HSA: Algebra

ALGEBERA

Mathematical Formulas

When given a mathematical formula, you may be asked to:

Evaluate the formula by substituting numbers for one or more variables (symbols representing a certain quantity).
Solve the formula for a variable.

View the attached list of mathematical formulas.

Patterns

Patterns can consist of numbers, algebraic expressions, or geometric figures. In order to recognize a pattern, look for the change from one term of the pattern to the next element. In number patterns, a sequence of numbers will be given and the assigned task may be to describe or continue the pattern. To discover the pattern, look for the difference between and the ratio of two consecutive numbers. Algebraic patterns are similar to number patterns except they use algebraic expressions. In geometric patterns, the patterns may be based on the number of shapes, the size of shapes, the position of the shapes, or their geometric properties. Sometimes a pattern may be presented in a word problem, shown in a table, or described using symbols or graphs.

Algebraic Expressions

Algebraic expressions are expressions that use numbers and/or variables. These expressions are similar to numbers because you can add, subtract, multiply, and divide with them. In order to solve an algebraic expression, choose the appropriate whether to add, subtract, multiply, or divide when combining the algebraic expressions based on the context of the problem.

Graphs

Graphs tell about the function that is being graphed. The maxima and minima are the greatest and least y-values on the graph of a function. To find the maximum of a graph, look where it has the greatest y-value. To find the minimum of a graph, look where it has the least y-value. Although the maxima and minima depend on the y-values of the graph, they are usually referred to be their x-coordinates. Note that a graph may have more than one maximum or minimum.

In any function, there are two related sets of values known as the domain and range. The domain is the set of x-values that the function is defined on. The range is the set of y-values that the function takes on.

A graph is continuous if the graphs does not have any skips or jumps. The graph is not continuous if there are any skips or jumps on it.

Zeros (or roots) of a function refer to where the graph intersects (crosses or touches) the y-axis.

The rate of change of a function is the same as its slope. The rate of change is positive when the graph is sloped upward. The rate of change is negative when the graph is sloped downward. The rate of change is 0 when the graph is horizontal. The rate of change is greatest in absolute value when the graph is steepest.

The graph of a linear function is always a straight line. However, many graphs applied to real-world problems include curves or combinations of lines and curves. These graphs are non-linear graphs.

Linear Equations

A linear equation is any equation that includes only constants and variables that are multiplied by coefficients. (An example of a linear equation is 2x -12=10.) The main rule to follow when solving linear equations is that whatever operation you perform on one side of the equation, you must also perform on the other side of the equation.

A straight line connects any two points in the coordinate plane on a graph. View the Mathematical Formulas sheet for ways to find the equation of the line.

There are two simple ways to graph linear equations:

If you know two points on the line, simply plot the two points and draw the straight line that goes through the two points.
If you know the slope and the y-intercept of the line, plot the y-intercept first. Then, use the "rise over run" interpretation of slope to plot a second point. Finally, draw the straight line that goes through the two points.

Inequalities

While some situations require exact answers, other times there may be a range of solutions. In mathematics, these types of situations are expressed by inequalities. Inequalities can be expressed using the symbols for "greater than", "greater than or equal to", "less than" or "less than or equal to."

When an inequality involves only a single variable, its solution can be graphed on a number line. An open circle on the number line means a number is not part of the solution set. A closed circle means that the number is part of the solution set.

When an inequality involves two variables, its solution is best represented on the coordinate plane. Graphs of inequalities with "greater than or equal to " or "less than or equal to" include the line. Graphs of inequalities with "greater than" or "less than" do not include the line.

Solving inequalities follows the same rules as solving equations with two exceptions:

When you multiply or divide both sides of an inequality by a negative number, the inequality symbol changes direction.
In real-world situations, answers will often need to be rounded up or down, depending on the context of the problem.

System of Equations

A system of equations is a set of two or more equations that must be solved as the same time. When solving problems that involve two linear equations, these problems are solved most easily be graphing the equations. The solution to a system of equations will be where the graphs intersect.

If you are unable to graph a system of equations, algebra can be used to find a solution. One of the simplest methods for solving a system of two linear equations includes the following steps:

Solve both equations for y.
Solve the two equations equal to each other, and solve for x.
Substitute the value of x into either equation to find the value of y.

Parallel lines on a graph never intersect. A system of equations that includes two parallel lines will not have any solution.

Two differently straight lines either intersect in one point (one solution) or never intersect (no solution). If two lines in a system of equations are the same line, the lines intersect at infinitely many points. Therefore, the system of equations will have infinitely many solutions. Lines that completely overlap are sometimes called coincident lines.

Matrix

A matrix is a mathematical way of representing information. A matrix often provides the same information as a table or chart, however mathematical operations can be performed with matrices.

Surveys and Data

Surveys gather information about the public. The population is entire set of individuals about whom you would like information. A sample is the smaller set of individuals from whom you collect information in a survey. A sample is always part of the larger population. Information from samples is used to make predictions or draw conclusion about the population. A representative sample is as similar to the population as possible.

In a random sample, all members of the population have the same probability of being chosen for a survey. In a simple random sample, all members of the population have the same probability of being chosen for a survey, and the probability of one member being chosen is not affected by whether or not another member is chosen. A simple random sample is the best way to obtain a representative sample.

Conclusions from data are frequently based on the mean, median, mode or range of a set of data.

The mean is also known as the average of a set of numbers. To find the mean of a set of numbers, add the number in the set together and divide them by how many numbers are in the set.

The median is the middle value of a set of numbers when organized from least to greatest. If a data set contains an even number of values, the median of the set is the mean of the two middle numbers.

The mode is the number that occurs most frequently in the data set.

Range is a measure of the spread of the data. When the range of a set of data is large, the data are spread out. When the range is small, the data are close together.

When data are divided into fourths, the divisions between the groups of data are called quartiles.

The first quartile is the median of the lower half of the data.
The second quartile is the median of all the data.
The third quartile is the median of the upper half of the data.
The interquartile range is the range between the the third and first quartiles.

Experimental probability is probability that is based on data. To calculate experimental probability:

Find the total number of times that all outcomes in the data occurred.
Find the number of times that the given outcome occurred.
Divide the second number by the first.

An inference is a prediction based on data.

A simulation is a method used to estimate probability. However, the results of a simulation will not always be accurate. The more trials in a simulation, the more reliable the results of the simulation will be.

Sample Tests

2009
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Answer Key 2009

2008
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2007
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2006
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2005
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2004
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2003
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2002
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