Exponential Functions

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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 in choice C has these properties. The answer is C.Extra 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 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 in A has these properties. The answer is A.Extra 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 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.
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Exercises 37 Looking at the graph, we see that the graph of f(x)=2x is shrunk vertically.The absolute value of a should be less than 1 due to the vertical compression. 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 k. 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 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 the 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 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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