Podcast
Questions and Answers
What are the transformations applied to the function $f(x) = \rac{1}{
qrt{x}}$ to obtain the new function $g(x) = 3f(x - 2)$?
What are the transformations applied to the function $f(x) = \rac{1}{
qrt{x}}$ to obtain the new function $g(x) = 3f(x - 2)$?
Horizontal translation by 2 units to the right and vertical stretching by a factor of 3
What is the domain and range of the function $h(x) = 2x^2 - 5$?
What is the domain and range of the function $h(x) = 2x^2 - 5$?
Domain: all real numbers; Range: $y \geq -5$
What is the expanded and simplified form of the expression $(x + 3)(x - 2)$?
What is the expanded and simplified form of the expression $(x + 3)(x - 2)$?
$x^2 + x - 6$
What is the simplified form of the rational expression $\frac{2x^2 - x - 6}{x^2 - 4}$ and are there any restrictions on the variable?
What is the simplified form of the rational expression $\frac{2x^2 - x - 6}{x^2 - 4}$ and are there any restrictions on the variable?
What are the solutions to the quadratic equation $x^2 - 4x + 3 = 0$ when factored?
What are the solutions to the quadratic equation $x^2 - 4x + 3 = 0$ when factored?
What is the inverse function of $y = 2x + 1$?
What is the inverse function of $y = 2x + 1$?
Find the time it takes for the ball to hit the ground.
Find the time it takes for the ball to hit the ground.
Simplify the expression $2^{3/2} \times 2^{-1/2}$.
Simplify the expression $2^{3/2} \times 2^{-1/2}$.
Determine the population after 10 years, given the exponential function $P(t) = 5000 \times 1.02^t$.
Determine the population after 10 years, given the exponential function $P(t) = 5000 \times 1.02^t$.
Solve the equation $\sin(x) = \frac{\sqrt{3}}{2}$ for $0 \leq x \leq 2\pi$.
Solve the equation $\sin(x) = \frac{\sqrt{3}}{2}$ for $0 \leq x \leq 2\pi$.
Determine the length of side AC in triangle ABC, given that angle A measures $45^\ ext{o}$ and side BC is 8 units long.
Determine the length of side AC in triangle ABC, given that angle A measures $45^\ ext{o}$ and side BC is 8 units long.
Determine the length of the ladder, given the angle of elevation is $60^\ ext{o}$ and the distance from the base of the ladder to the wall is 10 meters.
Determine the length of the ladder, given the angle of elevation is $60^\ ext{o}$ and the distance from the base of the ladder to the wall is 10 meters.
Apply the given transformations to the function $f(x) = \sin(x)$ and sketch the new graph.
Apply the given transformations to the function $f(x) = \sin(x)$ and sketch the new graph.
Sketch the graph of $y = 2 \sin(x + \frac{\pi}{4})$, indicating the amplitude, period, phase shift, and vertical shift.
Sketch the graph of $y = 2 \sin(x + \frac{\pi}{4})$, indicating the amplitude, period, phase shift, and vertical shift.
Determine the maximum and minimum temperatures, given the function $T(t) = 20 + 10 \sin(\frac{\pi}{6}t)$.
Determine the maximum and minimum temperatures, given the function $T(t) = 20 + 10 \sin(\frac{\pi}{6}t)$.
Prove the identity $\tan(x) \cdot \cot(x) = 1$.
Prove the identity $\tan(x) \cdot \cot(x) = 1$.
What are the domain and range of the function $h(x) = 2x^2 - 5$?
What are the domain and range of the function $h(x) = 2x^2 - 5$?
What is the result of expanding and simplifying the expression $(x + 3)(x - 2)$?
What is the result of expanding and simplifying the expression $(x + 3)(x - 2)$?
Simplify the rational expression $\frac{2x^2 - x - 6}{x^2 - 4}$ and state any restrictions on the variable.
Simplify the rational expression $\frac{2x^2 - x - 6}{x^2 - 4}$ and state any restrictions on the variable.
What is the inverse function of $y = 2x + 1$?
What is the inverse function of $y = 2x + 1$?
What is the solution to the quadratic equation $x^2 - 4x + 3 = 0$ when factored?
What is the solution to the quadratic equation $x^2 - 4x + 3 = 0$ when factored?
What are the transformations applied to the function $f(x) = \sqrt{x}$ to obtain the new function $g(x) = 3f(x)$?
What are the transformations applied to the function $f(x) = \sqrt{x}$ to obtain the new function $g(x) = 3f(x)$?
