graphing quadratic functions in 3 forms worksheet

Neter

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22-11-2024

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Quadratic functions are ubiquitous in mathematics and real-world applications. Understanding how to graph them is crucial. This article will guide you through graphing quadratic functions presented in three common forms: standard, vertex, and factored. We'll use a worksheet approach to solidify your understanding.

Understanding the Three Forms of Quadratic Functions

Before we delve into graphing, let's review the three forms of quadratic functions:

1. Standard Form: f(x) = ax² + bx + c

• a, b, and c are constants.
• The y-intercept is easily identified as c.
• Determining the vertex and x-intercepts requires more calculation.

2. Vertex Form: f(x) = a(x - h)² + k

• (h, k) represents the vertex of the parabola.
• a determines the parabola's vertical stretch or compression and whether it opens upwards (a > 0) or downwards (a < 0).
• Finding the x-intercepts requires solving a quadratic equation.

3. Factored Form (Intercept Form): f(x) = a(x - p)(x - q)

• p and q are the x-intercepts (or roots) of the quadratic.
• a again dictates the parabola's vertical stretch/compression and direction.
• The vertex can be found using the midpoint formula between the x-intercepts.

Graphing Quadratic Functions: A Worksheet Approach

Let's work through some examples, applying the knowledge above. Imagine this as your worksheet.

Problem 1: Standard Form

Graph the quadratic function: f(x) = x² - 4x + 3

Solution:

1. Identify the y-intercept: The y-intercept is 3 (c = 3). Plot the point (0, 3).
2. Find the vertex: The x-coordinate of the vertex is given by x = -b / 2a. In this case, x = -(-4) / 2(1) = 2. Substitute this into the function to find the y-coordinate: f(2) = (2)² - 4(2) + 3 = -1. The vertex is (2, -1). Plot this point.
3. Find the x-intercepts: Factor the quadratic: f(x) = (x - 1)(x - 3). The x-intercepts are 1 and 3. Plot the points (1, 0) and (3, 0).
4. Sketch the parabola: Draw a smooth curve through the plotted points, ensuring the parabola opens upwards since 'a' is positive.

[Insert image here showing a graph of the parabola for f(x) = x² - 4x + 3] Alt Text: Graph of the quadratic function x² - 4x + 3 showing vertex (2, -1) and x-intercepts (1,0) and (3,0).

Problem 2: Vertex Form

Graph the quadratic function: f(x) = 2(x + 1)² - 4

Solution:

1. Identify the vertex: The vertex is (-1, -4) because the equation is in vertex form. Plot this point.
2. Determine the direction and stretch: Since a = 2, the parabola opens upwards and is vertically stretched.
3. Find additional points: Choose x-values around the vertex and calculate their corresponding y-values. For example, if x = 0, f(0) = 2(1)² - 4 = -2. Plot (0,-2). Similarly, find and plot another point.
4. Sketch the parabola: Draw a smooth curve through the points.

[Insert image here showing a graph of the parabola for f(x) = 2(x + 1)² - 4] Alt Text: Graph of the quadratic function 2(x + 1)² - 4 showing vertex (-1,-4).

Problem 3: Factored Form

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

Solution:

1. Find the x-intercepts: The x-intercepts are 2 and -4. Plot (2, 0) and (-4, 0).
2. Determine the vertex: The x-coordinate of the vertex is the midpoint of the x-intercepts: (-4 + 2) / 2 = -1. The y-coordinate is found by substituting x = -1 into the function: f(-1) = -(-1 - 2)(-1 + 4) = 9. The vertex is (-1, 9). Plot this point.
3. Determine the direction: Since a = -1, the parabola opens downwards.
4. Sketch the parabola: Draw a smooth curve through the plotted points.

[Insert image here showing a graph of the parabola for f(x) = -(x - 2)(x + 4)] Alt Text: Graph of the quadratic function -(x-2)(x+4) showing vertex (-1,9) and x-intercepts (2,0) and (-4,0).

Practice Problems

Now, try graphing these quadratic functions yourself:

1. f(x) = x² + 2x - 8
2. f(x) = -3(x - 1)² + 5
3. f(x) = 1/2(x + 3)(x - 1)

Remember to identify key features like the vertex, intercepts, and direction of opening before you start plotting points. Using this worksheet approach will build a strong foundation in graphing quadratic functions. Remember to check your work by using graphing calculators or online graphing tools. Good luck!