graphing quadratic functions answer key

Lemon

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02-02-2025

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Quadratic functions, represented by the general equation f(x) = ax² + bx + c (where 'a', 'b', and 'c' are constants and a ≠ 0), form parabolic curves when graphed. Understanding how to graph these functions is crucial in algebra and beyond, finding applications in physics, engineering, and economics. This guide provides a step-by-step approach to graphing quadratic functions, along with examples and an answer key for practice problems.

Understanding Key Features of Quadratic Functions

Before diving into graphing, let's review some essential characteristics that shape the parabola:

1. The Parabola's Orientation:

• a > 0: The parabola opens upwards (U-shaped). The vertex represents the minimum value of the function.
• a < 0: The parabola opens downwards (∩-shaped). The vertex represents the maximum value of the function.

2. The Vertex: This is the turning point of the parabola. Its coordinates are given by:

x = -b / 2a y = f(-b / 2a) (Substitute the x-value back into the original equation)

3. The Axis of Symmetry: This is a vertical line passing through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b / 2a.

4. The y-intercept: This is the point where the parabola intersects the y-axis. It's found by setting x = 0 in the equation: y = c.

5. x-intercepts (Roots or Zeros): These are the points where the parabola intersects the x-axis. They are found by solving the quadratic equation ax² + bx + c = 0. This can be done using factoring, the quadratic formula, or completing the square. A parabola can have 0, 1, or 2 x-intercepts.

Step-by-Step Graphing Process

Let's graph the quadratic function f(x) = x² - 4x + 3 as an example.

1. Determine the Parabola's Orientation: Since a = 1 > 0, the parabola opens upwards.

2. Find the Vertex:

• x = -b / 2a = -(-4) / 2(1) = 2
• y = f(2) = (2)² - 4(2) + 3 = -1

The vertex is (2, -1).

3. Find the Axis of Symmetry: The axis of symmetry is x = 2.

4. Find the y-intercept: When x = 0, y = 3. The y-intercept is (0, 3).

5. Find the x-intercepts: We solve x² - 4x + 3 = 0. This factors to (x - 1)(x - 3) = 0, giving x-intercepts at (1, 0) and (3, 0).

6. Plot the Points and Sketch the Parabola: Plot the vertex, y-intercept, and x-intercepts. Use the axis of symmetry to help you sketch the symmetrical curve. You can also plot additional points by substituting x-values into the equation to get corresponding y-values.

Practice Problems with Answer Key

Instructions: Graph each quadratic function. Identify the vertex, axis of symmetry, y-intercept, and x-intercepts (if any).

Problem 1: f(x) = x² + 2x + 1

Problem 2: f(x) = -x² + 4x - 4

Problem 3: f(x) = 2x² - 8

Problem 4: f(x) = x² - 2x + 5

Answer Key

Problem 1: f(x) = x² + 2x + 1

• Vertex: (-1, 0)
• Axis of Symmetry: x = -1
• y-intercept: (0, 1)
• x-intercept: (-1, 0)

Problem 2: f(x) = -x² + 4x - 4

• Vertex: (2, 0)
• Axis of Symmetry: x = 2
• y-intercept: (0, -4)
• x-intercept: (2, 0)

Problem 3: f(x) = 2x² - 8

• Vertex: (0, -8)
• Axis of Symmetry: x = 0
• y-intercept: (0, -8)
• x-intercepts: (2, 0), (-2, 0)

Problem 4: f(x) = x² - 2x + 5

• Vertex: (1, 4)
• Axis of Symmetry: x = 1
• y-intercept: (0, 5)
• x-intercepts: None (the discriminant is negative)

This comprehensive guide provides a solid foundation for graphing quadratic functions. Remember to practice regularly to master this essential skill. By understanding the key features and following the step-by-step process, you can confidently graph any quadratic function.