unit 3 homework 4 graphing quadratic equations and inequalities answers

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02-02-2025

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Unit 3, Homework 4: Graphing Quadratic Equations and Inequalities – A Comprehensive Guide

This guide provides a thorough walkthrough of the concepts and solutions related to graphing quadratic equations and inequalities, commonly covered in Unit 3, Homework 4 of various algebra courses. We'll cover key aspects, including identifying key features, plotting points, and understanding the differences between equations and inequalities. Remember to consult your textbook and class notes for specific problem sets and variations.

Understanding Quadratic Equations and Inequalities

A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The graph of a quadratic equation is a parabola, a U-shaped curve. A quadratic inequality replaces the equals sign (=) with an inequality symbol (<, >, ≤, or ≥). This changes the nature of the solution, representing a region on the graph rather than a specific point.

Key Features of a Parabola

Before graphing, understanding the key features of a parabola is crucial. These include:

• Vertex: The highest or lowest point on the parabola. Its x-coordinate is given by x = -b / 2a.
• Axis of Symmetry: A vertical line passing through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b / 2a.
• x-intercepts (roots or zeros): The points where the parabola intersects the x-axis. These are found by solving the quadratic equation ax² + bx + c = 0 (using factoring, the quadratic formula, or completing the square).
• y-intercept: The point where the parabola intersects the y-axis. This is found by setting x = 0 in the equation, resulting in y = c.

Graphing Quadratic Equations

To graph a quadratic equation, follow these steps:

1. Find the vertex: Use the formula x = -b / 2a to find the x-coordinate of the vertex. Substitute this value back into the equation to find the y-coordinate.
2. Find the axis of symmetry: This is the vertical line x = -b / 2a.
3. Find the x-intercepts (if any): Solve the quadratic equation ax² + bx + c = 0.
4. Find the y-intercept: Set x = 0 and solve for y.
5. Plot the points: Plot the vertex, x-intercepts, y-intercept, and at least one or two additional points to ensure accuracy. Remember the parabola is symmetrical around the axis of symmetry, so you can use this to your advantage.
6. Sketch the parabola: Connect the points with a smooth, U-shaped curve.

Graphing Quadratic Inequalities

Graphing quadratic inequalities involves similar steps, but with an additional consideration:

1. Graph the related quadratic equation: Treat the inequality as an equation and graph it as described above. This will form the boundary of your solution region.
2. Determine the shading:
   • For inequalities of the form y > ax² + bx + c or y ≥ ax² + bx + c, shade the region above the parabola.
   • For inequalities of the form y < ax² + bx + c or y ≤ ax² + bx + c, shade the region below the parabola.
3. Consider the boundary:
   • If the inequality includes '≤' or '≥', the parabola itself is included in the solution, so draw a solid line.
   • If the inequality includes '<' or '>', the parabola is not included in the solution, so draw a dashed line.

Example

Let's consider the quadratic equation y = x² - 4x + 3.

1. Vertex: x = -(-4) / (2*1) = 2. y = (2)² - 4(2) + 3 = -1. Vertex is (2, -1).
2. Axis of symmetry: x = 2
3. x-intercepts: x² - 4x + 3 = 0 factors to (x-1)(x-3) = 0, so x-intercepts are (1, 0) and (3, 0).
4. y-intercept: When x = 0, y = 3. y-intercept is (0, 3).

Plot these points and sketch the parabola.

Conclusion

Mastering the graphing of quadratic equations and inequalities requires a solid understanding of parabolas and their characteristics. By following the steps outlined above and practicing with various examples, you can confidently tackle your Unit 3, Homework 4 assignment. Remember to always check your work and utilize online resources and your textbook to clarify any doubts. Good luck!