123 y0.25141664 Notice that the difference between each x-value is 1. Let's calculate the difference between consecutive y-values. 1−0.2516−4​=0.75,4−1=3,=12,64−16=48​ We can see that the differences are not constant, so the data cannot be modeled by a linear function. Let's determine the ratios between consecutive y-values. 0.251​416​​=4, 14​=4,=4,1664​=4​ Each ratio is equal to 4. Since the x-values are at regular intervals and the y-values differ by a positive common factor, the table represents an exponential function.
Exercises 14 We want to check whether the given table represents a linear or an exponential function. If the ratios of consecutive y-values are equal, then the data represents an exponential function. If the difference of consecutive y-values is constant, then the data represents a linear function. Consider the given table.x-30369 y101-8-17-26 Notice that the difference between each x-value is 3. Let's calculate the ratios between consecutive y-values. 101​-8-17​​=101​, 1-8​=-8=817​,-17-26​=1726​​ We can see that the ratios are not equal to each other, so the data cannot be modeled by an exponential function. Let's calculate the difference between consecutive y-values. 1−10-17−(-8)​=-9,  -8−1=-9=-9-26−(-17)=-9​ Each difference is equal to -9. Since the x-values are at regular intervals and the y-values differ by a constant, the table represents a linear function.
Exercises 15 To evaluate the given function for x=2, we should substitute 2 for x in the given function and then simplify. y=3xx=2y=32Calculate powery=9
Exercises 16 Using the function f(x), we want to evaluate for the given value, x=-1. To do this, we need to substitute -1 for x in each instance of the x-variable. f(x)=3(2)xx=-1f(-1)=3(2)-1a-1=a1​f(-1)=3(21​)a⋅b1​=ba​f(-1)=23​
Exercises 17 To evaluate the given function for x=2, we should substitute 2 for x in the given function and then simplify. y=-4(5)xx=2y=-4(5)2Calculate powery=-4(25)(-a)b=-aby=-100
Exercises 18 To evaluate the given function for x=-3, we should substitute -3 for x in the given function and then simplify. f(x)=0.5xx=-3f(-3)=0.5-3a=22⋅a​f(-3)=(21​)-3\FracToNegPowf(-3)=(12​)31a​=af(-3)=23Calculate powerf(-3)=8
Exercises 19 To evaluate the given function for x=3, we should substitute 3 for x in the given function and then simplify. f(x)=31​(6)xx=3f(3)=31​(6)3Calculate powerf(3)=31​⋅216b1​⋅a=ba​f(3)=3216​ba​=b/3a/3​f(3)=72
Exercises 20 Using the function y, we want to evaluate for the given value, x=23​. To do this, we need to substitute 23​ for x in each instance of the x-variable. y=41​(4)xx=23​y=41​(4)23​ Evaluate right-hand side ba​=b1​⋅ay=41​(4)21​(3)am⋅n=(am)ny=41​(421​)3a21​=a​y=41​(4​)3Calculate rooty=41​(2)3Calculate powery=41​(8)b1​⋅a=ba​y=48​Calculate quotient y=2
Exercises 21 Let's analyze the given function. f(x)=2(0.5)x​ We see that the initial value of the function and constant multiplier are 2 and 0.5. Therefore, it intercepts the y-axis at (0,2). Since 0<0.5<1, it is a decreasing function. The function f matches graph C.Extra info Graphing Exponential Functions When we draw an exponential function, y=abx, the value of a and b changes the shape of the graph. The graphs below show the possible graphs of the exponential functions.
Exercises 22 We see that there is a minus sign in front of the function. y=-2(0.5)x​ The graph of it is the graph of f(x)=2(0.5)x reflected across the x-axis. See the graph of the function f(x)=2(0.5)x. Therefore, the graph in B matches with the function y=-2(0.5)x.Extra info Graphing Exponential Functions When we draw an exponential function, y=abx, the value of a and b changes the shape of the graph. The graphs below show the possible graphs of the exponential functions.
Exercises 23 Let's analyze the given function. f(x)=2(2)x​ We see that the initial value of the function and constant multiplier are 2 and 2. Therefore, it intercepts the y-axis at (0,2). Since 2>1, it is an increasing function. The function f matches graph A.Extra info Graphing Exponential Functions When we draw an exponential function, y=abx, the value of a and b changes the shape of the graph. The graphs below show the possible graphs of the exponential functions.
Exercises 24 We see that there is a minus sign in front of the function. y=-2(2)x​ The graph of it is the graph of f(x)=2(2)x reflected across the x-axis. See the graph of the function f(x)=2(2)x. Therefore, the graph in D matches with the function y=-2(2)x.Extra info Graphing Exponential Functions When we draw an exponential function, y=abx, the value of a and b changes the shape of the graph. The graphs below show the possible graphs of the exponential functions.
