32 Questions
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 $h(x) = 2x^2 - 5$?
Domain: all real numbers; Range: $y \geq -5$
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?
$\frac{(2x - 3)(x + 2)}{(x - 2)(x + 2)}$; Restrictions: $x \neq 2, -2$
What are the solutions to the quadratic equation $x^2 - 4x + 3 = 0$ when factored?
$x = 1$ and $x = 3$
What is the inverse function of $y = 2x + 1$?
$f^{-1}(x) = \frac{x - 1}{2}$
Find the time it takes for the ball to hit the ground.
6 seconds
Simplify the expression $2^{3/2} \times 2^{-1/2}$.
$2$
Determine the population after 10 years, given the exponential function $P(t) = 5000 \times 1.02^t$.
6103.02
Solve the equation $\sin(x) = \frac{\sqrt{3}}{2}$ for $0 \leq x \leq 2\pi$.
$\frac{\pi}{3}, \frac{2\pi}{3}$
Determine the length of side AC in triangle ABC, given that angle A measures $45^\ ext{o}$ and side BC is 8 units long.
$8\sqrt{2}$
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.
20 meters
Apply the given transformations to the function $f(x) = \sin(x)$ and sketch the new graph.
Not provided
Sketch the graph of $y = 2 \sin(x + \frac{\pi}{4})$, indicating the amplitude, period, phase shift, and vertical shift.
Not provided
Determine the maximum and minimum temperatures, given the function $T(t) = 20 + 10 \sin(\frac{\pi}{6}t)$.
Maximum: 30 degrees, Minimum: 10 degrees
Prove the identity $\tan(x) \cdot \cot(x) = 1$.
Not provided
What are the domain and range of the function $h(x) = 2x^2 - 5$?
Domain: All real numbers; Range: $y ext{ such that } y geq -5$
What is the result of expanding and simplifying the expression $(x + 3)(x - 2)$?
$x^2 + x - 6$
Simplify the rational expression $\frac{2x^2 - x - 6}{x^2 - 4}$ and state any restrictions on the variable.
$\frac{2x + 3}{x + 2}$; $x \neq -2$
What is the inverse function of $y = 2x + 1$?
$x = \frac{y - 1}{2}$
What is the solution to the quadratic equation $x^2 - 4x + 3 = 0$ when factored?
$x = 1, 3$
What are the transformations applied to the function $f(x) = \sqrt{x}$ to obtain the new function $g(x) = 3f(x)$?
Vertical stretch by a factor of 3
What is the time it takes for the ball to hit the ground?
6 seconds
Simplify the expression $2^{3/2} imes 2^{-1/2}$.
2
What is the population after 10 years, given the exponential function $P(t) = 5000 imes 1.02^t$?
6103.02
What are the solutions to the equation $ ext{sin}(x) = rac{ ext{sqrt}{3}}{2}$ for $0 eq x eq 2 ext{pi}$?
$rac{ ext{pi}}{3}, rac{5 ext{pi}}{3}$
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?
8
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?
20 meters
What is the maximum and minimum temperatures, given the function $T(t) = 20 + 10 ext{sin}(rac{ ext{pi}}{6}t)$?
30, 10
Prove the identity $ ext{tan}(x) imes ext{cot}(x) = 1$.
It is an identity and holds true for all x where tan(x) and cot(x) are defined.
Find the sum of the first 10 terms of the arithmetic sequence $2, 6, 10, ...$
260
Determine the amount of money in the account after 3 years, if you invest $5000 in an account that earns 4% interest annually.
$5624.32
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 $A = 5000 \times (1 + 0.04)^3 \approx 6253.81$
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.
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