Which Of The Following Functions Is Quadratic

Which of the Following Functions is Quadratic? A Comprehensive Guide

Determining whether a function is quadratic requires a keen understanding of quadratic functions' defining characteristics. This comprehensive guide will not only explain how to identify quadratic functions but also delve into the properties that distinguish them from other types of functions, providing you with a robust understanding of this fundamental concept in algebra.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two. This means the highest power of the independent variable (usually x) is 2. The general form of a quadratic function is:

f(x) = ax² + bx + c

where a, b, and c are constants, and a is not equal to zero (a ≠ 0). If a were zero, the x² term would disappear, resulting in a linear function.

The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The parabola opens upwards if a > 0 and downwards if a < 0. The vertex of the parabola represents either the minimum (if a > 0) or maximum (if a < 0) value of the function.

Key Characteristics of Quadratic Functions:

• Degree 2: The highest exponent of the variable is 2.
• Parabola Shape: The graph is always a parabola.
• Axis of Symmetry: The parabola has a vertical line of symmetry that passes through the vertex.
• Vertex: The highest or lowest point on the parabola.
• Roots/Zeros/x-intercepts: The points where the parabola intersects the x-axis (where f(x) = 0). A quadratic function can have zero, one, or two real roots.

Identifying Quadratic Functions: A Step-by-Step Approach

Let's examine several functions and determine which ones are quadratic using the criteria outlined above. We'll consider various representations, including explicit formulas, tables of values, and graphs.

Example 1:

f(x) = 3x² - 5x + 2

This function is clearly quadratic. It's in the standard form ax² + bx + c, with a = 3, b = -5, and c = 2. The highest power of x is 2.

Example 2:

g(x) = x³ + 2x - 1

This function is not quadratic. The highest power of x is 3, making it a cubic function, not a quadratic function. Cubic functions have a different shape than parabolas; they can have multiple turning points.

Example 3:

h(x) = 5x - 7

This is a linear function, not a quadratic function. The highest power of x is 1. Linear functions graph as straight lines.

Example 4:

i(x) = 2(x - 1)² + 4

This function is quadratic, even though it's not initially in standard form. Expanding the expression gives:

i(x) = 2(x² - 2x + 1) + 4 = 2x² - 4x + 2 + 4 = 2x² - 4x + 6

Now, it's evident that it fits the ax² + bx + c form, with a = 2, b = -4, and c = 6.

Example 5:

j(x) = √x + 1

This function is not quadratic. It involves a square root, which is not a characteristic of polynomial functions, including quadratic functions.

Example 6: Identifying Quadratic Functions from Tables of Values

Consider the following table of values:

x	-2	-1	0	1	2
f(x)	8	3	0	-1	0

To determine if this represents a quadratic function, we look for a pattern in the second differences. Calculate the first differences (the difference between consecutive f(x) values):

• 3 - 8 = -5
• 0 - 3 = -3
• -1 - 0 = -1
• 0 - (-1) = 1

Now calculate the second differences (the difference between consecutive first differences):

• -3 - (-5) = 2
• -1 - (-3) = 2
• 1 - (-1) = 2

Since the second differences are constant, this table of values likely represents a quadratic function.

Example 7: Identifying Quadratic Functions from Graphs

If a graph shows a U-shaped curve (a parabola), it strongly suggests a quadratic function. Look for symmetry: if you can draw a vertical line that divides the parabola into two mirror images, this is further confirmation. However, visual inspection alone may not be conclusive, particularly if the graph is not accurately drawn or only shows a limited portion of the curve.

Distinguishing Quadratic Functions from Other Functions

It's crucial to differentiate quadratic functions from other types of functions, especially linear and cubic functions.

Quadratic vs. Linear Functions

• Linear functions have a degree of 1 (highest power of x is 1). Their graphs are straight lines. The equation is of the form f(x) = mx + c, where m is the slope and c is the y-intercept.
• Quadratic functions have a degree of 2 (highest power of x is 2). Their graphs are parabolas. The equation is of the form f(x) = ax² + bx + c, where a ≠ 0.

The key difference is the degree of the polynomial and the shape of the graph.

Quadratic vs. Cubic Functions

• Cubic functions have a degree of 3 (highest power of x is 3). Their graphs can have up to two turning points. The equation is of the form f(x) = ax³ + bx² + cx + d, where a ≠ 0.
• Quadratic functions have a degree of 2. Their graphs have only one turning point (the vertex).

Again, the degree of the polynomial and the number of turning points are the primary distinguishing features.

Real-World Applications of Quadratic Functions

Quadratic functions are not merely abstract mathematical concepts; they have numerous applications in various fields:

• Physics: Calculating projectile motion (the trajectory of a ball or rocket), understanding the relationship between force and distance in springs (Hooke's Law), and analyzing the path of a light beam through a lens.
• Engineering: Designing parabolic antennas and reflectors, optimizing structures for strength and stability, and modeling the behavior of electrical circuits.
• Economics: Modeling cost, revenue, and profit functions, particularly those involving economies of scale.
• Computer Graphics: Creating curved lines and shapes in computer-aided design (CAD) software.
• Statistics: Fitting quadratic models to data sets that show a parabolic trend.

Understanding and identifying quadratic functions is a fundamental skill in algebra and has significant practical implications across diverse scientific and engineering disciplines. By mastering the concepts outlined in this guide, you'll be well-equipped to confidently analyze and solve problems involving quadratic functions. Remember to always look for the defining characteristics: a degree of 2, a parabolic graph, and a constant second difference in a table of values. Careful examination, combined with a grasp of the theoretical underpinnings, will lead to accurate identification and effective problem-solving.