Summary Tradisional | Second Degree Function: Graph and Table

Contextualization

Quadratic functions play a vital role in mathematics, especially given their relevance to real-world applications across various fields. A quadratic function is defined by the polynomial equation f(x) = ax² + bx + c, where a, b, and c are constants, and a cannot be zero. The graph of this function resembles a parabola, opening either upward or downward based on the sign of the coefficient a. Understanding quadratic functions is key to modelling quadratic behaviours in both nature and human-made systems, such as the trajectory of a thrown basketball or the design of satellite dishes. Beyond theoretical use, quadratics have practical implications as well. For instance, the path of a ball thrown into the air follows a parabolic trajectory that can be described using a quadratic function. Additionally, in economics, these functions help illustrate the relationship between cost and production, ultimately aiding in process optimization and profit maximization. Mastering how to construct graphs and tables for these functions allows for effective visual and numerical analysis which supports data interpretation and decision-making.

To Remember!

Definition of Quadratic Function

A quadratic function is a polynomial function that can be expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a cannot equal zero. This representation is known as the standard form of the quadratic function. The coefficient a determines how the parabola behaves: if a is greater than zero, it opens upwards; if a is less than zero, it opens downwards. The coefficient b affects the slope of the parabola, while c shows the point where the graph intersects the y-axis. Quadratic functions are symmetrical about a vertical line called the axis of symmetry, which runs through the vertex of the parabola. This vertex indicates where the function reaches its maximum or minimum point, depending on whether a is positive or negative. The x-coordinate of the vertex can be found with x = -b/2a, and by substituting this value back into the function we can find the vertex's y-coordinate. The roots, or solutions, of the quadratic function—values of x that make f(x) = 0—are determined using the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, which indicates where the graph intersects the x-axis.

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

• The coefficient a determines the curvature of the parabola.

• The vertex is located at x = -b/2a.

• Roots of the function can be found using the quadratic formula.

Graph of the Quadratic Function

The graph of a quadratic function forms a parabola. This shape is symmetrical about its axis of symmetry and features a vertex, which denotes the maximum or minimum point of the graph. The sign of the coefficient a determines the direction of the parabola's opening: upwards if a is positive and downwards if a is negative. Understanding the vertex is important, as it reveals crucial information regarding the function's extreme values. To find the x-coordinate of the vertex, we use x = -b/2a, and we can determine the corresponding y-coordinate by substituting this x value back into the original function. The axis of symmetry is the vertical line that cuts through the vertex, represented by the equation x = -b/2a, and divides the parabola into mirrored halves. The roots of the function are points at which the graph intersects the x-axis and can be found by solving the equation f(x) = 0.

• The graph of a quadratic function forms a parabola.

• The curvature of the parabola is determined by the sign of the coefficient a.

• The vertex indicates the maximum or minimum point of the parabola.

• The axis of symmetry is the line that passes through the vertex, creating two equal halves of the parabola.

Value Table

A value table is an effective way to visualize the relationship between x and f(x) in a quadratic function. To create a value table, choose a series of x values, substitute them into the function's equation, and compute the corresponding f(x) values. These ordered pairs (x, f(x)) can be then plotted on the Cartesian plane to create the graph of the function. This table construction aids in pinpointing significant points of the parabola, such as its vertex and roots. Selecting x values close to the vertex allows for a better representation of the parabola's shape. Moreover, the value table reinforces the understanding of the parabola's symmetry relative to its axis of symmetry. For instance, let’s consider the function f(x) = x² - 4x + 3. By calculating f(x) for x values of -1, 0, 1, 2, 3, 4, and 5, we can create a value table. Plotting these pairs on the Cartesian plane helps sketch the graph, identifying the vertex, axis of symmetry, and roots.

• A value table visualizes the relationship between x and f(x).

• Select x values, substitute them into the equation, and calculate f(x).

• The table helps recognize key points like the vertex and roots.

Practical Example

To put the theory into action, let's look at the function f(x) = x² - 4x + 3. Initially, identify the coefficients: a = 1, b = -4, and c = 3. Next, build a value table by calculating the f(x) values for various x values: For x = -1, f(x) = (-1)² - 4(-1) + 3 = 8; for x = 0, f(x) = 0² - 4(0) + 3 = 3; for x = 1, f(x) = 1² - 4(1) + 3 = 0; for x = 2, f(x) = 2² - 4(2) + 3 = -1; for x = 3, f(x) = 3² - 4(3) + 3 = 0; for x = 4, f(x) = 4² - 4(4) + 3 = 3; for x = 5, f(x) = 5² - 4(5) + 3 = 8. With these values, we can set up a table and plot the points on the Cartesian plane. The vertex is found using the formula x = -b/2a, giving us x = 2. By substituting x = 2 back into the function, we find the y-coordinate of the vertex, which is -1. Thus, the vertex of the parabola is (2, -1), and the axis of symmetry is x = 2. The roots, identified by where the function crosses the x-axis, occur at x = 1 and x = 3. This practical example illustrates how we can apply the theory of quadratic functions in creating value tables and graphs while recognizing the vertex, axis of symmetry, and roots, enhancing our understanding of the parabola's characteristics.

• Example: f(x) = x² - 4x + 3.

• Create a value table with various x values.

• Identify the vertex, axis of symmetry, and the roots.

Key Terms

• Quadratic Function: A polynomial function represented as f(x) = ax² + bx + c, where a, b, and c are constants and a cannot equal zero.

• Parabola: The graph of a quadratic function, which can open upwards or downwards.

• Vertex: The maximum or minimum point of the parabola, calculated as x = -b/2a.

• Axis of Symmetry: The vertical line that passes through the vertex of the parabola, dividing it into two equal halves.

• Roots: The x-values that satisfy f(x) = 0, found using the quadratic formula.

• Coefficient a: Determines the curvature of the parabola (upward when a > 0, downward when a < 0).

• Value Table: A means of visualizing the relationship between x and f(x) in a quadratic function.

Important Conclusions

In this lesson, we explored the definition and characteristics of quadratic functions, with a focus on their graphical representation and the use of value tables. We established that a quadratic function is expressed as f(x) = ax² + bx + c, where 'a' regulates the parabola's curvature. The graph is characterized by its parabola shape, a vertex, an axis of symmetry, and roots determined by the quadratic formula. Crafting value tables is an essential tool for visualizing the relationship between x and f(x), assisting in sketching the graph of a quadratic function. This exercise helps us pinpoint crucial features of the parabola, including the vertex and roots, leading to a more nuanced understanding of the function's behaviour. We applied these concepts via a practical example, showing how theory translates into creating graphs and analysing functions. A solid grasp of quadratic functions is crucial for varied practical and theoretical purposes, ranging from studying projectile motion in physics to modeling financial dynamics. This knowledge equips us for effective visual and numerical analysis, enhancing our ability to interpret data for informed decision-making. We encourage further study to enrich your understanding and application of quadratic functions across different scenarios.

Study Tips

• Practice building value tables and sketching graphs for various quadratic functions to solidify your grasp of how coefficients affect the shape of the parabola.

• Review the quadratic formula regularly and practice solving quadratic equations to identify the roots, as this is pivotal for thorough function analysis.

• Investigate practical applications of quadratic functions in other disciplines, such as physics and economics, to appreciate the relevance of these concepts in everyday life while deepening your understanding of the subject.