Consider The Following Function F X 3x 2-5 Calculate

Introduction

The function f(x) = 3x² - 5x is a quadratic function, which means it's a second-degree polynomial. Quadratic functions are fundamental in mathematics and have wide applications in physics, engineering, and economics.

This guide will help you understand how to calculate values of this function, graph it, and interpret its key characteristics.

Function Definition

f(x) = 3x² - 5x

Calculating f(x)

To calculate the value of the function for any given x, simply substitute the x value into the equation and perform the arithmetic operations.

Example Calculation

Let's calculate f(2):

f(2) = 3(2)² - 5(2) = 3(4) - 10 = 12 - 10 = 2

You can use our calculator in the sidebar to compute f(x) for any value of x. Just enter the x value and click "Calculate".

Graphing the Function

The graph of a quadratic function is a parabola. For f(x) = 3x² - 5x, the parabola opens upwards because the coefficient of x² is positive.

The vertex of the parabola represents the minimum point of the function. You can find the vertex using the formula:

Vertex Formula

x-coordinate of vertex: x = -b/(2a)

For f(x) = 3x² - 5x, a = 3 and b = -5

x = -(-5)/(2*3) = 5/6 ≈ 0.833

The y-coordinate of the vertex is f(5/6). Using our calculator, you can find that f(5/6) ≈ -1.875.

Key Characteristics

Quadratic functions have several important characteristics:

Vertex: The highest or lowest point of the parabola
Axis of Symmetry: A vertical line that passes through the vertex
Y-intercept: The point where the graph crosses the y-axis (when x=0)
X-intercepts (Roots): The points where the graph crosses the x-axis (where f(x)=0)

Key Values for f(x) = 3x² - 5x

Vertex: (5/6, -1.875)
Axis of Symmetry: x = 5/6
Y-intercept: (0, 0)
X-intercepts: x = 0 and x = 5/3

Practical Applications

Quadratic functions are used in various real-world scenarios:

Projectile motion in physics
Optimization problems in business
Modeling growth and decay in biology
Engineering design and analysis

Understanding how to work with quadratic functions gives you the tools to solve many practical problems.

FAQ

▼ How do I find the vertex of a quadratic function?

The vertex of a quadratic function f(x) = ax² + bx + c can be found using the formula x = -b/(2a). The y-coordinate is then found by substituting this x value back into the function.

▼ What does it mean if the coefficient of x² is negative?

If the coefficient of x² is negative, the parabola opens downward. This means the function has a maximum point (the vertex) rather than a minimum point.

▼ How do I find the x-intercepts of a quadratic function?

The x-intercepts occur where f(x) = 0. You can solve the equation ax² + bx + c = 0 using the quadratic formula: x = [-b ± √(b² - 4ac)]/(2a).

▼ What is the difference between a quadratic function and a linear function?

A quadratic function has an x² term, while a linear function only has x terms. This means quadratic functions graph as parabolas while linear functions graph as straight lines. Quadratic functions can have a vertex and x-intercepts, while linear functions only have a y-intercept.