graphing quadratic functions worksheet with answer key

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04-02-2025

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This worksheet provides a thorough exploration of graphing quadratic functions, covering key concepts and offering ample practice problems with a complete answer key. Whether you're a student needing extra practice or a teacher looking for a resource, this worksheet will help solidify your understanding of quadratic functions and their graphical representations.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, generally expressed in 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, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0).

Key Features of a Parabola

Several key features help us understand and graph a parabola:

• Vertex: The lowest (minimum) or highest (maximum) point on the parabola. Its x-coordinate is given by -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/Zeros): The points where the parabola intersects the x-axis (where y = 0). These can be 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 (where x = 0). This is simply the value of 'c'.

Practice Problems

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

1. f(x) = x² + 2x - 3

2. f(x) = -x² + 4x

3. f(x) = 2x² - 8x + 6

4. f(x) = x² - 6x + 9

5. f(x) = -½x² + 2x + 1

Answer Key

1. f(x) = x² + 2x - 3

• Vertex: (-1, -4)
• Axis of Symmetry: x = -1
• x-intercepts: (-3, 0) and (1, 0)
• y-intercept: (0, -3)

2. f(x) = -x² + 4x

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

3. f(x) = 2x² - 8x + 6

• Vertex: (2, -2)
• Axis of Symmetry: x = 2
• x-intercepts: (1, 0) and (3, 0)
• y-intercept: (0, 6)

4. f(x) = x² - 6x + 9

• Vertex: (3, 0)
• Axis of Symmetry: x = 3
• x-intercept: (3, 0)
• y-intercept: (0, 9)

5. f(x) = -½x² + 2x + 1

• Vertex: (2, 3)
• Axis of Symmetry: x = 2
• x-intercepts: Approximately (-0.44, 0) and (4.44, 0) (Use quadratic formula for precise values)
• y-intercept: (0, 1)

Tips for Graphing

• Start with the vertex: Finding the vertex is crucial, as it provides the central point around which the parabola is symmetrical.
• Plot the intercepts: The x and y-intercepts provide additional key points on the graph.
• Use symmetry: Once you've plotted a few points, use the axis of symmetry to easily find corresponding points on the other side of the parabola.
• Consider the 'a' value: The value of 'a' determines the parabola's direction (upward or downward) and its width (larger |a| means narrower parabola).

This worksheet provides a solid foundation for understanding and graphing quadratic functions. Remember to practice regularly to master these concepts. Further exploration into transformations of quadratic functions can build upon this knowledge.