Quadratic Functions

Quadratic Functions
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In this mini-lesson, we will explore the world of quadratic functions in math. You will get to learn about the graphs of quadratic functions, quadratic functions formula, and other interesting facts around the topic. You can also check out the playful calculators to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

The word "Quadratic" is derived from the word "Quad" which means square.

In other words, a quadratic function is a “ polynomial function of degree 2.” 

There are many scenarios where quadratic functions are used.

Did you know that when a rocket is launched, its path is described by the solution of a quadratic function?

When a rocket is launched, its path is described by the solution of a quadratic function

Let's go ahead and learn more about this topic now.

Lesson Plan

What Is a Quadratic Function?

Quadratic Function: Definition

A function of the form \(f(x)=ax^2+bx+c\), where \(a \neq 0\) is called a quadratic function.

What is general form of Quadratic Function

Here are some quadratic functions examples.

  Functions Values of \(a\), \(b\) and \(c\)
1. \(f(x)=8x^2+7x-1\) \[\begin{align}a&=8 \\b&=7 \\c&=-1 \end{align}\]
2. \(f(x)=4x^2-9\) \[\begin{align}a&=4 \\b&=0 \\c&=-9 \end{align}\]
3. \(f(x)=x^2+x\) \[\begin{align}a&=1 \\b&=1 \\c&=0 \end{align}\]
4. \(f(x)=6x-8\) \[\begin{align}a&=0 \\b&=6 \\c&=-8 \end{align}\]

Look at Example 4

Here, \(a = 0\). Therefore, it is not a quadratic function.

The remaining functions are quadratic function examples.


What Is an Example of a Quadratic Function?

Sometimes, a quadratic function is not written in its standard form, \(f(x)=ax^2+bx+c\), and we may have to change it into the standard form.

Here are some examples of functions and their standard forms. 

Functions Steps to follow Standard Form
\(f(x)\)=\((x-1)(x+2)\) Multiply the factors \((x-1)\) and \((x+2)\) \(f(x)\)=\(x^2+x-2\)
\(f(x)-x^2\)=\(-3x+1\) Move the term \(-x^2\) to the right side of the \(=\) sign \(f(x)\)=\(x^2-3x+1\)
\(f(x)+5x(x+3)\)=\(12x\) Expand the bracket on the left side and move \(12x\) to the left side of the \(=\) sign \(f(x)\)=\(-5x^2-3x\)
\(f(x)+x^3\)=\(x(x^2+x-3)\) Expand the bracket on the right side and move \(x^3\) to right side of the \(=\) sign \(f(x)\)=\(x^2-3x\)

Graphs of Quadratic Functions

This is how the quadratic function is represented on a graph. 

This curve is called a parabola.

parabola curve of quadratic equation

Quadratic Functions: Calculator

Now let’s explore some quadratic equations on a graph using the quadratic function calculators.

Drag the values of \(a\) to check for the variations of the upward parabola and downward parabola.

Dragging the values of \(b\) will move the curve either right or left.

Dragging the values of \(c\) will move the curve either up or down.


Quadratic Functions: Standard Form

The standard form of a quadratic function is \(f(x)=a(x-h)^{2}+k\), where \((h,k)\) is the vertex of the parabola. 

standard form of a quadratic function

This shows the quadratic functions standard form.

If \(a>0\), then the parabola opens upward.

If \(a<0\), then the parabola opens downward.

The curve of a quadratic function is a parabola.

Quadratic Functions and Equations

An equation of the form \(ax^2+bx+c=0\), where \(a \neq 0\) is called a quadratic equation.

So, by a quadratic equation, we mean all the values of \(x\) for which the quadratic function is zero.

Difference in Quadratic equation and Quadratic function


What Is the Quadratic Formula?

Sometimes working with a hard quadratic function is difficult. But don’t worry! Quadranator is here to help.

Quadranator alone is enough to hunt down all the solutions of the quadratic functions.

The quadratic functions formula is:

What Is the Quadratic Formula

This is also called a quadratic formula.

Do you have any difficulty memorizing the quadratic formula?

Don't worry! A rhyme or a song is the best way to remember it.

So, just sing along to the tune of “Pop Goes the Weasel.”

