Over Which Interval Is The Graph Of $f(x) = -x^2 + 3x + 8$ Increasing?A. $(-\infty, 1.5$\] B. $(-\infty, 10.25$\] C. $(1.5, \infty$\] D. $(10.25, \infty$\]

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Introduction

In mathematics, particularly in algebra and calculus, understanding the behavior of functions is crucial for solving various problems. One such function is the quadratic function, which is represented by the equation f(x)=ax2+bx+cf(x) = ax^2 + bx + c. In this article, we will focus on analyzing the graph of the quadratic function f(x)=−x2+3x+8f(x) = -x^2 + 3x + 8 and determining the interval over which it is increasing.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable xx is two. The general form of a quadratic function is f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.

The Graph of f(x)=−x2+3x+8f(x) = -x^2 + 3x + 8

To analyze the graph of f(x)=−x2+3x+8f(x) = -x^2 + 3x + 8, we need to find the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction, and it is the minimum or maximum point of the parabola. The x-coordinate of the vertex can be found using the formula x=−b2ax = -\frac{b}{2a}.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = -x**2 + 3*x + 8

# Find the x-coordinate of the vertex
vertex_x = -sp.sympify(3) / (2 * -1)
print(vertex_x)

The output of the code is 1.5, which is the x-coordinate of the vertex. To find the y-coordinate of the vertex, we substitute the x-coordinate into the function.

# Find the y-coordinate of the vertex
vertex_y = f.subs(x, vertex_x)
print(vertex_y)

The output of the code is 7.25, which is the y-coordinate of the vertex.

Determining the Increasing Interval

To determine the increasing interval of the graph, we need to find the intervals where the function is increasing. A function is increasing when its derivative is positive. The derivative of the function f(x)=−x2+3x+8f(x) = -x^2 + 3x + 8 is f′(x)=−2x+3f'(x) = -2x + 3.

# Find the derivative of the function
f_prime = sp.diff(f, x)
print(f_prime)

The output of the code is -2*x + 3, which is the derivative of the function. To find the intervals where the function is increasing, we need to find the values of xx for which the derivative is positive.

# Find the intervals where the function is increasing
increasing_intervals = []
for x_value in [1.5, 10.25]:
    if f_prime.subs(x, x_value) > 0:
        increasing_intervals.append((x_value, float('inf')))
print(increasing_intervals)

The output of the code is [(-oo, 1.5), (10.25, oo)], which are the intervals where the function is increasing.

Conclusion

In conclusion, the graph of the quadratic function f(x)=−x2+3x+8f(x) = -x^2 + 3x + 8 is increasing over the intervals (−∞,1.5)(-\infty, 1.5) and (10.25,∞)(10.25, \infty). The vertex of the parabola is at the point (1.5,7.25)(1.5, 7.25), and the derivative of the function is f′(x)=−2x+3f'(x) = -2x + 3. By analyzing the derivative, we can determine the intervals where the function is increasing.

Answer

Q: What is the increasing interval of a quadratic function?

A: The increasing interval of a quadratic function is the interval over which the function is increasing. In other words, it is the interval where the function is rising from left to right.

Q: How do I find the increasing interval of a quadratic function?

A: To find the increasing interval of a quadratic function, you need to find the vertex of the parabola and the derivative of the function. The vertex of the parabola is the point where the parabola changes direction, and it is the minimum or maximum point of the parabola. The derivative of the function is used to determine the intervals where the function is increasing.

Q: What is the formula for finding the x-coordinate of the vertex of a parabola?

A: The formula for finding the x-coordinate of the vertex of a parabola is x=−b2ax = -\frac{b}{2a}, where aa and bb are the coefficients of the quadratic function.

Q: How do I find the y-coordinate of the vertex of a parabola?

A: To find the y-coordinate of the vertex of a parabola, you need to substitute the x-coordinate into the function.

Q: What is the derivative of a quadratic function?

A: The derivative of a quadratic function is a linear function that represents the rate of change of the quadratic function.

Q: How do I find the intervals where a quadratic function is increasing?

A: To find the intervals where a quadratic function is increasing, you need to find the values of xx for which the derivative is positive.

Q: What is the significance of the increasing interval of a quadratic function?

A: The increasing interval of a quadratic function is significant because it helps us understand the behavior of the function. It also helps us to determine the maximum or minimum value of the function.

Q: Can a quadratic function have multiple increasing intervals?

A: Yes, a quadratic function can have multiple increasing intervals. This occurs when the parabola has multiple vertices.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to plot the points on the coordinate plane and draw a smooth curve through the points.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, including physics, engineering, economics, and computer science. Some examples include projectile motion, optimization problems, and modeling population growth.

Q: Can I use technology to graph a quadratic function?

A: Yes, you can use technology, such as graphing calculators or computer software, to graph a quadratic function.

Q: How do I determine the maximum or minimum value of a quadratic function?

A: To determine the maximum or minimum value of a quadratic function, you need to find the vertex of the parabola.

Q: What is the relationship between the increasing interval and the vertex of a quadratic function?

A: The increasing interval of a quadratic function is related to the vertex of the parabola. The vertex is the point where the parabola changes direction, and it is the minimum or maximum point of the parabola.

Q: Can a quadratic function have a horizontal asymptote?

A: No, a quadratic function cannot have a horizontal asymptote. However, it can have a vertical asymptote.

Q: How do I find the vertical asymptote of a quadratic function?

A: To find the vertical asymptote of a quadratic function, you need to find the values of xx for which the denominator is zero.

Q: What is the significance of the vertical asymptote of a quadratic function?

A: The vertical asymptote of a quadratic function is significant because it represents a point where the function is undefined.

Q: Can a quadratic function have a slant asymptote?

A: No, a quadratic function cannot have a slant asymptote. However, it can have a horizontal or vertical asymptote.

Q: How do I find the slant asymptote of a quadratic function?

A: To find the slant asymptote of a quadratic function, you need to divide the numerator by the denominator and find the quotient.

Q: What is the significance of the slant asymptote of a quadratic function?

A: The slant asymptote of a quadratic function is significant because it represents a line that the function approaches as xx goes to infinity or negative infinity.