# Intermediate Algebra: Functions and Graphs

## SectionA.2Entering Expressions

### SubsectionA.2.1Parentheses

Order of Operations: The calculator follows the standard order of operations.

#### ExampleA.2.1.

Compute $$2+3\cdot 4\text{.}$$
Enter $$\qquad 2$$ + $$3$$ × $$4$$
Ans. $$14$$

#### ExampleA.2.2.

Compute $$(2+3)\cdot4\text{.}$$
Enter $$\qquad$$ ( $$2$$ + $$3$$ ) × $$4$$
Ans. $$20$$

### SubsectionA.2.2Fractions

#### ExampleA.2.3.

Compute $$\dfrac{1+3}{2}\text{.}$$
Enter $$\qquad$$ ( $$1$$ + $$3$$ ) ÷ $$2$$
Ans. $$2$$

#### CautionA.2.4.

GeoGebra displays "built-up" fractions like $$\dfrac{1}{2} \text{.}$$ Once we enter the numerator and select the ÷ key, the cursor is in the denominator and will stay there until we use an arrow key on the keyboard to move the cursor outside the fraction, or until we press . The arrow keys on the virtual keyboard are shown in the keyboard image A.0.1.

#### ExampleA.2.5.

Compute $$\dfrac{1}{2\cdot 3} \text{.}$$
Enter $$\qquad 1$$ ÷ $$2$$ × $$3$$
Ans. $$\dfrac{1}{6}$$
In the output history, there is an approximation icon that looks like the "$$\approx$$" symbol in a blue square. To get a decimal approximation of the fraction, click on that icon.
Ans. $$0.1666666666667$$

#### ExampleA.2.6.

Compute $$\dfrac{1}{2}\cdot 3 \text{.}$$
Enter $$\qquad 1$$ ÷ $$2$$ × $$3$$
Ans. $$\dfrac{3}{2}$$

### SubsectionA.2.3Exponents and Powers

The key for exponents and the key for squaring are shown in the keyboard image.

#### ExampleA.2.7.

Evaluate $$57^{2}\text{.}$$
Enter $$\quad 57\text{,}$$ then the squaring key, then
OR
Enter $$\quad 57\text{,}$$ then the exponent key, then $$2~$$
Ans. $$3249$$

#### ExampleA.2.8.

Evaluate $$2^{10}\text{.}$$
Enter $$\quad 2\text{,}$$ and then the exponent key, 10
Ans. $$1024$$

#### CautionA.2.9.

GeoGebra nicely displays powers such as $$10^7 \text{,}$$ with the exponent raised like a superscript. Once we enter the base and select the exponent key, the cursor is in the exponent and will stay there until we use the arrow keys to move the cursor outside the power, or until we press . The arrow keys on the virtual keyboard are shown in the keyboard image.

#### ExampleA.2.10.

Evaluate $$8^{2/3}\text{.}$$
Enter $$\quad 8\text{,}$$ then the exponent key, then $$2$$ ÷ $$3$$
Ans. $$4$$

#### ExampleA.2.11.

Evaluate $$\dfrac{8^2}{3}\text{.}$$
Enter $$\quad 8\text{,}$$ then the exponent key, then $$2$$ ÷ $$3$$
Ans. $$\dfrac{64}{3}$$

### SubsectionA.2.4Square Roots

The key for square roots is shown in the keyboard image A.0.1.

#### ExampleA.2.12.

Evaluate $$\sqrt{2} \text{.}$$
Select the square root key, then enter $$~ 2$$
Ans. $$1.414213562$$

#### CautionA.2.13.

GeoGebra nicely displays square roots such as $$\sqrt{9+16} \text{,}$$ with the radicand inside the square root "house". Once we select the square root key, the cursor is in under the radical and will stay there until we use the arrow keys to move the cursor outside, or until we press . The arrow keys on the virtual keyboard are shown in the keyboard image.

