Comparing Quadratic and Exponential Functions: Explanation

You might have studied the Quadratic and Exponential functions in your 9th or 11th-grade syllabus. However, studying these functions as part of your syllabus doesn’t necessarily give you a clear understanding of the difference between the two.

As part of your syllabus, you’re only required to solve equations and problems related to the two without speculating about the possible differences between them and their applications.

So in this article, I aim to educate you on the difference between the two with the help of graphs, equations, and examples so that you can comprehend the knowledge easily.

Let’s start.

What is Function in Math?

A function in math is best defined as a relation between inputs where each input has the same result, meaning that each input will return the same output.

A function in math is often shown by or represented by f(x). For example, f(x)=x^2. This function will give us the square of the number in the bracket, in this case, the number 2.

It will give us the same output no matter what the input in the function is. In this case, it will always return the square of the number in the bracket as the output.

Numerous functions in math are used to accomplish different tasks and are applied in various areas. However, the functions we will discuss in this article are quadratic and exponential. We will focus mainly on highlighting the difference between these two functions.

What Is a Quadratic Function?

A quadratic function is a polynomial function, which is any form of the equation ax^2+bx+c. It’s also called the polynomial of degree 2 because the maximum exponent can be 2.

The quadratic formula is used in various fields of science, such as engineering. It’s graphically represented through a parabola.

This parabola is used for different activities in our daily lives, such as throwing a ball or hitting a golf ball. Quadratic equations are also used to find missing variables in measurements, determine the velocity of any object, and calculate the profit of any item or product in commerce.

Here’s an example of a quadratic equation: 3x^2+5x+9 a:3 b:5 c:9

This is an example of a quadratic function in its standard form. The formula that is used to solve such equations is known as the quadratic formula, which is the following: (-b±√(b²-4ac))/(2a).

What Is an Exponential Function?

An exponential function in math is a function of f(x)=a^x where a is the base, a constant, and must always be greater than 0. It is denoted by f(x)=\exp or e^{x}.

The most widely used exponential base is base e, called the natural logarithm. It calculates the growth rate of various things, such as population and bacteria. An exponential function is arguably the most important function in math.

It’s very important because it is used in various areas, such as:

  • Science
  • Commerce.

For example, the interest rate on the money you deposit in a bank increases exponentially, which means it follows an exponential curve; thus, it can be calculated using exponential functions.

Moreover, debt growth also increases exponentially and follows an exponential curve. By using exponential functions, you can stop your debt from rising and have greater control over your finances.

In biology, it’s used to estimate the population growth of a specific area over a certain period.

Radioactivity, such as the decay of uranium, also follows exponential growth. Thus, this is another application of the exponential function.

In physics, all the waves, such as sin, cos, sound waves, and many other waves, can also be written in terms of exponential functions, so this function helps physicists research these waves.

What Is a Quadratic Graph?

This is a representation of a quadratic graph
This is a representation of a quadratic graph.

The graph of a quadratic function is a U-shaped parabola, as shown in the picture above. This parabola can open up like a smile or downwards like a frown. The way the parabola opens up depends on the coefficient: ”a” in the equation ax^2+bx+c. If the coefficient is a>0, the parabola opens up; if the coefficient is a<0, then the parabola opens down.

  • The highest or lowest point of a parabola is called a vertex.
  • The point the vertex represents, whether maximum or minimum, depends on how the parabola opens.

If it opens up, then the vertex represents the minimum point on the graph, and if it opens down, then the vertex represents the maximum point on the quadratic graph. Another feature of parabolas is the line of symmetry, a vertical line that passes through the vertex and is used to split the parabola into 2 equal and identical halves.

Graphical Analysis: Quadratic Graph Intercepts and Real-World Applications

It can be obtained using the following formula:  y=a(x−h)2+k. The quadratic graph has a y-intercept, the point where the parabola intersects the y-axis. This y-intercept only has one value meaning that the parabola only intersects the y-axis once. The x-intercept is the point where the parabola intercepts or crosses the x-axis.

The number of intercepts can be 0, 1, or 2. The maximum number of intercepts is 2 because a quadratic equation can only have up to 2 solutions or 2 roots. The quadratic graph is one way of solving quadratic equations. It’s called the graphical method of solving quadratic equations.

The quadratic graph is used in many areas of our daily lives, mainly in sports. Throwing a ball or jumping from a high platform are examples of situations that a quadratic graph could demonstrate. The quadratic graph could then determine the maximum or minimum points the ball or the person reached.

What are Exponential Graphs?

This is a representation of an exponential graph
This is a representation of an exponential graph.

Both algebraic and transcendental equations can often be solved by hand with the help of a calculator. However, when these algebraic and transcendental equations appear together, solving them by hand becomes difficult or impossible. Therefore to solve these two equations together, we use the exponential graph and solve it graphically.

The simplest exponential function is f(x) = ax, a>0, a≠1. In this function, the base a is always kept greater than 0 because if the base is anything less than 0, it could give us an unreal number.

If the base is 1, it would always return 1 regardless of its exponent, and it would be a very boring function. Certain restrictions are placed on the exponential function because of these reasons.

The graph of an exponential function displays different properties depending on whether the base is greater than 1 or less than 1 but greater than 0. It will display the following properties when the base is larger than 1. The domain will consist of only real numbers, the range will be y>0, the graph will constantly increase, and the graph will be continuous and smooth.

The exponential graph shows similar properties when the base is less than 1 but larger than 0. The only change in its properties is that the graph will be decreasing. Exponential graphs are used to represent the data obtained through exponential functions. The types of data and the application of exponential functions have been discussed previously.

Difference Between Exponential and Quadratic Functions (Use the content here as a table)

Now that a good understanding of quadratic and exponential functions has been developed, we shall discuss the differences between these very important functions.

Quadratic FunctionExponential Function
The variable is the base and the highest possible power is (ax^2+bx+c).The base is a constant and the power of that base is a variable.
The rate of change is constant which means that the graph increases at a constant rate and therefore it’s easy to calculate the change in the graph over a certain time period.In an exponential function, the rate of change is proportional to itself, and the graph increases at an increasing rate.
The quadratic graph will form a parabola when it reaches the vertex in an upward or downward direction.An exponential graph will continue to fall in one direction either up or down.
A quadratic graph curves when it reaches its maximum or minimum point.An exponential graph continues to curve from the very beginning.
Quadratic Function vs. Exponential Function


A brief explanation to fully understand the difference between the two
  • Quadratic and exponential functions are fundamental concepts in mathematics. They are often studied in various grade levels.
  • This article aims to clarify the disparities between these functions. It explains by delving into their graphs, equations, and practical applications.
  • A function in mathematics is a relationship between inputs and outputs. It ensures that each input yields a consistent result.
  • Quadratic functions are characterized by ax^2 + bx + c equations. They are associated with parabolic graphs. These are used in diverse fields such as sports and commerce.
  • Exponential functions are represented as f(x) = a^x, where ‘a’ is the base. These functions are vital in modeling growth, decay, and various natural phenomena.
  • Understanding the graphs and properties of these functions reveals their distinctive attributes.
  • Quadratic functions create parabolic curves with vertices that signify minimum or maximum points. In contrast, exponential graphs exhibit steady growth or decay.
  • Proficiency in quadratic and exponential functions is essential. It is because they find applications in fields like science, commerce, and everyday life.

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