Since 2008, when I started my private practice as a math tutor in Las Vegas, I have used a unified formula that captures all six of the possible transformations that can be applied to a function. Textbooks, by contrast, typically teach the transformations separately. The photo above was well received by a calculus student who utilized my math tutoring services for a refresher in precalculus. From left to right, the possible transformations are as follows:

1. If the leading coefficient is negative, reflect the function about the x-axis. In other words, flip it upside down.
2. If the leading coefficient has an absolute value greater than one, the graph stretches vertically. For example, 2 means that height of every point is doubled. If the leading coefficient has an absolute value less than one, the graph shrinks (compresses) vertically. For example, 1/2 means that every point is reduced to half of its original height.
3. If the leading coefficient inside of the grouping symbol is negative reflect about the y-axis. In other words, flip it over laterally.
4. If the leading coefficient inside of the grouping symbol is greater than one, the graph shrinks (compresses) horizonally. For example, 3 means that the graph shrinks to one-third of its original width. If the leading coefficient inside of the grouping symbol is 1/3, the graph stretches to three times its original width.
5. If the operation inside of the grouping symbol is subtraction, move the graph right by the number of units to the right of the subtraction symbol. If the operation inside of the grouping symbol is addition, move the graph left by the number of units to the right of the subtraction symbol.
6. If the operation outside of the grouping symbol is addition, and the number being added is positive, move the graph up by the number of the units being added to the rest of the function. If the operation outside of the grouping symbol is addition, and the number being added is negative, move the graph down by the number of the units being added to the rest of the function. If the number outside of the grouping symbol is subtracted from the function, move the graph down by the number of unites being added to the rest of the function.

This method requires the student to choose a row or column. It is best to choose the one which has the most entries of zero, if any, because those terms do not have to be computed. Take the non-zero entry in the row or column, multiply it by the appropriate sign, based on raising -1 to the power of the sum of row and column number. Then, multiply it by the determinant of the 2 x 2 minor matrix containing the entries outside of the chosen row and column. Repeat for any other non-zero entries in the chosen row or column. Finally, sum all of these products to obtain the determinant of the 3x3 matrix.

A less complicated alternative is to use the method of diagonals. Although it would be nice to use this to compute determinants of matrices of higher order, it unfortunately only works for 3x3 square matrix. However, this shortcut method can be used to compute the minor 3x3 matrices derived from higher order matrices. It is a technique worth learning.

The first step is to rewrite the first two columns of entries to the right of the determinant matrix. Then, start with the upper left entry, in the first row and first column, and take the product of the three diagonal entries, moving downard from left to right. Repeat with the second and third entries in the first row, then sum the three products.

Next, start with the lower left entry and compute the three diagonal products, moving upward from left to right. Sum the three products, then subtract this sum from the sum of the previous sum.

Based on my experience as a private math tutor in Las Vegas since 2008, the method of diagonals is normally taught at the high school level. It is a less complicated method, but the tradeoff is that it doesnʻt work for higher order square matrices. The method of minor matrices and coefficients is typically taught at the high school honors and college level.

A third method for solving matrix determinants is Cramer's Rule, which I will not cover in this post.