What is the time it takes for the ball to hit the ground?
What is the time it takes for the ball to hit the ground?
Simplify the expression $2^{3/2} imes 2^{-1/2}$.
Simplify the expression $2^{3/2} imes 2^{-1/2}$.
What is the population after 10 years, given the exponential function $P(t) = 5000 imes 1.02^t$?
What is the population after 10 years, given the exponential function $P(t) = 5000 imes 1.02^t$?
What are the solutions to the equation $ ext{sin}(x) = rac{ ext{sqrt}{3}}{2}$ for $0
eq x
eq 2 ext{pi}$?
What are the solutions to the equation $ ext{sin}(x) = rac{ ext{sqrt}{3}}{2}$ for $0 eq x eq 2 ext{pi}$?
What is the length of side AC in triangle ABC, given that angle A measures $45^ ext{o}$ and side BC is 8 units long?
What is the length of side AC in triangle ABC, given that angle A measures $45^ ext{o}$ and side BC is 8 units long?
What is the length of the ladder, given the angle of elevation is $60^ ext{o}$ and the distance from the base of the ladder to the wall is 10 meters?
What is the length of the ladder, given the angle of elevation is $60^ ext{o}$ and the distance from the base of the ladder to the wall is 10 meters?
What is the maximum and minimum temperatures, given the function $T(t) = 20 + 10 ext{sin}(rac{ ext{pi}}{6}t)$?
What is the maximum and minimum temperatures, given the function $T(t) = 20 + 10 ext{sin}(rac{ ext{pi}}{6}t)$?
Prove the identity $ ext{tan}(x) imes ext{cot}(x) = 1$.
Prove the identity $ ext{tan}(x) imes ext{cot}(x) = 1$.
Find the sum of the first 10 terms of the arithmetic sequence $2, 6, 10, ...$
Find the sum of the first 10 terms of the arithmetic sequence $2, 6, 10, ...$
Determine the amount of money in the account after 3 years, if you invest $5000 in an account that earns 4% interest annually.
Determine the amount of money in the account after 3 years, if you invest $5000 in an account that earns 4% interest annually.
Study Notes
Function Transformations
- Horizontal shift: $f(x) = \frac{1}{\sqrt{x}}$ to $g(x) = 3f(x - 2)$ involves a horizontal shift of 2 units to the right and a vertical stretch by a factor of 3
Quadratic Functions
- Domain and range of $h(x) = 2x^2 - 5$ are $(-\infty, \infty)$ and $[-5, \infty)$, respectively
- Solutions to the quadratic equation $x^2 - 4x + 3 = 0$ when factored are $x = 1$ and $x = 3$
Rational Expressions
- Simplified form of the rational expression $\frac{2x^2 - x - 6}{x^2 - 4}$ is $\frac{2x + 3}{x + 2}$
- Restrictions on the variable: $x \neq -2$ and $x \neq 2$
Trigonometry
- Inverse function of $y = 2x + 1$ is $y = \frac{x - 1}{2}$
- Solution to the equation $\sin(x) = \frac{\sqrt{3}}{2}$ for $0 \leq x \leq 2\pi$ is $x = \frac{\pi}{3}$ and $x = \frac{2\pi}{3}$
- Identity: $\tan(x) \cdot \cot(x) = 1$
Exponential Functions
- Population after 10 years, given the exponential function $P(t) = 5000 \times 1.02^t$, is $P(10) = 5000 \times 1.02^{10} \approx 6815.51$
Geometry
- Length of side AC in triangle ABC, given that angle A measures $45^{\circ}$ and side BC is 8 units long, is $AC = BC \times \sin(A) = 8 \times \sin(45^{\circ}) = 4\sqrt{2}$ units
- Length of the ladder, given the angle of elevation is $60^{\circ}$ and the distance from the base of the ladder to the wall is 10 meters, is $l = \frac{10}{\cos(60^{\circ})} = 20$ meters
Arithmetic Sequences
- Sum of the first 10 terms of the arithmetic sequence $2, 6, 10, ...$ is $S_{10} = \frac{10}{2}(2 + 46) = 240$
Compound Interest
- Amount of money in the account after 3 years, if you invest $5000 in an account that earns 4% interest annually, is $A = 5000 \times (1 + 0.04)^3 \approx 6253.81$
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Description
This quiz covers graphing transformed parent functions, including applying various transformations to the function and sketching the resulting graphs. Sample questions include modifying the function f(x) = √(x) according to specific transformations.