A song to learn Quadratic Equation Formula

 
important notes to remember
Important Notes
  1. The standard form of the quadratic function is \(f(x)=ax^2+bx+c\) where \(a\neq0\).
  2. The curve of the quadratic function is in the form of a parabola.
  3. The quadratic formula is given by \(\begin{align} \frac{-b \pm \sqrt{b^2-4ac}}{2a}\end{align}\).
  4. The discriminant is given by \(b^2-4ac\). This is used to determine the nature of the solutions of a quadratic function.
  5. If \(\alpha\) and \(\beta\) are the two solutions of a quadratic function, then the quadratic equation is given by \(x^2-(\alpha+\beta)x+\alpha\beta=0\).

Solved Examples 

Example 1

 

 

Determine the vertex of a quadratic function \(f(x)=2(x+3)^{2}-2\)

Solution

Let's express the given quadratic function in the standard function \(f(x)=a(x-h)^{2}+k\).

The function \(f(x)=2(x+3)^{2}-2\) can be written as \(f(x)=2(x-(-3))^{2}+(-2)\)

Here, \(h=-3\) and \(k=-2\)

\(\therefore\) The vertex of the function is (-3, -2)
Example 2

 

 

Jack shows a quadratic function.

Jack shows a quadratic function. Find the discriminant

Can you determine the discriminant?

Solution

The quadratic function is \(f(x)=x^{2}+3x-4\)

On comparing it with \(f(x)=ax^2+bx+c\), we get \(a=1\), \(b=3\), and \(c=-4\)

\[\begin{align}\text{Discriminant}&=b^2-4ac\\&=3^{2}-4(1)(-4)\\&=9+16\\&=25\end{align}\]

\(\therefore\), The discriminant is 25
Example 3

 

 

Mia is a fitness enthusiast who goes running every morning.

The park where she jogs is rectangular in shape and measures 12 feet by 8 feet.

Quadratic Equation Question to find width of a park's pathway

A nature restoration group plans to revamp the park and decides to build a pathway surrounding the park.

This would increase the total area to 140 sq. ft.

What will be the width of the pathway?

Solution

Let’s denote the width of the pathway as \(x\).

Then, the length and breadth of the outer rectangle is \((12+2x)\;\text{ft.}\) and \((8+2x)\;\text{ft.}\)

The area of the park,
\[\begin{align}(12+2x)(8+2x)&=140\\2(6+x)\cdot 2(4+x)&=140\\(x+6)(x+4)&=35\\x^2+10x-11&=0\end{align}\]

Now we have to find the solution of the quadratic function \(f(x)=x^2+10x-11\)

\[\begin{align}f(x)&=0\\x^2+11x-x-11&=0\\x(x+11)-(x+11)&=0\\(x+11)(x-1)&=0\\x&=1,-11\end{align}\]

Since length can’t be negative, we can only consider \(x=1\)

\(\therefore\), The width of the pathway will be 1 feet.
 
Thinking out of the box
Think Tank
  1. Is it possible to construct a rectangular park with perimeter \(60\;\text{m}\) and area \(200\;\text{sq. m}\). If yes, what will be its length and breadth?
  2. The sum of two numbers is 45. After subtracting 5 from both numbers, the product of the numbers is 124. What are the numbers?

Interactive Questions 

Here are a few activities for you to practice.

Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of the quadratic function. The math journey around the quadratic function starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath. 

We hope you enjoyed learning about quadratic functions and equations.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it problems, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. What is the general form of a quadratic function?

The general form of a quadratic function is \(f(x)=ax^2+bx+c\), where \(a \neq 0\)

2. Find the axis of symmetry with an example.

The axis of symmetry is the vertical line across from the vertex.

Look at the graph of the quadratic function \(f(x)=x^2-6x+5\)

Example of the axis of symmetry

The line \(x=3\) is the line of symmetry.

3. How can you find the intercept of the parabola?

\(x\) intercept of a parabola \(f(x)=a(x-h)^{2}+k\) is given by substituting 0 for \(x\) in \(f(x)\).

\(y\) intercept of a parabola \(f(x)=a(x-h)^{2}+k\) is given by solving \(f(x)\) for the variable \(x\).

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