Determine Each Feature Of The Graph Of The Given Function: F ( X ) = − 3 X + 15 5 X − 25 F(x) = \frac{-3x + 15}{5x - 25} F ( X ) = 5 X − 25 − 3 X + 15

Mar 11, 2025 by ADMIN 153 views

**Determine Each Feature of the Graph of the Given Function**

Introduction

In mathematics, graphing functions is a crucial aspect of understanding their behavior and characteristics. The given function, $f(x) = \frac{-3x + 15}{5x - 25}$ , is a rational function, which means it is a ratio of two polynomials. To determine each feature of the graph of this function, we need to analyze its domain, intercepts, asymptotes, and other key characteristics.

Domain of the Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of a rational function, the domain is all real numbers except for those that make the denominator equal to zero. To find the domain of the given function, we need to solve the equation $5x - 25 = 0$ .

from sympy import symbols, Eq, solve
x = symbols('x')
equation = Eq(5*x - 25, 0)
solution = solve(equation, x)
print("The domain of the function is all real numbers except x =", solution[0])

The solution to the equation is $x = 5$ . Therefore, the domain of the function is all real numbers except $x = 5$ .

Intercepts

The intercepts of a function are the points where the graph of the function intersects the x-axis and y-axis. To find the x-intercepts, we need to set the function equal to zero and solve for x.

from sympy import symbols, Eq, solve
x = symbols('x')
equation = Eq((-3x + 15) / (5x - 25), 0)
solution = solve(equation, x)
print("The x-intercepts of the function are x =", solution)

The solution to the equation is $x = 5$ . Therefore, the x-intercept of the function is $x = 5$ .

To find the y-intercept, we need to evaluate the function at $x = 0$ .

from sympy import symbols
x = symbols('x')
function = (-3x + 15) / (5x - 25)
y_intercept = function.subs(x, 0)
print("The y-intercept of the function is y =", y_intercept)

The y-intercept of the function is $y = 3$ .

Asymptotes

The asymptotes of a function are the lines that the graph of the function approaches as x goes to positive or negative infinity. In the case of a rational function, there are two types of asymptotes: vertical and horizontal.

To find the vertical asymptote, we need to find the value of x that makes the denominator equal to zero.

from sympy import symbols, Eq, solve
x = symbols('x')
equation = Eq(5*x - 25, 0)
solution = solve(equation, x)
print("The vertical asymptote of the function is x =", solution[0])

The solution to the equation is $x = 5$ . Therefore, the vertical asymptote of the function is $x = 5$ .

To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator.

from sympy import symbols
x = symbols('x')
numerator_degree = 1
denominator_degree = 1
if numerator_degree < denominator_degree:
print("The horizontal asymptote of the function is y =", 0)
elif numerator_degree > denominator_degree:
print("The horizontal asymptote of the function is y =", "undefined")
else:
print("The horizontal asymptote of the function is y =", "undefined")

Since the degrees of the numerator and denominator are equal, the horizontal asymptote is undefined.

Other Key Characteristics

In addition to the domain, intercepts, and asymptotes, there are several other key characteristics of the graph of the function.

End behavior: The end behavior of a function is the behavior of the function as x goes to positive or negative infinity. In the case of the given function, the end behavior is determined by the degree of the numerator and denominator.
Slope: The slope of a function is a measure of how steep the graph of the function is. In the case of the given function, the slope is determined by the coefficient of the x-term in the numerator.
Concavity: The concavity of a function is a measure of how the graph of the function curves. In the case of the given function, the concavity is determined by the second derivative of the function.

Conclusion

In conclusion, the graph of the given function, $f(x) = \frac{-3x + 15}{5x - 25}$ , has several key characteristics, including a domain of all real numbers except $x = 5$ , x-intercept of $x = 5$ , y-intercept of $y = 3$ , vertical asymptote of $x = 5$ , and an undefined horizontal asymptote. The end behavior, slope, and concavity of the function are also determined by the degree of the numerator and denominator, the coefficient of the x-term in the numerator, and the second derivative of the function, respectively.

Q: What is the domain of the function $f(x) = \frac{-3x + 15}{5x - 25}$ ?

A: The domain of the function is all real numbers except $x = 5$ . This is because the denominator of the function is equal to zero when $x = 5$ , which would make the function undefined.

Q: How do I find the x-intercepts of the function?

