Review for Test KEY PDF

## Review for Test ### Quiz 1 Graph the following exponential functions. Be sure to plot at least 4-5 points and to include the horizontal asymptote in your sketch. $f(x) = \frac{-3(3)^{x-1} + 5}{reflect}$ Asy $y=5$ | x | f(x) | |---|---| | 0 | 4 | | 1 | -2 | | 2 | -22 | | 3 | -67 | | 4 | -202...

## Review for Test ### Quiz 1 Graph the following exponential functions. Be sure to plot at least 4-5 points and to include the horizontal asymptote in your sketch. $f(x) = \frac{-3(3)^{x-1} + 5}{reflect}$ Asy $y=5$ | x | f(x) | |---|---| | 0 | 4 | | 1 | -2 | | 2 | -22 | | 3 | -67 | | 4 | -202 | Horizontal asymptote: $y=5$ Domain: (-∞, ∞) y-intercept: (0, 4) Range (-∞, 5) Transformation: Reflection over x, Translations up 5, dilates by 3. Write an exponential equation given 2 points in the form of $f(x) = a(b)^x$. 1. (1, 28) & (2,196) $f(x) = 4(7)^x$ 2. $a=2$ why a is the $y-intercept$ $28=ab$ $196=ab^2$ $196=4b^2$ $28=6b$ $7=b$ $28=a*7$ $a=4$ $f(x)=2(3)^x$ ### Quiz 2: Formulas: * Compounded $n$ times: $A = P(1+\frac{r}{n})^{nt}$ * Compounded continuously: $A = Pe^{rt}$ * Growth: $A = P(1+r)^{t}$ * Decay: $A = P(1-r)^{t}$ You deposit $500 into an account that pays 3.5% annual interest compounded quarterly ($n=4$). a. Write the exponential function for the amount of money in your account. $A=500(1+\frac{0.035}{4})^{4t}$ b. What is the balance after 5 years? $A = 500(1+\frac{0.035}{4})^{4(5)}$ $A \approx \$595.17$ You deposit $2000 into an account that pays 2.3% annual interest compounded continuously. a. Write the exponential function for the amount of money in your account. $A=2000(e)^{(0.023)t}$ b. What is the balance after 2 years? $A=2000(e)^{(0.023)(2)}$ $A \approx \$2094.15$ You buy a brand-new car for $38,000. The value of the car depreciates, by 4.8% each year. a. Write the exponential function that models the value of the car after $t$ years. $A=38,000(1-0.048)^t$ b. How much is the car worth after 4 years? $A=38,000(1-0.048)^4$ $A \approx \$31212.70$ ### Quiz 3: Rewrite the given equation in equivalent forms. | Logarithmic Form | Exponential Form | Exponential Form | Logarithmic Form | |---|---|---|---| | $log_7 49 = 2$ | $7^2 = 49$ | $10^{-3} = 0.001$ | $log(0.001) = -3$| | $log_4 16 = 2$ | $16 = 4^2$ | $13^2 = 169$ | $log_{13}(169) = 2$ | | $log_3 \frac{1}{9} = -2$ | $3^{-2} = \frac{1}{9}$ | $64^{\frac{1}{2}} = 8$ | $log_{64}(8) = \frac{1}{2}$ | Describe the properties of the logarithmic function graphed below. Domain: (0, ∞) Range: (-∞, ∞) Increasing Interval: (0, ∞) Decreasing Interval: *none* Vertical Asymptote: x=0 Solve the equations for x. Round your answers to three decimal places. * $2(4)^{2x+5} = 25$ $2(4)^{2x} = 20$ $4^{2x} = 10$ $log_4(10) = 2x$ $\frac{log_4(10)}{2} = x$ $x \approx 0.830$ * $log_4(10) = x$ $x \approx 0.830$ * $7^{x+1} + 8 = 73$ $7^{x+1} = 65$ $log_7 65 = x+1$ $log_7 65 - 1 = x$ $x \approx 1.258$ * $2(5)^x + 8 = 120$ $2(5)^{x} = 112$ $5^{x} = 56$ $log_5 56 = x$ $x \approx 2.501$ Solve the equations for $x$. * $9^{4x+1} = 27^{3x-2}$ $(3^2)^{4x+1} = (3^3)^{3x-2}$ $8x+2=9x-6$ $8=x$ Check: $9^{4(8)+1} = 27^{3(8)-2}$ $9^{33} = 27^{22}$ $3^{66} = 3^{66}$ * $8^{x+2} = (\frac{1}{8})^5$ $8^{x+2} = (8^{-1})^5$ $8^{x+2} = 8^{-5}$ $x+2=-5$ $x=-7$ Check: $8^{-7+2} = (\frac{1}{8})^5$ $8^{-5} = 8^{-5}$ * $2^{x+2} = 4^{x+4}$ $2^{x+2} = (2^2)^{x+4}$ $2^{x+2} = 2^{2x+8}$ $x+2=2x+8$ $-6=x$ Check: $2^{-6+2} = 4^{-6+4}$ $2^{-4} = 4^{-2}$ $\frac{1}{16} = \frac{1}{16}$ Solve the equations for x. * $(x+2)+10 = 12$ $x+2=2$ $x=0$ Check: $(0+2)+10 = 12$ $12=12$ * $3(2x-3) = 9$ $2x-3 = 3$ $2x = 6$ $x=3$ Check: $3(2(3)-3) = 9$ $3(6-3) = 9$ $3(3) = 9$ $9 = 9$