Graphing Quadratic Review Worksheet Answers

Graphing Quadratic Functions: A Comprehensive Review with Answers

This worksheet review covers graphing quadratic functions, a crucial topic in algebra. Understanding quadratics is key to grasping more advanced mathematical concepts. This guide will walk you through the essential elements, providing explanations and answers to help you master this skill. We'll cover everything from identifying key features like vertex, axis of symmetry, and intercepts to applying different graphing techniques. Let's dive in!

I. 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 several forms, each revealing different characteristics:

• Standard Form: f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. This form is useful for finding the y-intercept (the point where the graph intersects the y-axis). The y-intercept is simply (0, c).

• Vertex Form: f(x) = a(x - h)² + k, where (h, k) represents the vertex (the lowest or highest point on the parabola). This form is ideal for quickly identifying the vertex and axis of symmetry. The axis of symmetry is the vertical line x = h.

• Factored Form (or Intercept Form): f(x) = a(x - p)(x - q), where p and q are the x-intercepts (the points where the graph intersects the x-axis). This form is best for easily determining the x-intercepts.

The value of a determines the parabola's orientation and width:

• If a > 0: The parabola opens upwards (U-shaped), indicating a minimum value at the vertex.
• If a < 0: The parabola opens downwards (∩-shaped), indicating a maximum value at the vertex.
• The absolute value of a affects the parabola's width: A larger |a| results in a narrower parabola, while a smaller |a| results in a wider parabola.

II. Finding Key Features of Quadratic Functions

Before graphing, we need to identify the key features of the quadratic function:

1. Vertex:

• Using the Standard Form (f(x) = ax² + bx + c): The x-coordinate of the vertex is given by x = -b/(2a). Substitute this value back into the function to find the y-coordinate.

• Using the Vertex Form (f(x) = a(x - h)² + k): The vertex is directly given as (h, k).

2. Axis of Symmetry:

This is the vertical line that divides the parabola into two symmetrical halves. Its equation is always x = h, where h is the x-coordinate of the vertex.

3. x-intercepts (Roots or Zeros):

These are the points where the parabola intersects the x-axis (where y = 0).

• Using the Factored Form (f(x) = a(x - p)(x - q)): The x-intercepts are directly given as (p, 0) and (q, 0).

• Using the Standard Form (f(x) = ax² + bx + c): Set f(x) = 0 and solve the quadratic equation ax² + bx + c = 0. This can be done by factoring, using the quadratic formula, or completing the square. The quadratic formula is: x = [-b ± √(b² - 4ac)] / 2a

4. y-intercept:

This is the point where the parabola intersects the y-axis (where x = 0). In the standard form (f(x) = ax² + bx + c), the y-intercept is (0, c).

III. Graphing Quadratic Functions: Step-by-Step Guide

Let's illustrate the process with an example: Graph the quadratic function f(x) = 2x² - 8x + 6.

Step 1: Identify the Form:

This function is in standard form (f(x) = ax² + bx + c), where a = 2, b = -8, and c = 6.

Step 2: Find the Vertex:

• x-coordinate: x = -b / (2a) = -(-8) / (2 * 2) = 2
• y-coordinate: f(2) = 2(2)² - 8(2) + 6 = -2
• Vertex: (2, -2)

Step 3: Find the Axis of Symmetry:

The axis of symmetry is x = 2.

Step 4: Find the x-intercepts:

Set f(x) = 0: 2x² - 8x + 6 = 0. We can factor this equation: 2(x - 1)(x - 3) = 0. Therefore, the x-intercepts are (1, 0) and (3, 0).

Step 5: Find the y-intercept:

The y-intercept is (0, c) = (0, 6).

Step 6: Plot the Points and Sketch the Parabola:

Plot the vertex (2, -2), the x-intercepts (1, 0) and (3, 0), and the y-intercept (0, 6). Since a = 2 > 0, the parabola opens upwards. Sketch a smooth curve through these points, ensuring symmetry around the axis of symmetry (x = 2).

IV. Worked Examples and Answers to Practice Problems

Let's work through some more examples to solidify your understanding. Remember to follow the steps outlined above.

Example 1: Graph f(x) = -(x + 1)² + 4

• Vertex Form: This is already in vertex form, with vertex (-1, 4).
• Axis of Symmetry: x = -1
• x-intercepts: Set f(x) = 0: -(x + 1)² + 4 = 0. Solving for x gives x = 1 and x = -3. x-intercepts are (1,0) and (-3,0).
• y-intercept: f(0) = -(0 + 1)² + 4 = 3. y-intercept is (0,3).
• Parabola Orientation: Since a = -1 < 0, the parabola opens downwards.

Example 2: Graph f(x) = x² + 2x - 3

• Standard Form: This is in standard form, with a = 1, b = 2, c = -3.
• Vertex: x = -b/(2a) = -2/(2*1) = -1. f(-1) = (-1)² + 2(-1) - 3 = -4. Vertex is (-1, -4).
• Axis of Symmetry: x = -1
• x-intercepts: Factor the equation: (x + 3)(x - 1) = 0. x-intercepts are (-3, 0) and (1, 0).
• y-intercept: (0, -3)
• Parabola Orientation: Since a = 1 > 0, the parabola opens upwards.

Example 3: Graph f(x) = 0.5x² - 2

• Standard Form: a = 0.5, b = 0, c = -2.
• Vertex: x = -b/(2a) = 0. f(0) = -2. Vertex is (0, -2).
• Axis of Symmetry: x = 0
• x-intercepts: 0.5x² - 2 = 0 => x² = 4 => x = ±2. x-intercepts are (2,0) and (-2,0).
• y-intercept: (0,-2)
• Parabola Orientation: Since a = 0.5 > 0, the parabola opens upwards. Note that the parabola is wider than a parabola with a = 1.

V. Frequently Asked Questions (FAQ)

Q: What if I can't factor the quadratic equation to find the x-intercepts?

A: If factoring is difficult or impossible, use the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a. This will always provide the solutions (even if they are complex numbers, which would indicate no real x-intercepts).

Q: How do I determine the range of a quadratic function?

A: The range depends on whether the parabola opens upwards or downwards.

• Parabola opens upwards (a > 0): The range is [k, ∞), where k is the y-coordinate of the vertex.
• Parabola opens downwards (a < 0): The range is (-∞, k], where k is the y-coordinate of the vertex.

Q: What is the discriminant, and what does it tell us?

A: The discriminant is the expression inside the square root of the quadratic formula: b² - 4ac. It indicates the number and type of x-intercepts:

• b² - 4ac > 0: Two distinct real x-intercepts.
• b² - 4ac = 0: One real x-intercept (the vertex touches the x-axis).
• b² - 4ac < 0: No real x-intercepts (the parabola does not intersect the x-axis).

VI. Conclusion

Graphing quadratic functions is a fundamental skill in algebra. By understanding the different forms of quadratic equations, identifying key features like the vertex, axis of symmetry, and intercepts, and applying the steps outlined above, you can confidently graph any quadratic function. Remember to practice regularly to build your understanding and proficiency. The more you practice, the easier it will become to visualize the parabola and its key characteristics directly from the equation. Good luck!