# Math Tip of the Week: Parent Functions and Transformations

This topic is a popular one; it seems like at the beginning of most Algebra II and Pre-Calculus courses, parent functions and their transformations are discussed.

Here’s my table of the “most popular” parent functions and their graphs:

 Parent Function Graph Parent Function Graph $$y=x$$ Linear, Odd Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left( {-\infty ,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ $$y=\left| x \right|$$ Absolute Value, Even Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left[ {0,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to \infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ $$y={{x}^{2}}$$ Quadratic, Even Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left[ {0,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to \infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ $$y=\sqrt{x}$$ Radical (Square Root), Neither Domain: $$\left[ {0,\infty } \right)$$ Range: $$\left[ {0,\infty } \right)$$ End Behavior: $$\displaystyle \begin{array}{l}x\to 0,\,\,\,\,y\to 0\\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ $$y={{x}^{3}}$$ Cubic, Odd Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left( {-\infty ,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ $$y=\sqrt[3]{x}$$ Cube Root, Odd Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left( {-\infty ,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ $$\begin{array}{c}y={{b}^{x}},\,\,\,b>1\,\\(y={{2}^{x}})\end{array}$$Exponential, Neither Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\left( {0,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ Asymptote:  $$y=0$$ $$\begin{array}{c}y=lo{{g}_{b}}\left( x \right),\,\,b>1\,\,\,\\(y={{\log }_{2}}x)\end{array}$$Log, Neither Domain: $$\left( {0,\infty } \right)$$ Range: $$\left( {-\infty ,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to {{0}^{+}}\text{, }\,y\to -\infty \\x\to \infty \text{, }\,y\to \infty \end{array}$$ Asymptote:  $$x=0$$ $$\displaystyle y=\frac{1}{x}$$ Rational (Inverse), Odd Domain: $$\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)$$ Range: $$\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to 0\end{array}$$ Asymptotes: $$y=0,\,\,x=0$$ $$\displaystyle y=\frac{1}{{{{x}^{2}}}}$$ Rational (Inverse Squared), Even Domain: $$\left( {-\infty ,0} \right)\cup \left( {0,\infty } \right)$$ Range: $$\left( {0,\infty } \right)$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to 0\\x\to \infty \text{, }\,\,\,y\to 0\end{array}$$ Asymptotes:  $$y=0$$ $$y=int\left( x \right)=\left[ x \right]$$   Greatest Integer, Neither Domain:$$\left( {-\infty ,\infty } \right)$$ Range: $$\{y:y\in \mathbb{Z}\}\text{ (integers)}$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}$$ $$y=C$$   ($$y=2$$) Constant, Even Domain: $$\left( {-\infty ,\infty } \right)$$ Range: $$\{y:y=C\}$$ End Behavior: $$\begin{array}{l}x\to -\infty \text{, }\,y\to C\\x\to \infty \text{, }\,\,\,y\to C\end{array}$$

When functions are transformed on the outside of the $$f(x)$$ part, you move the function up and down and do the “regular” math, as we’ll see in the examples below.  These are vertical transformations.

When transformations are made on the inside of the $$f(x)$$ part, you move the function back and forth (but do the “opposite” math – basically since if you were to isolate the x, you’d move everything to the other side).  These are horizontal translations.

Please visit Parent Functions and Transformations to see more!

Hope your math classes are going well this year  😆

Lisa

# Math Tip of the Week – A Matrix Problem

Hi Math folks!  So sorry I haven’t been blogging recently; I’ve been working on getting my tutoring schedule worked out and have just been trying to catch up after a whirlwind trip to visit both of my sons “up north”  🙂

Here’s an interesting problem that someone just sent me:

Determine the solution by setting up and solving the matrix equation.
A nut distributor wants to determine the nutritional content of various mixtures of pecans, cashews, and almonds. Her supplier has provided the following nutrition information:
Almonds. Cashews. Pecans
Protein. 26.2g/cup. 21.0g/cup. 10.1g/cup
Carbs. 40.2g/cup. 44.8g/cup. 14.3g/cup
Fat. 71.9g/cup. 63.5g/cup. 82.8g/cup

Her first mixture, protein blend, contains 6 cups of almonds, 3 cups of cashews, and 1 cup of pecans. Her second mixture, low fat mix, contains 3 cups almonds, 6 cups cashews, and 1 cup of pecans. Her third mixture, low carb mix contains 3 cups almonds, 1 cup cashews, and 6 cups pecans. Determine the amount of protein, carbs, and fats in a 1 cup serving of each of the mixtures.

I have solved this by multiplying both of the matrices then dividing each element by 10 but that’s not the way I am supposed to solve this as there are no equations being set up.

### Solution:

Sometimes we can just put the information we have into matrices and see how we are going to go from there.  I knew to put the first group of data into a matrix with Almonds, Cashews, and Pecans as columns, and then put the second group of data into a matrix with information about Almonds, Cashews, and Pecans as rows.  This way the columns of the first matrix lined up with the rows of the second matrix, and I could perform matrix multiplication.  This way we get rid of the number of cups of Almonds, Cashews, and Pecans, which we don’t need:

Then I multiplied the matrices (using a graphing calculator) since I wanted to end up with the amount of Protein, Carbs, and Fat in each of the mixtures.  The product of the matrices consists of rows of Protein, Carbs, and Fat, and columns of the Protein, Low Fat, and Low Carb mixtures:

But we have to be careful, since these amounts are for 10 cups (add down to see we’ll get 10 cups for each mixture in the middle matrix above).  Also, notice how the cups unit “canceled out” when we did the matrix multiplication (grams/cup time cups = grams).

So to get the answers, I had to divide each answer by 10 to get grams per cup.  So the numbers in bold are my answers:

Learn more about matrices, how to use the graphing calculator with them,  and how to solve Systems of Equations using matrices in the The Matrix and Solving Systems with Matrices section!