Exercises 25 Let's graph and describe the domain and range of the given function first. Then we will compare the graph to the graph of the parent function.Graphing and Describing the Domain and Range To graph the given exponential function, we will first make a table of values.x3(0.5)xf(x)=3(0.5)x -43(0.5)-448 -23(0.5)-212 03(0.5)03 23(0.5)20.75 Let's now plot and connect the points (-4,48), (-2,12), (0,3), and (2,0.75) with a smooth curve.We can see in the graph that the range is all real numbers greater than zero. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers y>0​Comparing the Graph to the Parent Function The parent function is g(x)=0.5x. Let's graph it on the same coordinate plane. To do it, we will make a table of values first.x0.5xg(x)=0.5x -50.5-532 -20.5-24 00.501 20.520.25 Let's now plot and connect the points (-5,32), (-2,4), (0,1), and (2,0.25) with a smooth curve.We can tell that the graph of f is a vertical stretch by a factor of 3 of the graph of g. The y-intercept of the graph of f,3, is above the y-intercept of the graph of the parent function, 1.
Exercises 26 To graph the given exponential function, we will first make a table of values.x-4xf(x)=-4x -2-4-2-0.0625 -1-4-1-0.25 0-40-1 1-41-4 Let's now plot and connect the points (-2,-0.0625), (-1,-0.25), (0,-1), and (1,-4) with a smooth curve.We can see in the graph that the range is all real numbers less than zero. The domain of exponential functions is all real numbers. Let's compare the graph of f with the graph of its parent function, g(x)=4x.As we can see, the graph of f is a reflection across the x-axis of the graph of g.
Exercises 27 To graph the given exponential function, we will first make a table of values.x-2(7)xf(x)=-2(7)x -1-2(7)-1-0.2857 0-2(7)0-2 1-2(7)1-14 2-2(7)2-98 Let's now plot and connect the points (-1,-0.2857), (0,-2), (1,-14), and (2,-98) with a smooth curve.We can see in the graph that the range is all real numbers less than zero. The domain of exponential functions is all real numbers. Let's compare the graph of f with the graph of its parent function, g(x)=7x. It is useful to also consider the intermediate function h(x)=2(7)x.As we can see, the graph of f is a reflection across the x-axis of the graph of h, while the graph of h is a vertical stretch by a factor of 2 of graph of g. Therefore, the graph of f is a reflection across the x-axis followed by a vertical stretch by a factor of 2 of the graph of its parent function.
Exercises 28 To graph the given exponential function, we will first make a table of values.x6(31​)xf(x)=6(31​)x -16(31​)-118 06(31​)06 16(31​)12 26(31​)2≈0.67 Let's now plot and connect the points (-1,18), (0,6), (1,2), and (2,0.67) with a smooth curve.We can see in the graph that the range is all real numbers greater than zero. The domain of exponential functions is all real numbers. Let's compare the graph of f with the graph of its parent function, g(x)=(31​)x.As we can see, the graph of f is a vertical stretch by a factor of 6 of the graph of g.
Exercises 29 Let's graph and describe the domain and range of the given function first. Then we will compare the graph to the graph of the parent function.Graphing and Describing the Domain and Range To graph the given exponential function, we will first make a table of values.x21​(8)xf(x)=21​(8)x -221​(8)-2≈0.0078 021​(8)00.5 121​(8)14 221​(8)232 Let's now plot and connect the points (-2,0.0078), (0,0.5), (1,4), and (2,32) with a smooth curve.We can see in the graph that the range is all real numbers greater than zero. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers y>0​Comparing the Graph to the Parent Function The parent function is g(x)=8x. Let's graph it on the same coordinate plane. To do it, we will make a table of values first.x8xg(x)=8x -28-2≈0.016 0801 1818 1.581.5≈23 Let's now plot and connect the points (-2,0.016), (0,1), (1,8), and (1.5,23) with a smooth curve.We can tell that the graph of f is a vertical shrink by a factor of 21​ of the graph of g. The y-intercept of the graph of f,0.5, is under the y-intercept of the graph of the parent function, 1.
Exercises 30 Let's graph and describe the domain and range of the given function first. Then we will compare the graph to the graph of the parent function.Graphing and Describing the Domain and Range To graph the given exponential function, we will first make a table of values.x23​(0.25)xf(x)=23​(0.25)x -223​(0.25)-224 -123​(0.25)-16 023​(0.25)01.5 223​(0.25)20.09375 Let's now plot and connect the points (-2,24), (-1,6), (0,1.5), and (2,0.09375) with a smooth curve.We can see in the graph that the range is all real numbers greater than zero. Domain:Range:​ All real numbers y>0​Comparing the Graph to the Parent Function The parent function is g(x)=0.25x. Let's graph it on the same coordinate plane. To do it, we will make a table of values first.x0.25xg(x)=0.25x -2.50.25-2.532 -20.25-216 00.2501 20.2520.0625 Let's now plot and connect the points (-2.5,32), (-2,16), (0,1), and (2,0.0625) with a smooth curve.We can tell that the graph of f is a vertical stretch by a factor of 23​ of the graph of g. The y-intercept of the graph of f,1.5, is above the y-intercept of the graph of the parent function, 1.
