This comprehensive guide provides answers and explanations for a typical Algebra 2 worksheet on graphing quadratic functions. While I cannot provide answers to a specific worksheet you haven't shared, I will cover the key concepts and examples to help you solve problems on your own worksheet. Remember to always check your work against the provided answer key and understand the underlying principles.
Understanding Quadratic Functions
Before we dive into specific examples, let's review the fundamental concepts:
A quadratic function is a function of the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola.
Key Features of a Parabola:
- Vertex: The highest or lowest point on the parabola. The x-coordinate of the vertex is given by
x = -b / 2a. Substitute this x-value back into the function to find the y-coordinate. - Axis of Symmetry: A vertical line that passes 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 (where y = 0). These can be found by factoring the quadratic equation, using the quadratic formula, or completing the square.
- y-intercept: The point where the parabola intersects the y-axis (where x = 0). This is simply the value of 'c' in the equation
f(x) = ax² + bx + c. - Concavity: The parabola opens upwards (U-shaped) if 'a' > 0 and opens downwards (∩-shaped) if 'a' < 0.
Example Problems and Solutions
Let's work through some typical problems you might encounter on your worksheet. Remember to replace these examples with the actual problems from your worksheet.
Example 1: Graphing a Quadratic Function in Standard Form
Problem: Graph the quadratic function f(x) = x² - 4x + 3.
Solution:
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Find the vertex: a = 1, b = -4, c = 3. The x-coordinate of the vertex is x = -(-4) / (2 * 1) = 2. The y-coordinate is f(2) = 2² - 4(2) + 3 = -1. Therefore, the vertex is (2, -1).
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Find the axis of symmetry: The axis of symmetry is x = 2.
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Find the x-intercepts: Set f(x) = 0: x² - 4x + 3 = 0. This factors to (x - 1)(x - 3) = 0. The x-intercepts are (1, 0) and (3, 0).
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Find the y-intercept: The y-intercept is (0, 3) (since c = 3).
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Determine concavity: Since a = 1 > 0, the parabola opens upwards.
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Plot the points and sketch the parabola: Plot the vertex, x-intercepts, y-intercept, and a few additional points if needed to accurately sketch the parabola.
Example 2: Graphing a Quadratic Function in Vertex Form
Problem: Graph the quadratic function f(x) = 2(x + 1)² - 4.
Solution:
This equation is in vertex form, f(x) = a(x - h)² + k, where (h, k) is the vertex.
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Identify the vertex: The vertex is (-1, -4).
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Find the axis of symmetry: The axis of symmetry is x = -1.
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Determine concavity: Since a = 2 > 0, the parabola opens upwards.
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Find the x-intercepts (optional): Set f(x) = 0 and solve for x. This will involve solving a simple quadratic equation.
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Find the y-intercept (optional): Substitute x = 0 into the equation to find the y-intercept.
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Plot the points and sketch the parabola: Plot the vertex, x-intercepts (if found), y-intercept (if found), and other points as needed to sketch the parabola.
Solving Other Quadratic Function Problems
Your worksheet might include problems requiring you to:
- Write a quadratic function given its graph: Determine the vertex and another point on the parabola, then use the vertex form to write the equation.
- Find the maximum or minimum value of a quadratic function: This is the y-coordinate of the vertex.
- Solve quadratic inequalities: This involves finding the intervals where the quadratic function is positive or negative.
Remember to thoroughly understand the concepts and apply the appropriate formulas and techniques to solve each problem. If you have specific questions about particular problems on your worksheet, feel free to provide them, and I'll do my best to help.
