Which of the following functions is graphed below apex? This question opens up a vast landscape of mathematical exploration, where we delve into the intricacies of functions and their graphical representations. By examining the key features, properties, and applications of different functions, we embark on a journey to uncover the secrets hidden within their graphs.
In this comprehensive analysis, we will dissect the graph in question, identifying its key points, domain, range, and behavior. We will compare and contrast it with related functions, exploring their similarities and differences. Furthermore, we will delve into the historical development of the function, tracing its origins and significance in the world of mathematics.
Function Graph Analysis
A function graph visually represents the relationship between the input and output values of a function. It provides insights into the function’s behavior, including its key features, such as intercepts, extrema, and symmetry.
Consider the graph below, which represents the function y = x 2– 4:
The graph is a parabola that opens upwards. The vertex of the parabola, located at (0,-4), represents the minimum point of the function. The graph intersects the x-axis at x = ±2, indicating the x-intercepts of the function.
Function Properties
The domain of the function y = x 2– 4 is all real numbers, as the function can be evaluated for any input value. The range of the function is all non-negative real numbers, as the output value is always greater than or equal to -4.
The function is increasing for x > 0 and decreasing for x < 0. This is because the slope of the graph is positive for x > 0 and negative for x< 0.
The function is concave up for all values of x. This is because the second derivative of the function is always positive.
The function is symmetric with respect to the y-axis. This is because for any input value x, the output value-x also satisfies the function equation.
Comparison with Other Functions
The function y = x 2– 4 is a quadratic function. Quadratic functions share similar characteristics, such as their parabolic shape and the presence of a vertex. However, they can differ in their domain, range, and specific properties.
For example, the function y = x 2+ 4 has the same domain and range as y = x 2– 4, but it is shifted up by 4 units. The function y = 2x 2– 4 has a narrower range than y = x 2– 4, as it is multiplied by a factor of 2.
Applications and Examples
Quadratic functions have numerous applications in various fields.
- In physics, the trajectory of a projectile can be modeled by a quadratic function, where the height of the projectile is a function of its horizontal distance.
- In economics, the profit of a company can be modeled by a quadratic function, where the profit is a function of the number of units sold.
- In engineering, the stress on a beam can be modeled by a quadratic function, where the stress is a function of the distance from the center of the beam.
Historical Context and Development, Which of the following functions is graphed below apex
Quadratic functions have been studied for centuries. The ancient Greeks, such as Euclid and Archimedes, used quadratic equations to solve geometric problems.
In the 17th century, René Descartes developed analytic geometry, which allowed quadratic functions to be represented graphically. This led to a deeper understanding of their properties and applications.
Today, quadratic functions are an essential tool in many branches of mathematics, science, and engineering.
FAQ Section: Which Of The Following Functions Is Graphed Below Apex
What is the domain of the function?
The domain of a function is the set of all possible input values for which the function is defined.
What is the range of the function?
The range of a function is the set of all possible output values that the function can produce.
How can I determine whether a function is increasing or decreasing?
A function is increasing if its output values increase as its input values increase. A function is decreasing if its output values decrease as its input values increase.
