Problem 54 - Algebra 2

Question

A quadrilateral has coordinates \(\left[\begin{array}{cccc}{3} & {4} & {-3} & {-4} \\ {5} & {4} & {-4} & {-3}\end{array}\right]\) a. Graph the quadrilateral. b. Find the product \(\left[\begin{array}{rr}{0} & {-1} \\ {1} & {0}\end{array}\right]\left[\begin{array}{cccc}{3} & {4} & {-3} & {-4} \\ {5} & {4} & {-4} & {-3}\end{array}\right]\) c. Graph the result as a new quadrilateral. d. What is the relationship between the quadrilaterals in parts (a) and (c)?

Step-by-Step Solution

Answer

The quadrilateral, when multiplied by the matrix, rotates 90 degrees counterclockwise around the origin. The resulting quadrilateral is the original one rotated.

Step 1 - Graphing the Original Quadrilateral

Plot the points of the original quadrilateral on a Cartesian plane. The points are (3,5), (4,4), (-3,-4), and (-4,-3). Connect these points in order, forming a quadrilateral.

Step 2 - Multiplying Matrices

Multiply the 2x2 matrix by the 2x4 matrix to obtain the coordinates of the transformed quadrilateral. Perform matrix multiplication by taking the dot product of the rows of the first matrix with the columns of the second matrix.

Step 3 - Calculating the Product

For each column of the second matrix, calculate the product with the first matrix. The result is a new 2x4 matrix that represents the coordinates of the transformed quadrilateral.

Step 4 - Graphing the Transformed Quadrilateral

Graph the resulting quadrilateral formed by the new coordinates. Plot the points on the same Cartesian plane as the original quadrilateral.

Step 5 - Analyzing the Relationship

Compare the original quadrilateral and the transformed quadrilateral to determine their relationship, which could be rotation, reflection, translation, or dilation.

Key Concepts

Coordinate Plane

The coordinate plane, also known as a Cartesian plane, is a two-dimensional surface defined by two number lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants, which allow us to plot points given by coordinates.The coordinates are written as ordered pairs \( (x, y) \). The first number, or x-coordinate, indicates the horizontal position relative to the origin \( (0,0) \), while the second number, or y-coordinate, indicates the vertical position. To graph a quadrilateral on this plane, we plot its vertices using their coordinates and connect them in sequence to form the shape.Understanding how to graph on a coordinate plane is crucial for visualizing many mathematical concepts, including the transformation of shapes.

Matrix Multiplication

Matrix multiplication is a fundamental operation that can represent various transformations in a coordinate plane. Unlike element-wise multiplication, matrix multiplication involves taking the dot product of the rows of the first matrix with the columns of the second.The process can be visualized as follows:

• Select the first row of matrix A and the first column of matrix B.
• Multiply each corresponding element and sum the products to get the element of the resulting matrix.
• Repeat this process for each row of matrix A and column of matrix B.

It's important for students to ensure the number of columns in the first matrix matches the number of rows in the second, or else the multiplication isn't defined.

Transformation of Shapes

When working with shapes on a coordinate plane, we often perform transformations to change their position, size, or orientation. There are four main types of transformations:

• Translation: Sliding the shape without rotating or flipping it.
• Rotation: Turning the shape around a fixed point.
• Reflection: Flipping the shape over a line to create a mirror image.
• Dilation: Changing the size of the shape while maintaining its proportions.

To achieve these transformations algebraically, we can multiply the coordinates of a shape by a specific matrix. Each transformation type has a corresponding matrix that encodes how to move each point in the shape.

Cartesian Plane Graphing

Graphing on a Cartesian plane is an essential skill for visualizing and understanding relationships between algebraic equations and geometric figures. To effectively graph on the Cartesian plane, it is helpful to:

• Clearly mark the x-axis and y-axis along with their respective scales.
• Identify the quadrants where specific points or shapes will appear based on their coordinates.
• Use a ruler or straightedge to draw the sides of geometric figures, like quadrilaterals, for precision.

When students graph the original and transformed quadrilaterals as in the exercise, they can observe how matrix multiplication corresponds to rotation, reflecting the real-world application of matrices in graphics and animation.