This worksheet provides a thorough exploration of graphing quadratic functions, progressing from foundational concepts to more complex applications. Understanding quadratic functions is crucial in algebra and beyond, forming the basis for understanding parabolas and their real-world applications in physics, engineering, and economics. This worksheet aims to solidify your understanding through a range of exercises.
Understanding Quadratic Functions
A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (typically x) is 2. It can be expressed in the general form:
f(x) = ax² + bx + c
where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if a > 0 and downwards if a < 0.
Key Features of a Parabola:
- Vertex: The highest or lowest point on the parabola. The x-coordinate of the vertex 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 or Zeros): The points where the parabola intersects the x-axis (where y = 0). These are found by solving the quadratic equation ax² + bx + c = 0.
- y-intercept: The point where the parabola intersects the y-axis (where x = 0). This is simply the value of c.
Practice Problems: Graphing Quadratic Functions
Instructions: For each quadratic function below, identify the vertex, axis of symmetry, x-intercepts (if any), and y-intercept. Then, sketch the graph of the function. Show your work.
1. f(x) = x² + 2x - 3
2. f(x) = -x² + 4x
3. f(x) = 2x² - 8
4. f(x) = x² - 6x + 9
5. f(x) = -2x² + 4x - 2
Advanced Problems (Challenge Questions):
6. A ball is thrown upwards with an initial velocity of 40 m/s from a height of 1.5 meters. Its height (h) in meters after t seconds is given by the equation h(t) = -4.9t² + 40t + 1.5. Find the maximum height reached by the ball and the time it takes to reach that height. Sketch the graph of the function.
7. Find the equation of a parabola with vertex (2, -1) and passing through the point (0, 3).
Answer Key:
(Note: The following provides answers. Students should complete the graphing portion independently to ensure understanding.)
1. f(x) = x² + 2x - 3 * Vertex: (-1, -4) * Axis of Symmetry: x = -1 * x-intercepts: (-3, 0), (1, 0) * y-intercept: (0, -3)
2. f(x) = -x² + 4x * Vertex: (2, 4) * Axis of Symmetry: x = 2 * x-intercepts: (0, 0), (4, 0) * y-intercept: (0, 0)
3. f(x) = 2x² - 8 * Vertex: (0, -8) * Axis of Symmetry: x = 0 * x-intercepts: (-2, 0), (2, 0) * y-intercept: (0, -8)
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) = -2x² + 4x - 2 * Vertex: (1, 0) * Axis of Symmetry: x = 1 * x-intercept: (1, 0) * y-intercept: (0, -2)
6. (Requires Calculus or completing the square for precise solution) Maximum height is approximately 81.6 meters. The time to reach maximum height is approximately 4.1 seconds.
7. f(x) = a(x - h)² + k, where (h, k) is the vertex. Substituting (2, -1) for (h, k) and (0, 3) for (x, f(x)), we get the equation: f(x) = x² - 4x + 3
This worksheet provides a robust foundation for understanding and graphing quadratic functions. Remember to practice regularly and seek clarification on any concepts that remain unclear. Mastering quadratic functions opens doors to more advanced mathematical concepts.
