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Home Question Bank Math Pre Calculus Archive 2025 January 10

Pre-calculus Questions from Jan 10,2025

Browse the Pre-calculus Q&A Archive for Jan 10,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Sketch on the same set of axes the graphs of \( f(x)=-2 x^{2}-4 x+6 \) and \( g(x)=-2 \cdot 2^{x-1}+1 \) Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s). 2. Find the polar form of the following complex numbers and represent each complex number graphically on an Argand diagram: \( \begin{array}{lll}\text { a) } z=5+3 i & \text { b) } z=4 i-2 & \text { c) } z=-5 \\ \text { d) } z=-3 i & \text { e) } z=3 \sqrt[3]{8}-5 i & \end{array} \) 2. Si \( f(x)=2^{x} \) demostror que a) \( f(x+3)-f(x-1)=\frac{15}{2} f(x) \) b) \( \frac{f(x+3)}{f(x-1)}=f(4) \) 2. Si \( f(x)=2^{x} \) demostror que a.) \( f(x+3)-f(x-1)=\frac{15}{2} f(x) \) b) \( \frac{f(x+3)}{f(x-1)}=f(4) \) Consider the function \( f(x) = \sqrt{x} \). Describe how you would graph its inverse, and explain any transformations that occur. 11. Calculate the value of each expression: \( \begin{array}{llll}\text { (a) } 3125^{\frac{1}{5}} & \text { (b) } 128^{\frac{3}{7}} & \text { (c) } 81^{-\frac{3}{4}} & \text { (d) }\left(\frac{81}{144}\right)^{-\frac{3}{2}} \\ \text { 2. Determine the value of each expression: } \\ \begin{array}{llll}\text { (a) } 36^{\frac{3}{2}} & \text { (b) } 243 \frac{2}{5} & \text { (c) } 100^{-\frac{5}{2}} & \text { (d) }\left(\frac{64}{729}\right)^{-\frac{5}{6}}\end{array}\end{array} \) The population of a town is decreasing by \( 3 \% \) per year. The current population of the town is 10741 . The equation that represents \( P(t) \), the population of the town after \( t \) years, can be written in the form \( P(t)=a(b)^{\frac{t}{p}} \) Numeric Response 13. The population of the town, rounded to the nearest whole person, remaining after 6 years is (Record your answer in the numerical-response section below.) Your answer. Numeric Response 14. The number of years, rounded to the nearest tenth, before the population of the town reaches 6500 is (Record your answer in the numerical-response section below) Your answer. 8. Calculate the value of each expression: \( \begin{array}{llll}\text { (a) } 25^{\frac{3}{2}} & \text { (b) } 27^{\frac{4}{3}} & \text { (c) } 27^{-\frac{2}{3}} & \text { (d) }\left(\frac{27}{8}\right)^{-\frac{2}{3}}\end{array} \) Function \( f(x)=\log _{3} x \) was transformed by only a vertical translation and the transformed function was written in the form, \( g(x)-7=\log _{3}(9 x) \). 11. Complete the following statement. Function \( g(x) \) is the result of vertically translating \( f(x) \) \( \square 7 \) units down. \( \square 9 \) units up. \( \square 9 \) units down. \( \square 7 \) units up. \( \square \) 10. A student made the following statements about the graph of the exponential function \( y=3(2)^{x-20}-5 \). The two true statements are \( \square \) The graph has a domain of \( \{x \mid x \in R\} \) and a range of \( \{y \mid y>-5\} \) \( \square \) The graph has a horizontal asymptote at \( y=-5 \). \( \square \) The graph passes through the point \( (20,-5) \). \( \square \) The graph may be obtained by stretching the graph of \( y=2^{x} \) by a factor of 6 about the \( x \)-axis and then translating 20 units right and 5 units down. \( \square \)
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