The language of algebra | Practice Problems, Examples & Solutions

Algebraic Expressions

Precalculus : Building Concepts and Connections

Use the verbal description to find an algebraic expression for the function.
The graph of the function $h(t)$ is formed by scaling the graph of $f(t)=|t|$ vertically by a factor of $\frac{1}{2}$ and shifting it up 4 units.

More About Functions and Equations

Transformations of the Graph of a Function

00:37

Precalculus : Building Concepts and Connections

Use the verbal description to find an algebraic expression for the function.
The graph of the function $g(t)$ is formed by vertically scaling the graph of $f(t)=|t|$ by a factor of -2 and moving it to the left by 5 units.

More About Functions and Equations

Transformations of the Graph of a Function

01:15

Precalculus : Building Concepts and Connections

Evaluate $g(-x), g(2 x),$ and $g(a+h)$.
$$g(x)=2 x-3$$

Functions, Graphs, and Applications

Functions

Evaluating Algebraic Expressions and Formulas

574 Practice Problems

02:37

Precalculus : Building Concepts and Connections

State the domain, range, $x$ -intercept $(s),$ and $y$ -intercept, and find the expression for $f(x)$.
(GRAPH CAN'T COPY)

Functions, Graphs, and Applications

Piecewise-Defined Functions

01:16

Precalculus : Building Concepts and Connections

Graph the function by hand.
$$f(x)=\left\{\begin{array}{ll}
-2, & x<-1 \\
|x|, & -1 \leq x \leq 2 \\
2, & 2<x \leq 4
\end{array}\right.$$

Functions, Graphs, and Applications

Piecewise-Defined Functions

01:07

Precalculus : Building Concepts and Connections

Graph the function by hand.
$$F(x)=\left\{\begin{array}{ll}
0, & x \leq 1 \\
2, & x>1
\end{array}\right.$$

Functions, Graphs, and Applications

Piecewise-Defined Functions

Simplifying Algebraic Expressions and the Distributive Property

448 Practice Problems

01:09

Precalculus

Work each problem. Round to the nearest tenth of a degree, if necessary.
Temperature in Haiti The average annual temperature in Port-au-Prince, Haiti, is approximately $28.1^{\circ} \mathrm{C} .$ What is the corresponding Fahrenheit temperature?

Equations and Inequalities

Linear Equations

01:09

Precalculus

Work each problem. Round to the nearest tenth of a degree, if necessary.
Temperature of Venus Venus is the hottest planet, with a surface temperature of $867^{\circ} \mathrm{F}$. What is this temperature in Celsius? (Source: World Almanac and Book of Facts.

Equations and Inequalities

Linear Equations

01:07

Precalculus

In the metric system of weights and measures, temperature is measured in degrees Celsius ("C) instead of degrees Fahrenheit ( $^{\circ} \mathrm{F}$ ). To convert between the two systems, we use the equations
$$C=\frac{5}{9}(F-32) \text { and } F=\frac{9}{5} C+32$$
In each exercise, convert to the other system. Round answers to the nearest tenth of a degree if necessary.
$$20^{\circ} \mathrm{C}$$

Equations and Inequalities

Linear Equations

Combining Like Terms

199 Practice Problems

00:50

Precalculus

Using a graphing utility, plot $y_{1}=\sqrt{1-x}, y_{2}=x^{2}+2$ and $y_{3}=y_{1}^{2}+2 .$ If $y_{1}$ represents a function $f$ and $y_{2}$ represents a function $g$, then $y_{3}$ represents the composite function $g \circ f .$ The graph of $y_{3}$ is only defined for the domain of $g \circ f .$ State the domain of g of.

Functions and Their Graphs

Combining Functions

01:02

Precalculus

For the functions $f(x)=x+a$ and $g(x)=\frac{1}{x-a},$ find $g \circ f$ and state its domain.

Functions and Their Graphs

Combining Functions

01:11

Precalculus

A couple are about to put their house up for sale. They bought the house for $\$ 172,000$ a few years ago; if they list it with a realtor, they will pay a $6 \%$ commission. Write a function that represents the amount of money they will make on their home as a function of the asking price $p$

Functions and Their Graphs

Combining Functions

Simplifying Expressions to Solve Equations

1947 Practice Problems

02:34

Introductory and Intermediate Algebra for College Students

Will help you prepare for the material covered in the next section.
a. Solve by factoring: $8 x^{2}+2 x-1=0$
b. The quadratic equation in part (a) is in the standard form $a x^{2}+b x+c=0$. Compute $b^{2}-4 a c .$ Is $b^{2}-4 a c$ a perfect square?

Quadratic Equations and Functions

The Square Root Property and Completing the Square

02:59

Introductory and Intermediate Algebra for College Students

Solve by completing the square:
\[
x^{2}+x+c=0
\]

Quadratic Equations and Functions

The Square Root Property and Completing the Square

01:34

Introductory and Intermediate Algebra for College Students

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The graph of $y=(x-2)^{2}+3$ cannot have $x$ -intercepts.

Quadratic Equations and Functions

The Square Root Property and Completing the Square

Using Equations to Solve Application Problems

345 Practice Problems

00:56

Precalculus

Give two parametric representations for the equation of each parabola.
$$y=(x+3)^{2}-1$$

Applications of Trigonometry

Parametric Equations, Graphs, and Applications

03:31

Precalculus

Graph each plane curve defined by the parametric equations for $t$ in $[0,2 \pi] .$ Then find
a rectangular equation for the plane curve.
$$x=3 \cos t, y=3 \sin t$$

Applications of Trigonometry

Parametric Equations, Graphs, and Applications

12:50

Precalculus

The height of a certain tree in feet after $x$ years is modeled by
$$
f(x)=\frac{50}{1+47.5 e^{-0.22 x}}
$$
(a) Make a table for $f$ starting at $x=10,$ and incrementing by $10 .$ What appears to be the maximum height of the tree?
(b) Graph $f$ and identify the horizontal asymptote. Explain its significance.
(c) After how many years was the tree 30 ft tall? Round to the nearest tenth.

Inverse, Exponential, and Logarithmic Functions

Applications and Models of Exponential Growth and Decay