College Algebra 1314 Flashcards

Finding Intercepts

• x-intercepts are found by setting y to zero and solving for x.
• y-intercepts are determined by setting x to zero and solving for y.

Linear and Quadratic Equations

• A linear equation in one variable is represented as ax + b = 0, where a and b are real numbers and a ≠ 0.
• A quadratic equation, also known as a second-degree polynomial, is expressed as ax² + bx + c = 0, with a, b, and c as real numbers and a ≠ 0.

Zero-Factor Property

• The property states that if ab = 0, then a = 0 or b = 0.
• Utilized for solving quadratic equations by factoring and setting each factor to zero.

Square Root Solutions

• For the equation u² = d (where d > 0), the solutions are u = ±√d, indicating two possible values differing only by sign.
• Extracting square roots allows for solving equations without the need for factoring.

Completing the Square

• To solve a non-factorable quadratic equation, complete the square to facilitate extracting roots.
• The process involves rewriting the equation in a specific square form, particularly:
  • x² + bx + (b/2)² = (x + b/2)².

Quadratic Formula

• The formula for solving quadratic equations is given by:
  • x = [ -b ± √(b² - 4ac) ] / 2a.
• The discriminant (b² - 4ac) indicates the nature of the solutions.

Nature of Solutions Based on Discriminant

• A positive discriminant results in two distinct real solutions, with the graph showing two x-intercepts.
• A zero discriminant produces one repeated real solution, reflected in the graph with one x-intercept.
• A negative discriminant indicates no real solutions, and the graph features no x-intercepts, implying complex solutions exist.

Real Numbers and Graph Symmetry

• Real numbers are displayed on a number line, with positive numbers to the right of 0 and negative numbers to the left.
• Graphs can exhibit symmetry about the x-axis, y-axis, or the origin.

Graph Symmetry Conditions

• Symmetry with respect to the x-axis means (x, y) implies (x, -y) is also on the graph.
• Symmetry with respect to the y-axis is indicated by (x, y) suggesting (-x, y) must exist as well.
• Symmetry with respect to the origin implies (x, y) indicates (-x, -y) as part of the graph.

Algebraic Tests for Symmetry

• To demonstrate symmetry about the x-axis, replacing y with -y in the equation must yield an equivalent equation.
• For symmetry about the y-axis, substituting x with -x should result in an equivalent equation.
• Symmetry about the origin is verified if replacing x with -x and y with -y leads to an equivalent equation.