Questions and Answers
What is the definition of variables and expressions?
To solve an expression isolate the variable.
What must you do when you perform an operation on one side of an equation?
You must do the same to the other side.
What is the first step in solving absolute value equations?
Isolate the variable.
What happens to the inequality when you multiply or divide both sides by a negative number?
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What are continuous graphs comprised of?
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What is required for a relation to be considered a function?
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What is the slope-intercept form of a linear function?
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What does the correlation coefficient indicate?
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What do parallel lines have in common regarding slope?
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What should be done to the slope when reflecting a line?
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What is a system of linear equations?
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What indicates that a system of linear equations has infinite solutions?
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What is the form of a linear inequality?
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What is the result of raising a number to the power of zero?
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What does a fractional exponent indicate?
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What defines a polynomial?
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What do you do when adding and subtracting polynomials?
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What does FOIL stand for?
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What is a perfect square trinomial?
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What characterizes a set of numbers that is closed under an operation?
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What is the general form of a quadratic function?
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What indicates the vertex of a parabola?
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What is true about a parabola if the value of a is positive?
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Study Notes
Variables and Expressions
- Isolating the variable is essential for solving an expression.
Solving Equations
- Perform the same operation on both sides of the equation to maintain equality.
Absolute Value Equations
- Absolute value |x| represents distance from 0 and is always non-negative.
- Steps to solve:
- Isolate the variable
- Rewrite as two cases (e.g., |x+2| = 6 leads to x+2 = 6 and x+2 = -6)
- Solve each case
- If the result for y is negative, there are no solutions.
Solving Inequalities
- Multiplying or dividing by a negative number necessitates flipping the inequality sign.
- A negative sign alone does not automatically trigger a sign flip.
Graphing Relations
- Continuous graphs encompass all real numbers while discrete graphs consist of whole numbers.
Function
- Each x (domain) must correspond to a unique y (range).
- Domain includes x-values (independent input), while range includes y-values (dependent output).
Linear Function
- Slope-intercept form is given by y=mx+b, with m as slope and b as y-intercept.
- A linear function maintains constant changes in x and y.
Rate of Change
- The slope is calculated using the formula: (y2 - y1) / (x2 - x1).
- Slope represents the change in y over the change in x, often stated as "rise over run."
Slope
- A line with a negative slope descends from left to right.
- A vertical line indicates no slope.
Direct Variation
- The relationship y = kx demonstrates direct variation with k as a nonzero constant.
- Examples include y = 420x, while y = 69x + 2 does not indicate direct variation.
Point Slope Form
- Utilized when provided with a slope and a specific point on a line.
Correlation Coefficient
- A value (r) close to 1 indicates strong positive correlation; values near -1 represent strong negative correlation.
- Increased data spread results in weaker correlation.
Slopes of Parallel and Perpendicular Lines
- Parallel lines have identical slopes.
- Perpendicular lines have slopes that are opposite reciprocals.
Transformations
- Vertical shifts are achieved by modifying the y-intercept (b).
- Adjusting the slope (m) results in rotation.
- Reflecting a line changes the slope's sign.
System of Linear Equations (SLE)
- Consists of two or more linear equations with two or more variables.
- Solutions exist where the lines intersect.
Solving SLE
- Methods include graphing, substitution, and elimination.
Solutions
- Consistent Independent: unique solution; lines intersect at one point.
- Consistent Dependent: infinite solutions; lines overlap.
- Inconsistent: no solution; lines are parallel.
Linear Inequality
- A linear equation framed with an inequality symbol (e.g., <, >).
Integer Exponents
- Any nonzero number raised to the power of 0 equals 1.
- Negative exponents result in reciprocal fractions (e.g., 3^-2 = 1/3^2).
Fractional Exponents
- Indicates taking the root; for example, x^(1/n) signifies the nth root of x.
Polynomial
- A polynomial may be a monomial or a sum of monomials.
- The degree of a polynomial is defined by the highest sum of its exponents, arranged in standard form.
Adding and Subtracting Polynomials
- Distributing the sign is crucial before combining like terms.
FOIL
- A technique for multiplying two binomials.
Special Products of Binomial
- Perfect square trinomial identity: a^2 + 2ab + b^2.
- Difference of squares identity: a^2 - b^2 = (a + b)(a - b).
Closure
- A set is closed if the result of operations on any two numbers remains within the set.
Factoring ax^2 + bx + c
- Method 1: Guess and check.
- Method 2: Create a table for possible factors.
Choosing Best Factoring Method
- Assess for Gross Common Factor (GCF).
- Look for identifiable patterns.
- Apply other factoring strategies as needed.
Quadratic Function
- A function defined by ax^2 + bx + c where a ≠ 0.
- Determine the vertex by -b/(2a) and examine the second differences for concavity.
Graphing Parabola
- Locate the axis of symmetry at -b/(2a), including the vertex and y-intercept (c).
- The parabola's appearance varies based on the sign and magnitude of 'a'. Positive leads to a "happy" face; negative results in a "sad" face; the value of |a| influences the narrowness.
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This quiz consists of flashcards focused on key concepts in Algebra 1, including variables, expressions, and solving equations. Each card provides definitions and insights to help reinforce understanding of these foundational topics. Perfect for studying or quick revision before tests!
