## 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?

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What is the Zero-Factor Property?

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What does extracting square roots involve?

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What does the quadratic formula do?

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What does a positive discriminant indicate?

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What does a zero discriminant indicate?

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What does a negative discriminant indicate?

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What are real numbers?

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

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

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

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

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

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

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## 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.

## Studying That Suits You

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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.