9-1 Skills Practice Graphing Quadratic Functions Answer Key

9-1 Skills Practice: Graphing Quadratic Functions – Answer Key & Comprehensive Guide

This comprehensive guide provides detailed answers and explanations for the 9-1 Skills Practice worksheet on graphing quadratic functions. We'll delve into the core concepts, offering a step-by-step approach to understanding and mastering these essential skills. This isn't just an answer key; it's a learning resource designed to solidify your comprehension of quadratic functions and their graphical representations.

Understanding Quadratic Functions

Before we jump into the answers, let's refresh our understanding of quadratic functions. A quadratic function is a function of 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. The value of 'a' determines the parabola's direction (opens upwards if a > 0, downwards if a < 0) and its width (larger |a| means narrower parabola).

Key Features of a Parabola

Several key features help us understand and graph quadratic functions:

• Vertex: The highest or lowest point on the parabola. Its x-coordinate is given by -b/(2a).
• Axis of Symmetry: A vertical line that passes 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'.

Step-by-Step Approach to Graphing Quadratic Functions

To graph a quadratic function effectively, follow these steps:

1. Identify 'a', 'b', and 'c': Write the quadratic function in the standard form (ax² + bx + c) and identify the values of a, b, and c.

2. Find the Vertex: Use the formula x = -b/(2a) to find the x-coordinate of the vertex. Substitute this x-value back into the quadratic function to find the y-coordinate.

3. Determine the Axis of Symmetry: The axis of symmetry is a vertical line passing through the vertex. Its equation is x = (x-coordinate of the vertex).

4. Find the y-intercept: The y-intercept is the point where the graph crosses the y-axis (x=0). Substitute x = 0 into the function to find the y-coordinate.

5. Find the x-intercepts (if any): Solve the quadratic equation ax² + bx + c = 0 using factoring, the quadratic formula, or completing the square. The solutions are the x-coordinates of the x-intercepts.

6. Plot the points: Plot the vertex, y-intercept, and x-intercepts on a coordinate plane. If there are no x-intercepts, find additional points by substituting other x-values into the function and plotting the resulting (x, y) pairs. Remember to maintain symmetry around the axis of symmetry.

7. Sketch the parabola: Connect the plotted points with a smooth, U-shaped curve to complete the graph. Remember to indicate the direction of the parabola (upward or downward) based on the sign of 'a'.

Sample Problems and Solutions (Illustrative Examples)

Let's work through a few examples to illustrate the process. These are similar to the problems you might find in your 9-1 Skills Practice worksheet. Remember, this section focuses on demonstrating the method; your specific worksheet problems might have different numerical values.

Example 1: Graph the quadratic function f(x) = x² + 2x - 3

1. Identify a, b, and c: a = 1, b = 2, c = -3

2. Find the Vertex: x = -b/(2a) = -2/(2*1) = -1. f(-1) = (-1)² + 2(-1) - 3 = -4. Vertex: (-1, -4)

3. Axis of Symmetry: x = -1

4. Find the y-intercept: When x = 0, f(0) = -3. y-intercept: (0, -3)

5. Find the x-intercepts: Solve x² + 2x - 3 = 0. This factors to (x + 3)(x - 1) = 0. Therefore, x = -3 and x = 1. x-intercepts: (-3, 0) and (1, 0)

6. Plot the points: Plot the vertex (-1, -4), y-intercept (0, -3), and x-intercepts (-3, 0) and (1, 0).

7. Sketch the parabola: Draw a smooth U-shaped curve through these points, opening upwards since a = 1 > 0.

Example 2: Graph the quadratic function f(x) = -x² + 4x - 4

1. Identify a, b, and c: a = -1, b = 4, c = -4

2. Find the Vertex: x = -b/(2a) = -4/(2*(-1)) = 2. f(2) = -(2)² + 4(2) - 4 = 0. Vertex: (2, 0)

3. Axis of Symmetry: x = 2

4. Find the y-intercept: When x = 0, f(0) = -4. y-intercept: (0, -4)

5. Find the x-intercepts: Solve -x² + 4x - 4 = 0. This factors to -(x - 2)² = 0, so x = 2. There is only one x-intercept, which is the vertex.

6. Plot the points: Plot the vertex (2, 0) and y-intercept (0, -4). Find an additional point, for example, when x = 1, f(1) = -1. Plot (1, -1). Remember symmetry around x = 2.

7. Sketch the parabola: Draw a smooth U-shaped curve through these points, opening downwards since a = -1 < 0.

Addressing Specific Challenges in Graphing Quadratic Functions

Students often face specific challenges when graphing quadratic functions. Let's address some common difficulties:

• Understanding the significance of 'a', 'b', and 'c': Remember that 'a' determines the direction and width, 'b' influences the position of the vertex, and 'c' is the y-intercept. Practice identifying these values quickly and accurately.

• Solving quadratic equations: Mastering methods like factoring, the quadratic formula, and completing the square is crucial for finding x-intercepts. Practice different types of quadratic equations to improve your skills.

• Dealing with non-integer solutions: Sometimes, the vertex or x-intercepts have non-integer coordinates. Use a calculator or approximate values to plot these points accurately. Ensure your graph accurately represents the decimal values.

• Interpreting graphs: Once you've graphed a function, practice interpreting its features. Identify the vertex, axis of symmetry, x-intercepts, and y-intercept. Determine the range and domain.

Advanced Concepts and Extensions

Once you have mastered the basics, you can explore more advanced concepts related to quadratic functions:

• Maximum and Minimum Values: The y-coordinate of the vertex represents the maximum or minimum value of the function.

• Applications of Quadratic Functions: Quadratic functions have numerous real-world applications in areas like projectile motion, optimization problems, and modeling curves.

• Transformations of Quadratic Functions: Learn how changes in 'a', 'b', and 'c' affect the graph's position and shape.

This comprehensive guide, coupled with diligent practice, will help you achieve mastery over graphing quadratic functions. Remember to thoroughly review the steps outlined above and work through numerous practice problems to solidify your understanding. Good luck!