algebra 2 graphing quadratics using vertex form answer key

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04-02-2025

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Graphing quadratic functions is a fundamental skill in Algebra 2. Understanding the vertex form of a quadratic equation is crucial for efficiently and accurately plotting parabolas. This guide provides an answer key to common graphing problems, along with a detailed explanation of the process, ensuring a thorough understanding of the concepts involved.

Understanding Vertex Form

The vertex form of a quadratic equation is written as:

f(x) = a(x - h)² + k

Where:

• (h, k) represents the coordinates of the vertex of the parabola. The vertex is the parabola's highest or lowest point, depending on whether the parabola opens upwards (a > 0) or downwards (a < 0).
• 'a' determines the parabola's vertical stretch or compression and its direction. If |a| > 1, the parabola is narrower than the parent function y = x². If 0 < |a| < 1, the parabola is wider. If a is negative, the parabola opens downwards.

Graphing Quadratics: A Step-by-Step Approach

Let's illustrate the process with an example:

Example: Graph the quadratic function f(x) = 2(x - 3)² + 1

Step 1: Identify the Vertex

The vertex (h, k) is directly identifiable from the vertex form. In this case, h = 3 and k = 1. Therefore, the vertex is (3, 1).

Step 2: Determine the Direction and Stretch/Compression

The value of 'a' is 2. Since a > 0, the parabola opens upwards. Since |a| = 2 > 1, the parabola is narrower than the parent function y = x².

Step 3: Plot the Vertex and Additional Points

Plot the vertex (3, 1) on the coordinate plane. To find additional points, choose x-values on either side of the vertex and substitute them into the equation to find their corresponding y-values.

For example:

• If x = 2, f(2) = 2(2 - 3)² + 1 = 3
• If x = 4, f(4) = 2(4 - 3)² + 1 = 3
• If x = 1, f(1) = 2(1 - 3)² + 1 = 9
• If x = 5, f(5) = 2(5 - 3)² + 1 = 9

Plot these points (2, 3), (4, 3), (1, 9), (5, 9) on the coordinate plane.

Step 4: Sketch the Parabola

Connect the plotted points with a smooth, U-shaped curve to form the parabola. Remember to show the parabola extending infinitely in both directions.

Answer Key Examples (with Solutions)

Here are a few more examples to practice, along with the solutions:

1. f(x) = -1(x + 2)² - 3

• Vertex: (-2, -3)
• Direction: Opens downwards (a = -1)
• Stretch/Compression: Same width as the parent function (|a| = 1)

2. f(x) = 1/2(x - 1)² + 4

• Vertex: (1, 4)
• Direction: Opens upwards (a = 1/2)
• Stretch/Compression: Wider than the parent function (|a| = 1/2)

3. f(x) = 3(x + 5)² - 2

• Vertex: (-5, -2)
• Direction: Opens upwards (a = 3)
• Stretch/Compression: Narrower than the parent function (|a| = 3)

Remember to always follow the steps outlined above. Practice with various examples to solidify your understanding of graphing quadratics in vertex form. The more you practice, the more proficient you'll become. This comprehensive guide should provide you with the tools and understanding to excel in graphing quadratic functions.