Key Features Of Quadratic Functions: Worksheet Answers

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11-16-2024

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Quadratic functions are a fundamental concept in algebra and have various applications in mathematics and real-world situations. Understanding their key features is essential for students and learners alike. In this blog post, we will delve into the key features of quadratic functions, accompanied by worksheet answers for a better grasp of the subject. 📈

Understanding Quadratic Functions

A quadratic function is a type of polynomial function that can be expressed in the standard form:

[ f(x) = ax^2 + bx + c ]

where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of ( a ).

Key Features of Quadratic Functions

1. Vertex 🌟

   • The vertex is the highest or lowest point on the graph of a quadratic function. It can be found using the formula: [ x = -\frac{b}{2a} ]
   • After finding the x-coordinate of the vertex, substitute it back into the function to get the y-coordinate.

2. Axis of Symmetry 🔄

   • The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It is given by the equation: [ x = -\frac{b}{2a} ]
   • This axis helps to determine the location of the vertex.

3. Direction of Opening ⬆️⬇️

   • The direction in which the parabola opens depends on the coefficient ( a ):
     • If ( a > 0 ): The parabola opens upwards.
     • If ( a < 0 ): The parabola opens downwards.

4. Y-Intercept 📏

   • The y-intercept is the point where the graph intersects the y-axis. It can be found by substituting ( x = 0 ): [ y = c ]

5. X-Intercepts (Roots) 🌐

   • The x-intercepts (also known as roots or zeros) are the points where the graph intersects the x-axis. These can be found using the quadratic formula: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
   • The discriminant ( (b^2 - 4ac) ) determines the nature of the roots:
     • If positive: Two distinct real roots.
     • If zero: One real root (the vertex touches the x-axis).
     • If negative: No real roots.

Summary of Key Features

Here’s a quick summary table for the key features of quadratic functions:

<table> <tr> <th>Feature</th> <th>Description</th> <th>Formula</th> </tr> <tr> <td>Vertex</td> <td>Highest/lowest point on the graph</td> <td>x = -b/2a</td> </tr> <tr> <td>Axis of Symmetry</td> <td>Divides the parabola into two equal halves</td> <td>x = -b/2a</td> </tr> <tr> <td>Direction of Opening</td> <td>Indicates whether the parabola opens up or down</td> <td>Check sign of 'a'</td> </tr> <tr> <td>Y-Intercept</td> <td>Point where the graph intersects the y-axis</td> <td>y = c</td> </tr> <tr> <td>X-Intercepts</td> <td>Points where the graph intersects the x-axis</td> <td>x = (-b ± √(b²-4ac)) / 2a</td> </tr> </table>

Example Problems and Worksheet Answers

Let’s consider some example quadratic functions and their key features.

Example 1: ( f(x) = 2x^2 - 4x + 1 )

1. Find the vertex: [ x = -\frac{-4}{2 \times 2} = 1 ] [ f(1) = 2(1^2) - 4(1) + 1 = -1 \quad \text{(Vertex: (1, -1))} ]

2. Find the axis of symmetry: [ x = 1 ]

3. Determine the direction of opening: Since ( a = 2 > 0 ), the parabola opens upwards.

4. Find the y-intercept: [ f(0) = 1 \quad \text{(Y-Intercept: (0, 1))} ]

5. Find the x-intercepts: [ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \times 2 \times 1}}{2 \times 2} = \frac{4 \pm \sqrt{16 - 8}}{4} = \frac{4 \pm \sqrt{8}}{4} = 1 \pm \frac{\sqrt{2}}{2 ]

   • X-Intercepts: ( (1 + \frac{\sqrt{2}}{2}, 0) ) and ( (1 - \frac{\sqrt{2}}{2}, 0) )

Example 2: ( f(x) = -x^2 + 6x - 8 )

1. Find the vertex: [ x = -\frac{6}{2 \times -1} = 3 ] [ f(3) = -3^2 + 6 \times 3 - 8 = 1 \quad \text{(Vertex: (3, 1))} ]

2. Find the axis of symmetry: [ x = 3 ]

3. Determine the direction of opening: Since ( a = -1 < 0 ), the parabola opens downwards.

4. Find the y-intercept: [ f(0) = -8 \quad \text{(Y-Intercept: (0, -8))} ]

5. Find the x-intercepts: [ x = \frac{-6 \pm \sqrt{6^2 - 4 \times -1 \times -8}}{2 \times -1} = \frac{-6 \pm \sqrt{36 - 32}}{-2} = \frac{-6 \pm \sqrt{4}}{-2} = \frac{-6 \pm 2}{-2} ]

   • X-Intercepts: ( (2, 0) ) and ( (4, 0) )

Important Notes 📝

• When graphing a quadratic function, always plot the vertex, intercepts, and the axis of symmetry to ensure accuracy.
• The discriminant provides useful information about the number and nature of the roots without needing to solve the quadratic equation fully.

By understanding these features, students can better analyze quadratic functions and use them effectively in various mathematical contexts. Mastery of these concepts lays a strong foundation for further studies in algebra and beyond. Happy learning! 🎓