Exercises 31 To graph the given exponential function, we will first make a table of values.x3x−1f(x)=3x−1 -43-4−1≈-0.99 030−10 232−18 333−126 Let's now plot and connect the points (-4,-0.99), (0,0), (2,8), and (3,26) with a smooth curve.We can see in the graph that the range is all real numbers greater than negative one. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers y>-1​
Exercises 32 To graph the given exponential function, we will first make a table of values.x4x+3f(x)=4x+3 -54-5+30.0625 -34-3+31 -24-2+34 -14-1+316 Let's now plot and connect the points (-5,0.0625), (-3,1), (-2,4), and (-1,16) with a smooth curve.We can see in the graph that the range is all real numbers greater than zero. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers y>0​
Exercises 33 To graph the given function, we will first make a table of values.x5x−2+7y=5x−2+7 050−2+77.04 151−2+77.2 252−2+78 353−2+712 Let's now plot and connect the points (0,7.04), (1,7.2), and (2,8), (3,12) with a smooth curve.We can see that the graph approaches the line y=7 but does not intersect it. Therefore, the range is all real numbers greater than 7. The domain of the function is all real numbers. Domain:Range:​ All real numbers y>7​
Exercises 34 To graph the given function, we will first make a table of values.x-(21​)x+1−3y=-(21​)x+1−3 -3-(21​)-3+1−3-7 -2-(21​)-2+1−3-5 -1-(21​)-1+1−3-4 0-(21​)0+1−3-3.5 1-(21​)1+1−3-3.25 Let's now plot and connect the points (-3,-7), (-2,-5), (-1,-4), (0,-3.5), and (1,-3.25) with a smooth curve.We can see that the graph approaches the line y=-3 but does not intersect it. Therefore, the range is all real numbers less than -3. The domain of the function is all real numbers. Domain:Range:​ All real numbers y<-3​
Exercises 35 To graph the given exponential function, we will first make a table of values.x-8(0.75)x+2−2y=-8(0.75)x+2−2 -2-8(0.75)-2+2−2-10 -1-8(0.75)-1+2−2-8 0-8(0.75)0+2−2-6.5 1-8(0.75)1+2−2-5.375 Let's now plot and connect the points (-2,-10), (-1,-8), (0,-6.5), and (1,-5.375) with a smooth curve.We can see in the graph that the range is all real numbers less than -2. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers y<-2​
Exercises 36 To graph the given exponential function, we will first make a table of values.x3(6)x−1−5f(x)=3(6)x−1−5 -13(6)-1−1−5≈-4.92 03(6)0−1−5-4.5 13(6)1−1−5-2 23(6)2−1−513 Let's now plot and connect the points (-1,-4.92), (0,-4.5), (1,-2), and (2,13) with a smooth curve.We can see in the graph that the range is all real numbers greater than -5. The domain of exponential functions is all real numbers. Domain:Range:​ All real numbers f(x)>-5​
Exercises 37 Looking at the graph, we see that the graph of f(x)=2x is vertically shrunk.The absolute value of a should be less than 1 due to the vertical shrink. g(x)=a(2)x​ We see that the point (2,2) is on the graph of g(x)=a(2)x. Let's substitute it and solve for a. g(x)=a(2)xx=2, y=22=a(2)2 Solve for a Calculate power2=a(4)LHS/4=RHS/442​=aba​=b/2a/2​21​=aRearrange equation a=21​ The value of a is 21​.
Exercises 38 Looking at the graph, we see that the graph of f(x)=0.25x is translated k units up.Notice that the point (0,4) is on the graph of g(x)=0.25x+k. Let's substitute it and solve for k. g(x)=0.25x+kx=0, y=44=0.250+k Solve for k a0=14=1+kLHS−1=RHS−13=kRearrange equation k=3 The value of k is 3.
Exercises 39 Looking at the graph, we can say that the graph of f(x)=-3x is translated h units to the right.Notice that the point (4,-1) is on the graph of g(x)=-3x−h. Let's substitute it and solve for h. g(x)=-3x−hx=4, y=-1-1=-34−h Solve for h LHS/(-1)=RHS/(-1)1=34−ham−n=anam​1=3h34​LHS⋅3h=RHS⋅3h3h=34Equate exponents h=4 The value of h is 4.
