Rotation Guide
This guide explains how to rotate a set of points around a fixed center point. By following these steps, you will understand how to visually rotate points or shapes by a specified angle, either clockwise or counterclockwise, around a chosen center of rotation.
Step 1: Set Up the Rotation
Identify and plot the center of rotation along with each point that will be rotated. This setup helps establish the initial positions relative to the center.
- Identify and plot the center of rotation, labeled as P, on your graph paper.
- Plot each point you want to rotate around P (for example, points A, B, etc.).
Step 2: Draw Initial Guidelines
Draw guidelines from the center to each point to visualize the initial position of each point in relation to the center of rotation.
- For each point, draw a straight guideline from P to that point.
- These lines help show the direction and distance each point will rotate around P.
Step 3: Position the Protractor at the Center
Set up the protractor on P to measure the desired rotation angle accurately for each point.
- Place the center hole of the protractor directly on P.
- Align the baseline of the protractor with each guideline to prepare for measuring the angle.
Step 4: Measure the Rotation Angle
Using the protractor, determine the new angle for each point based on the specified rotation.
- For clockwise rotation, measure the angle clockwise from each guideline. For counterclockwise, measure counterclockwise.
- Mark the rotation angle on each guideline to show the rotated position for each point.
Step 5: Draw Rotated Guidelines
Draw a new line from the center through each marked angle to indicate the direction of each rotated point.
- For each point, draw a new line from P through the angle mark created in Step 4.
Step 6: Measure the Original Distance
Measure the original distance from P to each point. The rotated points will maintain this distance from the center.
- Using a ruler, measure the original distance from P to each point along the guideline.
Step 7: Mark the Rotated Points
Place each rotated point along the new guideline at the same distance from P as the original point to complete the rotation.
- From P, measure out the original distance along each rotated guideline and mark this as the rotated point.
- Label the rotated points as A’, B’, etc., to show the new positions.
Dilation Guide
This guide explains how to perform a dilation transformation, which either enlarges or reduces a set of points relative to a fixed center point. By following these steps, you will be able to scale points or shapes by a specific factor, expanding or contracting their distances from a chosen center.
Step 1: Set Up the Dilation
Identify the center of dilation and each point to be dilated. This establishes the initial setup for the transformation.
- Identify and plot the center of dilation, labeled as P, on your graph paper.
- Plot each point you want to dilate around P.
Step 2: Draw Initial Guidelines
Draw lines from the center to each point to show the direction each point will move along during dilation.
- For each point, draw a straight guideline from P to that point.
- These guidelines help visualize how each point will move closer to or further from P during dilation.
Step 3: Measure Original Distances
Measure the original distance from P to each point along the guideline. This distance will be scaled by the dilation factor in the next step.
- Use a ruler to measure the distance from P to each point along the guideline.
- Record these distances, as each will be multiplied by the dilation factor.
Step 4: Apply the Dilation Factor
Multiply each distance from P to each point by the dilation factor to determine the new distance each point should be from P.
- If the dilation factor is greater than 1, each point will move further from P.
- If the dilation factor is less than 1, each point will move closer to P.
Step 5: Mark the Dilated Points
Use the new scaled distances to locate each point along each guideline and mark their new positions after dilation.
- From P, measure out the new distance along each guideline and mark the point.
- Label these as A’, B’, etc., to show the dilated positions.
Step 6: Connect the Dilated Points (if forming a shape)
If you’re dilating a shape, connect the dilated points in the same order as the original points to complete the new shape.
- Connect each new point in the same sequence as the original points to form the dilated shape.
Translation Guide
This guide explains how to perform a translation transformation, which moves a set of points by a fixed distance in a specific direction. By following these steps, you’ll be able to translate points or shapes according to a given vector, shifting their positions on the plane while preserving their orientation and shape.
Step 1: Identify the Translation Vector
Determine the vector that specifies the direction and distance each point will move. This vector will guide the translation of each point in the shape.
- Identify the translation vector v = (a, b), where “a” represents the horizontal shift and “b” represents the vertical shift.
- For example, a translation vector of (3, -2) means each point will move 3 units to the right and 2 units down.
Step 2: Plot All Points to be Translated
With the translation vector identified, plot the initial positions of each point on a coordinate grid. This establishes the starting reference points for the translation.
- Plot each original point (e.g., A, B) on graph paper or a coordinate plane.
Step 3: Draw Guidelines for the Translation
Draw a guideline in the direction specified by the translation vector for each point. This visual guide shows the direction each point will move along.
- For each point, use a ruler to draw a guideline extending from the point in the direction indicated by the translation vector.
- These guidelines will help visualize the direction in which each point will be translated.
Step 4: Measure the Translation Distance
Use the translation vector’s components to determine the new position of each point along the guideline.
- Move each point “a” units horizontally (right if “a” is positive, left if “a” is negative) and “b” units vertically (up if “b” is positive, down if “b” is negative).
- This new location is the translated position of each point according to the translation vector.
Step 5: Mark the Translated Points
Mark the new position of each point after translation to complete this part of the transformation process.
- Mark each translated position and label it with a new name, such as A’, B’, etc.
Step 6: Connect the Translated Points (if forming a shape)
If translating a shape, connect the translated points in the same order as the original points to preserve the shape and orientation.
- Connect each new point in the same sequence as the original points to complete the translated shape.
Effective Rate Calculation for Doubling an Investment
Problem Statement:
Given the compound interest formula:
A = P (1 + r/n)nt
where:
- P is the principal (initial amount),
- r is the annual interest rate,
- n is the number of compounding periods per year,
- t is the number of years.
We want to determine the effective rate per compounding period, r/n, required to double the initial amount P (i.e., A = 2P), while keeping n and t constant.
Solution:
To find r/n that will double P over time t with n compounding periods per year, start by setting up the equation:
2P = P (1 + r/n)nt
Dividing both sides by P:
2 = (1 + r/n)nt
Next, take the nt-th root of both sides to remove the exponent:
1 + r/n = 21/(nt)
Then, isolate r/n by subtracting 1 from both sides:
r/n = 21/(nt) – 1
Conclusion:
Thus, the effective interest rate per compounding period required to double the principal P is:
r/n = 21/(nt) – 1
This formula provides the rate that applies at the end of each compounding period to double the initial investment over the specified time.