#### ExampleA.2.14.

Evaluate $$\sqrt{9+16} \text{.}$$
Select the square root key, then enter $$~9~$$ + $$~16~$$
Ans. $$5$$

#### ExampleA.2.15.

Evaluate $$\sqrt{9}+16 \text{.}$$
Select the square root key, then enter $$~9~$$ + $$~16~$$
Ans. $$19$$

### SubsectionA.2.5Other Roots

If we click on the "f(x)" on the keyboard in the Algebra View, we see other built-in keys.

#### ExampleA.2.17.

Compute $$\sqrt[3]{1728} \text{.}$$
Choose the f(x) keyboard, select the nth root key, go back to the standard 123 keyboard, then enter $$~3~$$ $$~1728~$$
Ans. $$12$$

#### ExampleA.2.18.

Compute $$\sqrt[10]{2\cdot 512} \text{.}$$
Choose the f(x) keyboard, select the nth root key, go back to the standard 123 keyboard, then enter $$~10~$$ $$~2~$$ × $$~512~$$
Ans. $$2$$

### SubsectionA.2.6Absolute Value

GeoGebra nicely displays the vertical bars of the standard absolute value notation, such as in the expression $$\abs{21\cdot 54 - 81} \text{.}$$ Once we select the absolute value key, the cursor is in between the vertical bars and will stay there until we use the arrow keys to move the cursor outside, or until we press . The arrow keys on the virtual keyboard are shown in the keyboard image.

#### ExampleA.2.19.

Evaluate $$\dfrac{\abs{21\cdot 54 - 81}}{-9} \text{.}$$
Select the absolute value key, then enter $$~21~$$ × $$~54~$$ - $$~81~$$ ÷ $$~{-9}~$$
Ans. $$-117$$

### SubsectionA.2.7Scientific Notation

The GeoGebra calculator does not display scientific notation by default. The calculator can display scientific notation in the Grahics View by using a ScientificText command.

#### ExampleA.2.20.

Compute $$123,456,789^2 \text{.}$$
Enter $$123456789$$ x^2
Ans. $$15241578750190522$$
We will enter ScientificText in the input box.
• On the computer, typing the first three letters (sci) is enough for the full command to appear as an option.
• On a phone app, we tap on the overflow menu (three horizontal dots) on the top right of the virtual keyboard (see keyboard image), tap in the Search in All Commands field, then start to input ScientificText until the full command appears as an option.
We click on ScientificText, and the cursor is appropriately positioned, ready for us to enter an expression. We can either repeat the calculation we entered above, or, more efficiently, select the last answer key ans and .
The result $$1.52415787501905 \times 10^{16}\text{,}$$ appears on the grid in the Graphics View. We can drag the text within the Graphics View, change its color, and/or delete it.
We could also type in the entire command scientifictext using the ABC keyboard, then use the 123 keyboard to enter the parentheses and numbers.

### SubsectionA.2.8Editing an Entry

We can edit an expression without starting again. The backspace key will remove the character to the left of the cursor. We can move the cursor within an input line using the arrow keys or by clicking in the appropriate place.
We can recall any previous entry by finding it in the output history (scrolling up if necessary), tapping on the kebab menu icon (three vertical dots), and selecting Duplicate. The command will be copied into the current input box, which we can edit before pressing .

#### NoteA.2.21.If the menu does not appear.

Sometimes when using the GeoGebra app embedded in this textbook, the menu does not appear when we tap on the kebab menu icon. This may occur when we are at a place in the section far from the top of the webpage. Try scrolling within the webpage to the top, and tap on the kebab menu icon again.
If we just want to use the most recent result in the current command, we use the ans key.

#### ExampleA.2.22.

Evaluate $$5^2 + 12^2$$ and then take the square root of the sum.
Enter $$5\text{,}$$ then the square key, then + $$12\text{,}$$ the square key again, then
Ans. $$169$$
Now select the square root key, then ans
Ans. $$13$$