Summary Tradisional | Second Degree Function: Graph and Table

Contextualization

Quadratic functions hold a special place in mathematics, owing much to their wide-ranging applications in everyday problems. Represented by the equation f(x) = ax² + bx + c (with a, b, and c being constants and a ≠ 0), these functions produce a curve called a parabola which can open either upwards or downwards, depending on the sign of a. This concept is fundamental for understanding various quadratic phenomena and for modelling situations ranging from the flight of a cricket ball to the design of satellite dishes. Besides its theoretical importance, quadratic functions come in handy in practical situations too. For instance, the trajectory of a ball thrown in the air follows a parabolic path that can be described using a quadratic equation. In fields like economics, these functions also help in analysing the relationship between cost and production, thereby aiding decision-making to improve efficiency and maximise profits. Learning how to plot graphs and create value tables for these functions not only enhances our visual grasp of the concepts but also supports more informed data analysis.

To Remember!

Definition of Quadratic Function

A quadratic function is essentially a polynomial expressed as f(x) = ax² + bx + c, where a, b, and c remain constant and a is not zero. This representation is commonly referred to as the standard form. The coefficient a plays a key role in determining whether the parabola opens upwards (a > 0) or downwards (a < 0). Meanwhile, the coefficient b affects the tilt of the parabola, and the constant c indicates where the graph meets the y-axis. A notable feature of quadratic functions is their symmetry about a vertical line, known as the axis of symmetry, which passes through the vertex – the highest or lowest point of the parabola. To find the vertex, the x-coordinate is given by x = -b/2a, and by plugging this back into the function, we arrive at the y-coordinate. The values of x that make f(x) zero, referred to as the roots or zeros, are determined using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. These roots pinpoint the exact location where the parabola crosses the x-axis.

• The quadratic function is represented as f(x) = ax² + bx + c.

• The sign of a controls whether the parabola opens upwards or downwards.

• The vertex is computed using the formula x = -b/2a.

• The quadratic formula helps to find the roots of the function.

Graph of the Quadratic Function

The graph of any quadratic function is a parabola. Its symmetrical nature is visible as it mirrors itself about its axis of symmetry, with the vertex marking the function's maximum or minimum point. The orientation of the parabola – whether it opens upwards or downwards – depends on the sign of a. When a is greater than zero, it opens upwards; when a is less, it opens downwards. The vertex, which is central to identifying the extreme values of the function, is calculated using x = -b/2a, and its corresponding y-value is obtained by substituting x into the function. The vertical line x = -b/2a is known as the axis of symmetry and evenly divides the parabola into two mirror image parts. Additionally, the points at which the graph cuts the x-axis (the roots) are found by solving f(x) = 0.

• The graph of a quadratic function takes the shape of a parabola.

• The orientation of the parabola is determined by the coefficient a.

• The vertex represents either the maximum or minimum point of the graph.

• The line passing through the vertex, x = -b/2a, serves as the axis of symmetry.

Value Table

A value table is an effective tool for demonstrating how x and f(x) interact in a quadratic function. By selecting different values of x, substituting these into the function, and calculating f(x), we create ordered pairs (x, f(x)) that can be plotted on a graph. This approach not only helps in sketching the curve but also in pinpointing crucial features like the vertex and the roots. Careful selection of x-values, especially near the vertex, provides a clear and accurate picture of the parabola's form. Moreover, the value table also highlights the symmetry with respect to the axis of symmetry. For example, if we consider the function f(x) = x² - 4x + 3 and choose values for x such as -1, 0, 1, 2, 3, 4, and 5, we can compute the corresponding f(x) values and eventually sketch the graph, thereby identifying important characteristics like the vertex and the points where the graph meets the x-axis.

• A value table is useful for visualising how x and f(x) relate.

• Different x-values are substituted into the equation to compute f(x).

• This table aids in identifying key points such as the vertex and the roots.

Practical Example

To bring the theory to life, let us examine the function f(x) = x² - 4x + 3. Here, the coefficients are a = 1, b = -4, and c = 3. After choosing a variety of x-values and calculating the corresponding values of f(x), the value table is as follows: For x = -1, f(x) computes to 8; for x = 0, it becomes 3; for x = 1, f(x) is 0; for x = 2, f(x) equals -1; for x = 3, it comes out as 0; for x = 4, f(x) is back to 3; and for x = 5, it again results in 8. Plotting these pairs on the Cartesian plane reveals the parabola’s shape. The vertex is found using the formula x = -b/2a which gives x = 2, and plugging this value into the function yields y = -1, so the vertex is located at (2, -1). The line x = 2 acts as the axis of symmetry, and the graph crosses the x-axis at the roots, x = 1 and x = 3. This example neatly illustrates how theory is applied in practice to construct value tables and graphs, making it easier to understand the behaviour and main features of a quadratic function.

• For the function f(x) = x² - 4x + 3, construct a value table using various x-values.

• Use this table to locate the vertex, the axis of symmetry, and the roots.

• This example shows how to integrate theory with practical graph plotting.

Key Terms

• Quadratic Function: A polynomial represented as f(x) = ax² + bx + c, with a, b, and c as constants and a ≠ 0.

• Parabola: The characteristic curve of a quadratic function, which may open upwards or downwards.

• Vertex: The highest or lowest point of the parabola, determined by x = -b/2a.

• Axis of Symmetry: The vertical line through the vertex that divides the parabola into two equal halves.

• Roots: The solutions for x when f(x) = 0, typically found using the quadratic formula.

• Coefficient a: This defines the direction of the parabola; if a > 0, it opens upwards, and if a < 0, downwards.

• Value Table: A method to list values of x and f(x) for easier graph plotting and analysis.

Important Conclusions

In today’s session, we delved into the definition and features of quadratic functions, focusing on graph plotting and the creation of value tables. We saw that a quadratic function is expressed as f(x) = ax² + bx + c, with the coefficient a determining whether the parabola opens upwards or downwards. The graph, a parabola, highlights key features such as the vertex, the axis of symmetry, and the roots which are computed using the quadratic formula. Building value tables is an essential practice that not only makes plotting easier but also clarifies the curve’s nature by identifying critical points like the vertex and the roots. A hands-on example using the function f(x) = x² - 4x + 3 helped bring these concepts to life. Understanding quadratic functions is not only vital from a theoretical standpoint but also for practical applications, ranging from analysing projectile motions in physics to modelling economic trends. This knowledge is key for a comprehensive and practical approach to problem-solving in both academic and real-life scenarios.

Study Tips

• Regular practice in constructing value tables and sketching graphs for different quadratic functions will consolidate your understanding of how the coefficients influence the curve.

• Revise the quadratic formula and undertake multiple exercises on solving quadratic equations to reliably identify the roots of these functions.

• Investigate real-world applications of quadratic functions in various fields like physics and economics. This will help you appreciate the practical relevance of these concepts.