Summary Tradisional | Second Degree Function: Introduction

Contextualization

The quadratic function, often referred to as the second-degree function, is one of the basic forms of polynomial functions that we study in mathematics. It is represented by the equation y = ax² + bx + c, where a, b, and c are constants, with a not equal to zero. The standout feature of this function is its graph – a parabola that either opens upwards or downwards depending on the sign of ‘a’. A good understanding of quadratic functions is vital for grasping more advanced mathematical concepts and for their practical use in fields like physics and economics.

Unlike first-degree (linear) functions that give straight-line graphs, quadratic functions produce a curved graph (parabola). This curve has a special point called the vertex, which gives the maximum or minimum value of the function. Additionally, the points where the graph meets the x-axis – known as the roots – can be found using the quadratic formula. Familiarising oneself with these aspects is essential for solving problems on projectile motion, optimisation, and overall function behaviour.

To Remember!

General Form of the Quadratic Function

The general form of a quadratic function is given by y = ax² + bx + c, where a, b, and c are constants and a is not zero. This form is the cornerstone for working with quadratic functions. The constant ‘a’ determines the curvature of the parabola, while ‘b’ and ‘c’ control its position on the Cartesian plane. In a nutshell, adding the ax² term transforms a linear, straight-line function into a curved parabola. This understanding is key to deriving other important aspects such as the vertex, the roots, and the overall shape of the graph.

• The quadratic function is given by y = ax² + bx + c, ensuring that a ≠ 0.

• The ax² term sets quadratic functions apart from linear functions.

• The coefficients a, b, and c influence the shape and location of the parabola on the graph.

Coefficient 'a' and the Concavity of the Parabola

The coefficient ‘a’ significantly affects the concavity of the parabola. When a > 0, the parabola opens upwards, meaning the vertex is the minimum point. Alternatively, if a < 0, the parabola opens downwards, making the vertex the maximum point. Furthermore, the absolute value of a determines how narrow or wide the parabola appears – a larger |a| results in a narrower graph, while a smaller |a| gives a wider spread. This concept is crucial for sketching the graph accurately and understanding the function’s behaviour.

• If a > 0, the parabola opens upwards (vertex is a minimum).

• If a < 0, the parabola opens downwards (vertex is a maximum).

• The absolute value of a decides the graph's width: higher |a| makes it narrower.

Vertex of the Parabola

The vertex of a parabola is the point that indicates the highest or lowest value of the quadratic function. It can be calculated using the formulas x = -b/(2a) and y = -Δ/(4a), where Δ (delta) is the discriminant computed as b² - 4ac. The vertex is important as it shows where the graph changes direction and gives us insight into its overall shape. In practical terms, the vertex might represent the point of maximum profit in an economic model or the peak height of a projectile in physics. Being able to locate and interpret the vertex is fundamental for a thorough analysis of quadratic functions.

• The vertex is the highest or lowest point of the parabola.

• It can be calculated using x = -b/(2a) and y = -Δ/(4a).

• The vertex helps indicate the parabola’s position and concavity on the graph.

Roots of the Quadratic Function

The roots of a quadratic function are the x-values at which y becomes zero – essentially, the points where the parabola cuts the x-axis. These can be found using the quadratic formula: x = (-b ± √Δ) / (2a), where Δ is the discriminant computed as b² - 4ac. The value of Δ tells us about the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is a single (double) real root; and if Δ < 0, the equation has no real roots. Knowing about the roots is important for solving equations and understanding how the graph behaves.

• Roots are the solutions for x when y = 0.

• They are found using the formula x = (-b ± √Δ) / (2a).

• The discriminant Δ indicates whether the roots are distinct, repeated, or non-real.

Key Terms

• Quadratic Function: A second-degree polynomial function represented by y = ax² + bx + c.

• Parabola: The curved graph that is the visual representation of a quadratic function.

• Coefficient 'a': The constant that impacts the curvature and the direction (concavity) of the parabola.

• Vertex: The maximum or minimum point of the parabola, calculated using x = -b/(2a) and y = -Δ/(4a).

• Roots: The values of x for which y = 0, determined using the quadratic formula.

• Discriminant (Delta): The value Δ = b² - 4ac, which decides the nature of the roots.

• Quadratic Formula: The formula used to find the roots, given as x = (-b ± √Δ) / (2a).

• Concavity: Describes the direction in which the parabola opens, decided by the sign of the coefficient ‘a’.

Important Conclusions

In this session, we explored the quadratic function, looking at its general form and how it contrasts with first-degree (linear) functions. We learned that a quadratic function is represented by y = ax² + bx + c (with a ≠ 0) and that this naturally produces a parabolic graph. We examined how the coefficient ‘a’ controls both the direction in which the parabola opens and its width. Additionally, we looked closely at the vertex, how to calculate it, and its significance, as well as the method to find the roots using the quadratic formula along with the discriminant Δ.

A solid understanding of quadratic functions is crucial not just in mathematics but also in practical scenarios like projectile motion in physics or profit analysis in economics. It lays the groundwork for delving into more complex topics. I encourage everyone to practice various problems related to quadratic functions, as regular practice is the key to mastery.

Remember, building a strong foundation in these basics will open up numerous academic and professional opportunities in the future.

Study Tips

• Practice solving quadratic equations using the quadratic formula to strengthen your understanding of roots and the discriminant Δ.

• Sketch graphs of different quadratic functions, noting the vertex, roots, and concavity to better visualise their behaviour.

• Apply these concepts to real-world problems such as projectile motion in physics or profit maximisation in economics for a deeper grasp.