Practice Makes Perfect Algebra - Carolyn Wheater [31]
Figure 12.1 Exponential growth and decay.
Plotting a few key points can help you shape the graph of an exponential equation. Always look for the y-intercept. If y = abx, when x = 0, b0 = 1, so the y-intercept will be a. Plugging in 1 and −1 for x will give you two more points that will easily set the shape:(1, ab) and . The graph of y = 3(2)x has a y-intercept of(0, 3) and passes through the points(1, 6) and , as shown in Figure 12.2.
Figure 12.2 Graph of exponential growth.
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EXERCISE 12.3
Graph each exponential function by making a table of values and plotting points.
1. y = 2(1.5)x
2. y = 4(.75)x
3. y = 2x
5. y = 3(2)x
6. y = -1(3)x
8. y = −2(1.5)x + 4
9. y = 5(.9)x + 2
10. y = 10(1.1)x − 3
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Matrix algebra
In mathematics, a matrix is a rectangular arrangement of numbers. On the coordinate plane shown in Figure 13.1, the triangle has vertices at the points(1, 2),(−1, −1), and(2, −1). It can be represented by the matrix
Figure 13.1 The vertices of a triangle can be organized in a matrix.
Numbers are often organized into matrices because they represent similar pieces of information or because the same calculation must be performed on all of them. Matrices are generally enclosed in square brackets, although sometimes other enclosures are used. Any rectangular arrangement may be considered a matrix, even if it is not enclosed(see Figure 13.2).
Figure 13.2 Matrices can be written in different formats.
Rows and columns
The matrix as a whole can be named by a single capital letter, and the individual numbers within the matrix are called elements. We can say . Every matrix is organized into rows, which are horizontal lines of elements, and columns, which are vertical stacks of numbers. The matrix has two rows, each containing five elements. The matrix has three rows, each containing two elements. The matrix has five columns of two elements, and the matrix has two columns of three elements.
The dimension, or order, of a matrix is a description of its size, giving first the number of rows and then the number of columns. The matrix is a 2 × 5 matrix, meaning that it has two rows and five columns. Because the matrix has three rows and two columns, we say its dimension is 3 × 2. Matrix has three rows and three columns, so it has dimension, or order, 3 × 3. Because the number of rows and columns are the same in matrix A, we say that matrix A is square. Matrix is also a square matrix and its dimension is 2 × 2.
A matrix with only one row is called a row matrix and a matrix with only one column is called a column matrix. Row matrices and column matrices are sometimes referred to as vectors.
Although the purpose of grouping elements into a matrix is generally to allow repetitive calculations to be performed in one operation, there are times when we want to refer to one specific element of the matrix. To do this, we indicate the row and column in which the element sits. If we wanted to point out the 4 in the matrix , we would talk about m(2, 3). The lowercase m indicates that we’re talking about an element of matrix M, the 2 indicates the row, and the 3 denotes the column. The element at the intersection of the second row and the third column is 4, so we write m(2, 3) = 4. If instead we refer to m(1, 5), we indicate the 7 in the first row, fifth column. To refer to the 5 in , we could write b(2, 1)=5.
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EXERCISE 13.1
Give the dimension of each matrix.
6. b(1, 3)
7. b(3, 2)
8. b(2, 3)
9. b(1, 1)
10. b(2, 1)
11. A small sporting goods shop keeps records of the types of purchases made by its customers. These records are organized into categories of equipment, clothing, accessories, and books, and then each category is divided by sport. Organize the records from a typical day, below, into a matrix.
Equipment: 24 tennis,