This worksheet guides you through graphing quadratic functions presented in three common forms: standard, vertex, and factored. Understanding these forms is crucial for mastering quadratic functions and their applications in algebra and beyond. We'll explore how each form reveals key characteristics of the parabola, allowing for efficient and accurate graphing.
Understanding the Three Forms
Quadratic functions are always of the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. However, this standard form doesn't immediately reveal crucial information like the vertex or x-intercepts. Therefore, alternative forms are beneficial.
1. Standard Form: f(x) = ax² + bx + c
- Advantages: This form is useful for finding the y-intercept (the point where the parabola crosses the y-axis), which is simply the value of 'c'.
- Disadvantages: Finding the vertex and x-intercepts requires more work (using the vertex formula or the quadratic formula).
2. Vertex Form: f(x) = a(x - h)² + k
- Advantages: This form directly gives the vertex of the parabola, (h, k). The value of 'a' still indicates whether the parabola opens upwards (a > 0) or downwards (a < 0).
- Disadvantages: The x-intercepts are not directly apparent and require solving the equation for x.
3. Factored Form: f(x) = a(x - r₁)(x - r₂)
- Advantages: This form readily reveals the x-intercepts (roots) of the quadratic function, which are r₁ and r₂. The parabola crosses the x-axis at these points.
- Disadvantages: Finding the vertex requires more calculation, involving finding the midpoint between the x-intercepts.
Graphing Techniques for Each Form
Let's delve into the specific graphing techniques for each form:
Graphing from Standard Form: f(x) = ax² + bx + c
- Find the y-intercept: The y-intercept is (0, c).
- Find the x-intercepts (roots): Use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a. If the discriminant (b² - 4ac) is negative, there are no real x-intercepts. - Find the vertex: Use the vertex formula:
x = -b / 2a. Substitute this x-value back into the equation to find the y-coordinate of the vertex. - Plot the points and sketch the parabola: Plot the y-intercept, x-intercepts (if any), and the vertex. Remember that the parabola is symmetrical around the vertical line passing through the vertex.
Graphing from Vertex Form: f(x) = a(x - h)² + k
- Identify the vertex: The vertex is (h, k).
- Determine the direction of opening: If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
- Find additional points: Choose x-values on either side of the vertex and substitute them into the equation to find their corresponding y-values.
- Plot the points and sketch the parabola: Plot the vertex and the additional points, keeping in mind the parabola's symmetry.
Graphing from Factored Form: f(x) = a(x - r₁)(x - r₂)
- Identify the x-intercepts: The x-intercepts are (r₁, 0) and (r₂, 0).
- Find the x-coordinate of the vertex: The x-coordinate of the vertex is the average of the x-intercepts:
x = (r₁ + r₂) / 2. - Find the y-coordinate of the vertex: Substitute the x-coordinate of the vertex into the equation to find the y-coordinate.
- Determine the direction of opening: If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
- Plot the points and sketch the parabola: Plot the x-intercepts and the vertex. Add additional points as needed for accuracy.
Practice Problems
Now, let's put this knowledge into practice! Try graphing the following quadratic functions:
f(x) = x² - 4x + 3(Standard Form)f(x) = 2(x - 1)² - 8(Vertex Form)f(x) = -(x + 2)(x - 4)(Factored Form)
By working through these problems, you'll solidify your understanding of graphing quadratic functions in different forms. Remember to pay close attention to the 'a' value, which dictates the parabola's orientation and width. This worksheet provides a solid foundation for further exploration of quadratic functions and their applications in various mathematical contexts.
