Transformation Review 1

\[ \definecolor{ucrblue}{rgb}{0.0627,0.3843,0.6039} \definecolor{ucrgold}{rgb}{0.9686,0.6941,0.2824} \definecolor{ucrred}{rgb}{0.8941,0,0.1686} \definecolor{ucrgreen}{rgb}{0.4706,0.7451,0.1255} \definecolor{ucraccent}{rgb}{1.0000,0.9569,0.8392} \DeclareMathOperator*{\LO}{O} \DeclareMathOperator*{\Lo}{o} \DeclareMathOperator*{\Recip}{Recip} \DeclareMathOperator*{\abs}{abs} \DeclareMathOperator{\pow}{pow} \]

Notes

You can reference these sections to review.

• Chapter 2.1: Vectors and Translation
• Chapter 2.2: Scaling Vectors and Subsets of the Plane
• Chapter 2.3: Movement along Lines
• Chapter 2.4: Orthogonality and Reflection
• Chapter 2.5: Inverse Functions
• Chapter 2.6: Describing Rotation in Cartesian Coordinates
• Chapter 2.7: Rotations
• Chapter 2.8: Involution

Knowledge Checks

1. Identify a vector that moves \((3,4)\) to \((5,7)\).
2. Take \(V\) to be the vector given by \[V=\langle 2,-1\rangle.\] Determine \(2V\), \(-V\), \(\|V\|\), and \(\hat{V}\) and sketch a picture to illustrate these quantities.
3. A particle moves at constant velocity. It is at the point \((-1,5)\) at time \(3\) and at the point \((4,8)\) at time \(9\). Write an equation that models the motion of the particle
4. Write the following functions using set notation: \[ g(x)=x^3+x,\quad h(x)=\log_2(x-2),\quad k(x)=2^x-1. \]
5. Sketch the function \(f\) that is given by \[f(x)=\left(\frac{1}{2}\right)^{x+2}+1.\] Identify any asymptotes of the function.
6. Sketch the function \(f\) that is given by \[f(x)=\log_2(x-1)-2.\] Identify any asymptotes of the function.
7. Calculate \(\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\star\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)\) and explain its meaning.
8. Take \(p\) to be the point given by \(p=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\). Calculate \(p^{-1}\) and explain its meaning.
9. Rotate the point \((2,4)\) about \((3,3)\) by \(\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)\).
10. A particle rotates counterclockwise around the point \((0,0)\) at constant speed. It is at the point \((3,0)\) at time \(0\). Write an equation that models the position of the particle at time \(t\).
11. A particle rotates counterclockwise around the point \((1,2)\) at constant speed. It is at the point \((5,2)\) at time \(0\) and at the point \((1,6)\) at time \(3\). Write an equation that models the position of the particle at time \(t\).
12. Take \(A\) and \(B\) to be two real numbers so that \[\cos(A)=-\frac{1}{3},\quad\sin(A)=\frac{2\sqrt{2}}{3},\quad \cos(B)=\frac{2}{5}\quad \text{and}\quad \sin(B)=-\frac{\sqrt{21}}{5}.\] Determine \(\cos(A+B)\) and \(\sin(A+B)\).
13. Take \(f\) to be the function given by \[f(x)=(x-4)^2.\] Sketch \(\Recip(f)\).
14. Take \(f\) to be the function whose sketch is given below. Sketch \(\Recip(f)\).

• Knowledge Checks PDF
• Knowledge Checks Solutions

Remember, do the knowledge checks before checking the solutions.

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