Exercises 40 Looking at the graph, we can say that the graph of f(x)=31​(6)x is translated h units to the left.Notice that the point (-1,2) is on the graph of g(x)=31​(6)x−h. Let's substitute it and solve for h. g(x)=31​(6)x−hx=-1, y=22=31​(6)-1−h Solve for h LHS⋅3=RHS⋅36=6-1−ha=a161=6-1−hEquate exponents1=-1−hLHS+1=RHS+12=-hLHS/(-1)=RHS/(-1)-2=hRearrange equation h=-2 The value of h is -2.
Exercises 41 Let's try to find the value of g(x) when x=-2. g(x)=6(0.5)xx=-2g(-2)=6(0.5)-2Calculate powerg(-2)=6⋅4Multiplyg(-2)=24 The function is equal to 24 when x=-2. In the given solution, the multiplication is performed before calculating the power of 0.5. Correct:Incorrect:​  6 (0.5)-2=6 (4) ✓  6 (0.5)-2=3-2×​
Exercises 42 We want to find the domain and range of the given exponential function. y=-(0.5)x−1⇓y=-1(0.5)x−1​ Looking at the function, we can see that its parent function has been reflected across the x-axis, and translated 1 unit down. Therefore, the asymptote of the function is the line y=-1.The domain of the function is all real numbers. The function cannot be greater than -1, so the range of the function is all real numbers less than -1, not 0. Domain:Range:​ -∞<x<∞ y<-1​
Exercises 43 To draw the graph of g(x), we will first make a table of values. We know that g(0)=8, and each term is 2.5 times the previous term.The ordered pairs (-1,3.2), (0,8), (1,20), and (2,50) all lie on the function. Now, we will plot and connect these points with a smooth curve.Now let's draw the functions f(x)=0.5(4)x and g(x) on the same axes and compare them. We see that both functions have the different y-intercepts when x=0. The value of g(x) is greater than the value of f(x) over the rest of the interval. Showing Our Work info Graphing f(x) To graph the exponential function, f(x)=0.5(4)x, we will first make a table of values.x0.5(4)xf(x)=0.5(4)x -10.5(4)-10.125 00.5(4)00.5 10.5(4)12 20.5(4)28 30.5(4)332 Let's now plot and connect the points (-1,0.125), (0,0.5), (1,2), (2,8), and (3,32) with a smooth curve.
Exercises 44 To draw the graph of h(x), we will first make a table of values. We know that h(0)=32, and each term is 21​ times the previous term.The ordered pairs (-1,64), (0,32), (1,16), and (2,8) all lie on the function. Now, we will plot and connect these points with a smooth curve.Now let's draw the functions f(x)=0.5(4)x and g(x) on the same axes and compare them. We see that both functions have different y-intercepts when x=0. The value of h(x) is greater than the value of f(x) when 0<x<2. They have the same value when x=2. Showing Our Work info Graphing f(x) To graph the exponential function, f(x)=0.5(4)x, we will first make a table of values.x0.5(4)xf(x)=0.5(4)x -10.5(4)-10.125 00.5(4)00.5 10.5(4)12 20.5(4)28 30.5(4)332 Let's now plot and connect the points (-1,0.125), (0,0.5), (1,2), (2,8), and (3,32) with a smooth curve.
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Exercises 47 We want to write an exponential function represented by the given table. To do so, let's calculate the increase in the x-values and the ratio between consecutive y-values.The y-values increase by a factor of 7 as x increases by 1. Moreover, note that when x=0, we have that y=2. This means the y-intercept is 2. We have enough information to write the exponential function. y=2(7)x​
Exercises 48 We want to write an exponential function represented by the given table. To do so, let's calculate the increase in the x-values and the ratio between consecutive y-values.The y-values change by a factor of 0.2 as x increases by 1. Moreover, note that when x=0, we have that y=-50. This means the y-intercept is -50. We have enough information to write the exponential function. y=-50(0.2)x​
Exercises 49 We want to write an exponential function represented by the given graph.To do so, we will start by making a table of the x- and y-values marked on the given graph. Then, we will calculate the increase in the x-values and the ratio between consecutive y-values.The y-values change by a factor of 2 as x increases by 1. Moreover, note that when x=0, we have that y=-0.5. This means the y-intercept is -0.5. We have enough information to write the exponential function. y=-0.5(2)x​
Exercises 50 We want to write an exponential function represented by the given graph.To do so, we will start by making a table of the x- and y-values marked on the given graph. Then, we will calculate the increase in the x-values and the ratio between consecutive y-values.The y-values change by a factor of 0.5 as x increases by 1. Moreover, note that when x=0, we have that y=8. This means the y-intercept is 8. We have enough information to write the exponential function. y=8(0.5)x​
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