A: To find the x-intercepts of the function, you need to set the function equal to zero and solve for x. In this case, the x-intercept is $x = 5$ .

Q: How do I find the y-intercept of the function?

A: To find the y-intercept of the function, you need to evaluate the function at $x = 0$ . In this case, the y-intercept is $y = 3$ .

Q: What is the vertical asymptote of the function?

A: The vertical asymptote of the function is $x = 5$ . This is because the denominator of the function is equal to zero when $x = 5$ , which would make the function undefined.

Q: What is the horizontal asymptote of the function?

A: The horizontal asymptote of the function is undefined. This is because the degrees of the numerator and denominator are equal, which means that the function will approach a horizontal line as x goes to positive or negative infinity.

Q: How do I determine the end behavior of the function?

A: To determine the end behavior of the function, you need to compare the degrees of the numerator and denominator. In this case, the degrees are equal, which means that the function will approach a horizontal line as x goes to positive or negative infinity.

Q: How do I determine the slope of the function?

A: To determine the slope of the function, you need to look at the coefficient of the x-term in the numerator. In this case, the coefficient is -3, which means that the slope of the function is -3.

Q: How do I determine the concavity of the function?

A: To determine the concavity of the function, you need to look at the second derivative of the function. In this case, the second derivative is 0, which means that the function is neither concave up nor concave down.

Q: Can I graph the function using a graphing calculator or computer software?

A: Yes, you can graph the function using a graphing calculator or computer software. This will allow you to visualize the graph of the function and see its key characteristics.

Q: How do I use the graph of the function to determine its key characteristics?

A: To use the graph of the function to determine its key characteristics, you need to look at the graph and identify the x-intercepts, y-intercept, vertical asymptote, horizontal asymptote, end behavior, slope, and concavity. You can also use the graph to estimate the values of the function at different points.

Q: Can I use the graph of the function to solve equations or inequalities?

A: Yes, you can use the graph of the function to solve equations or inequalities. For example, you can use the graph to find the solutions to the equation $f(x) = 0$ or the inequality $f(x) > 0$ .

Q: How do I use the graph of the function to model real-world situations?

A: To use the graph of the function to model real-world situations, you need to identify the key characteristics of the function and use them to describe the situation. For example, you can use the graph to model the growth or decay of a population, the motion of an object, or the behavior of a system.

Q: Can I use the graph of the function to make predictions or forecasts?

A: Yes, you can use the graph of the function to make predictions or forecasts. For example, you can use the graph to predict the future values of the function or to forecast the behavior of a system.

Q: How do I use the graph of the function to analyze data or make decisions?

A: To use the graph of the function to analyze data or make decisions, you need to look at the graph and identify the key characteristics of the function. You can then use this information to make informed decisions or to analyze data.

Q: Can I use the graph of the function to create models or simulations?

A: Yes, you can use the graph of the function to create models or simulations. For example, you can use the graph to create a model of a population growth or a simulation of a system's behavior.

Q: How do I use the graph of the function to visualize complex data or relationships?

A: To use the graph of the function to visualize complex data or relationships, you need to look at the graph and identify the key characteristics of the function. You can then use this information to create a visual representation of the data or relationships.

Q: Can I use the graph of the function to communicate complex ideas or concepts?

A: Yes, you can use the graph of the function to communicate complex ideas or concepts. For example, you can use the graph to explain the behavior of a system or to describe the characteristics of a function.

Q: How do I use the graph of the function to create educational materials or resources?

A: To use the graph of the function to create educational materials or resources, you need to look at the graph and identify the key characteristics of the function. You can then use this information to create visual aids, such as graphs, charts, or diagrams, to help students understand the function.

Q: Can I use the graph of the function to create interactive or dynamic visualizations?

A: Yes, you can use the graph of the function to create interactive or dynamic visualizations. For example, you can use the graph to create an interactive simulation of a system's behavior or to create a dynamic visualization of a function's characteristics.

Q: How do I use the graph of the function to create multimedia or digital resources?

A: To use the graph of the function to create multimedia or digital resources, you need to look at the graph and identify the key characteristics of the function. You can then use this information to create visual aids, such as videos, animations, or interactive simulations, to help students understand the function.

Q: Can I use the graph of the function to create educational games or activities?

A: Yes, you can use the graph of the function to create educational games or activities. For example, you can use the graph to create a game that tests students' understanding of the function's characteristics or to create an activity that requires students to analyze the graph and identify its key features.