Graphing Quadratics Review Worksheet Answers

Graphing Quadratics: A Comprehensive Review with Worked Examples

This worksheet review focuses on graphing quadratic functions. We'll cover identifying key features like the vertex, axis of symmetry, x-intercepts (roots or zeros), y-intercept, and concavity, and then use these to accurately sketch the parabola. Understanding these elements is crucial for solving quadratic equations and understanding their real-world applications in areas like projectile motion and optimization problems. This guide will provide detailed explanations and worked examples to solidify your understanding.

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's generally represented by the equation:

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 either upwards (if a > 0) or downwards (if a < 0). The shape and position of the parabola are determined by the values of a, b, and c.

II. Key Features of a Parabola

Before we dive into graphing, let's define the crucial elements we'll be focusing on:

• Vertex: The highest or lowest point on the parabola. It represents the minimum or maximum value of the function. The x-coordinate of the vertex is given by x = -b / 2a. Substitute this x-value back into the quadratic equation to find the y-coordinate.

• Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves. Its equation is x = -b / 2a, the same as the x-coordinate of the vertex.

• 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. This can be done 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 in the equation f(x) = ax² + bx + c.

• Concavity: Whether the parabola opens upwards (concave up) or downwards (concave down). This is determined by the value of a:

  • a > 0: Parabola opens upwards (concave up).
  • a < 0: Parabola opens downwards (concave down).

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

Let's use a step-by-step approach to graph the quadratic function: f(x) = x² - 4x + 3

Step 1: Identify a, b, and c

In our example, a = 1, b = -4, and c = 3. Since a > 0, the parabola opens upwards.

Step 2: Find the Vertex

The x-coordinate of the vertex is: x = -b / 2a = -(-4) / 2(1) = 2

Substitute x = 2 into the equation to find the y-coordinate: f(2) = (2)² - 4(2) + 3 = -1

Therefore, the vertex is (2, -1).

Step 3: Find the Axis of Symmetry

The axis of symmetry is a vertical line passing through the vertex. Its equation is x = 2.

Step 4: Find the Y-intercept

The y-intercept is the value of c, which is 3. Therefore, the y-intercept is (0, 3).

Step 5: Find the X-intercepts (Roots)

To find the x-intercepts, set f(x) = 0 and solve for x:

x² - 4x + 3 = 0

This quadratic equation can be factored as: (x - 1)(x - 3) = 0

Therefore, the x-intercepts are (1, 0) and (3, 0).

Step 6: Plot the Points and Sketch the Parabola

Plot the vertex (2, -1), the y-intercept (0, 3), and the x-intercepts (1, 0) and (3, 0) on a coordinate plane. Remember that the parabola is symmetric about the axis of symmetry (x = 2). Use these points to sketch a smooth, U-shaped curve. The parabola should pass through all the plotted points.

IV. Worked Examples: Different Forms of Quadratic Equations

Quadratic equations can be presented in different forms: standard form (ax² + bx + c), vertex form (a(x-h)² + k), and factored form ((x-r₁)(x-r₂)). Let's review graphing with examples in these various forms.

Example 1: Vertex Form

Consider the equation: f(x) = 2(x + 1)² - 4

• Vertex: The vertex is (-1, -4). The equation is already in vertex form, a(x - h)² + k, where (h, k) is the vertex.

• Axis of Symmetry: x = -1

• Y-intercept: Substitute x = 0: f(0) = 2(0 + 1)² - 4 = -2. The y-intercept is (0, -2).

• X-intercepts: Set f(x) = 0 and solve for x: 2(x + 1)² - 4 = 0 (x + 1)² = 2 x + 1 = ±√2 x = -1 ± √2

• Concavity: Since a = 2 > 0, the parabola opens upwards.

Plot these points and sketch the parabola.

Example 2: Factored Form

Consider the equation: f(x) = -(x - 2)(x + 4)

• X-intercepts: The x-intercepts are (2, 0) and (-4, 0). The factored form directly gives the roots.

• Vertex: The x-coordinate of the vertex is the average of the x-intercepts: x = (-4 + 2) / 2 = -1. Substitute x = -1 into the equation to find the y-coordinate: f(-1) = -(-1 - 2)(-1 + 4) = -(-3)(3) = 9. The vertex is (-1, 9).

• Axis of Symmetry: x = -1

• Y-intercept: Substitute x = 0: f(0) = -(0 - 2)(0 + 4) = 8. The y-intercept is (0, 8).

• Concavity: Since a = -1 < 0, the parabola opens downwards.

Example 3: Standard Form (requiring the quadratic formula)

Consider the equation: f(x) = x² + 2x - 2

• Vertex: x = -b / 2a = -2 / 2(1) = -1. y = (-1)² + 2(-1) - 2 = -3. Vertex is (-1, -3).

• Axis of Symmetry: x = -1

• Y-intercept: (0, -2)

• X-intercepts: Use the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a x = [-2 ± √(2² - 4(1)(-2))] / 2(1) x = [-2 ± √12] / 2 x = -1 ± √3

V. Solving Quadratic Equations using Graphs

The graph of a quadratic function can be used to solve quadratic equations. The x-intercepts of the graph represent the solutions (roots) of the equation ax² + bx + c = 0. If the parabola does not intersect the x-axis, the equation has no real solutions.

VI. Frequently Asked Questions (FAQ)

Q1: What if the quadratic equation doesn't factor easily?

A1: Use the quadratic formula, which works for all quadratic equations: x = [-b ± √(b² - 4ac)] / 2a

Q2: How do I determine the number of x-intercepts?

A2: This is determined by the discriminant (b² - 4ac) in the quadratic formula:

• 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).

Q3: Can I use technology to graph quadratics?

A3: Yes! Graphing calculators and online graphing tools can make graphing much easier and more accurate, especially for equations that are difficult to graph manually. These tools can also help you find the exact coordinates of the vertex and intercepts.

Q4: What are some real-world applications of graphing quadratics?

A4: Quadratic functions model many real-world phenomena, including:

• Projectile motion: The trajectory of a ball thrown in the air.
• Area optimization: Finding the maximum area of a rectangle given a fixed perimeter.
• Engineering design: Designing parabolic reflectors for antennas or telescopes.

VII. Conclusion

Graphing quadratic functions is a fundamental skill in algebra. By understanding the key features of a parabola – the vertex, axis of symmetry, intercepts, and concavity – you can accurately sketch the graph and use it to solve related equations. Remember to practice regularly, using different forms of quadratic equations to strengthen your understanding. Mastering this skill will not only help you excel in your math class but also provide a solid foundation for more advanced mathematical concepts. This review aims to provide you with the tools and knowledge to tackle any graphing quadratics problem with confidence. Remember to practice with various examples to solidify your understanding!