College Algebra 1314 Flashcards

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Questions and Answers

How do you find x-intercepts?

Set y equal to zero and solve the equation for x.

How do you find y-intercepts?

Set x equal to zero and solve the equation for y.

What is a linear equation in one variable?

ax + b = 0

What is a quadratic equation?

ax^2 + bx + c = 0 Signup and view all the answers

What is the Zero-Factor Property?

If ab = 0, then a = 0 or b = 0. Signup and view all the answers

What does extracting square roots involve?

The equation u^2 = d has solutions: u = √d and u = -√d. Signup and view all the answers

What does the quadratic formula do?

-b ± √(b² - 4ac) / 2a Signup and view all the answers

What does a positive discriminant indicate?

The quadratic equation has two distinct real solutions. Signup and view all the answers

What does a zero discriminant indicate?

The quadratic equation has one repeated real solution. Signup and view all the answers

What does a negative discriminant indicate?

The quadratic equation has no real solutions. Signup and view all the answers

What are real numbers?

Numbers represented graphically on the real number line. Signup and view all the answers

What does it mean for a graph to be symmetric with respect to the x-axis?

If whenever (x, y) is on the graph, (x, -y) is also on the graph. Signup and view all the answers

What does it mean for a graph to be symmetric with respect to the y-axis?

If whenever (x, y) is on the graph, (-x, y) is also on the graph. Signup and view all the answers

What does it mean for a graph to be symmetric with respect to the origin?

If whenever (x, y) is on the graph, (-x, -y) is also on the graph. Signup and view all the answers

How can you algebraically test for symmetry with respect to the x-axis?

Replace y with -y and see if the equation is equivalent. Signup and view all the answers

How can you algebraically test for symmetry with respect to the y-axis?

Replace x with -x and see if the equation is equivalent. Signup and view all the answers

How can you algebraically test for symmetry with respect to the origin?

Replace x with -x and y with -y and see if the equation is equivalent. Signup and view all the answers

Study Notes

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.

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Description

Test your knowledge of key concepts in College Algebra 1314 HCC with these flashcards. From finding x and y intercepts to understanding linear equations, this quiz covers essential definitions and terms. Perfect for review and reinforcement of algebra principles.