Here is a rocket getting launched and it can go as high as 1000 ft!
It will go up for a few seconds after the engine is ignited.
As the engine burns out, the rocket will be in freefall.
Now, Cooper will investigate the time and height attained by the rocket.
He describes the path by a quadratic function.
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.”
In this mini-lesson, we will explore the world of graphs of quadratic functions. You will get to learn the domain and range of quadratic function, quadratic graph, vertex form of quadratic function, graphing quadratic functions in vertex form, standard form of a quadratic function, graphing quadratic functions in standard form, quadratic function graph, 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.
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.
Domain and Range of Quadratic Function
A quadratic function is a polynomial function that is defined for all real numbers.
So, the domain of a quadratic function is the set of real numbers, that is, \(\mathbb{R}\).
The range of quadratic function depends on the graph's opening side and vertex.
So, look for the lowermost and uppermost \(y\) values on the graph of the function to determine the range of the quadratic function.
Standard Form of a Quadratic Function
The standard form of a quadratic function is \(f(x)=ax^2+bx+c\), where \(a \neq 0\).
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 |
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\(f(x)\)= \((x-1)(x+2)\) |
Multiply the factors \((x-1)\) and \((x+2)\) |
\(f(x)\)= \(x^2+x-2\) |
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\(f(x)-x^2\)= \(-3x+1\) |
Move the term \(-x^2\) to the right side of the \(=\) sign |
\(f(x)\)= \(x^2-3x+1\) |
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\(f(x)+5x(x+3)\) =\(12x\) |
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 |
\(f(x)\)= \(x^2-3x\) |
Vertex Form of Quadratic Function
The vertex form of quadratic function is \(f(x)=a(x-h)^{2}+k\), where \((h,k)\) is the vertex of the parabola.
This shows the vertex form of a quadratic function.
How to Graph Quadratic Functions?
Graphing Quadratic Functions in Vertex Form
Now, in terms of graphing quadratic functions, we will understand a step-by-step procedure to plot the graph of any quadratic function.
Consider the general quadratic function \(f\left( x \right) = a{x^2} + bx + c\).
First, we rearrange it (by the method of completion of squares) to the following form:
\[f\left( x \right) = a{\left( {x + \frac{b}{{2a}}}\right)^2} - \frac{D}{{4a}}\]
The term D is the discriminant, given by \(D = {b^2} - 4ac\).
If you are unsure about this rearrangement, you are urged to refer to the chapter on quadratic equations.
Now, to plot the graph of \(f(x)\), we start by taking the graph of \({x^2}\) ,and applying a series of transformations on it:
Step-1: \({x^2} \to a{x^2}\): This will imply a vertical scaling of the original parabola. If \(a\) is negative, the parabola will also flip its mouth from the positive to the negative side. The magnitude of the scaling depends upon the magnitude of \(a\).
Step-2: \(a{x^2} \to a{\left( {x + \dfrac{b}{{2a}}}\right)^2}\): This is a horizontal shift of magnitude \(\left| {\dfrac{b}{{2a}}} \right|\) units. The direction of the shift will be decided by the sign of \(\dfrac{b}{{2a}}\). The new vertex of the parabola will be at \(\left( { -\dfrac{b}{{2a}},0} \right)\). The following figure shows an example shift:
Step-3: \(a{\left( {x + \dfrac{b}{{2a}}} \right)^2} \to a{\left( {x + \dfrac{b}{{2a}}} \right)^2} - \dfrac{D}{{4a}}\): This transformation is a vertical shift of magnitude \(\left| {\dfrac{D}{{4a}}} \right|\) units. The direction of the shift will be decided by the sign of \(\dfrac{D}{{4a}}\). The final vertex of the parabola will be at \(\left( { - \dfrac{b}{{2a}},- \dfrac{D}{{4a}}} \right)\). The following figure shows an example shift:
Sometimes working with a hard quadratic function is difficult. But don’t worry!
Here are some tips for you to decide to which side the graph makes a frown or a smile.
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Let the vertex form of quadratic function be \(f(x)=a(x-h)^{2}+k\).
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Quadratic Function Grapher
This is how the quadratic function is represented on a graph.
This curve is called a parabola.
Now let’s explore some quadratic equations on a graph using the quadratic function graph.
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.
- The standard form of the quadratic function is \(f(x)=ax^2+bx+c\) where \(a\neq0\).
- The vertex form of the quadratic function is \(f(x)=a(x-h)^{2}+k\), where \((h,k)\) is the vertex of the parabola.
- The curve of the quadratic function is in the form of a parabola.
- The \(x\)-intercepts of the graph shows the roots of quadratic functions.
Solved Examples
| Example 1 |
Jack shows a quadratic function to his friend.
Can you plot the graph of this function?
Solution
We have:
\[\begin{array}{l}a = - 3,\,\,\,b = - 2,\,\,\,c = 1\\ \Rightarrow \,\,\,D = {b^2} - 4ac =16\end{array}\]
The coordinates of the vertex are.
\[V \equiv \left( { - \frac{b}{{2a}}, -\frac{D}{{4a}}} \right) = \left( { - \frac{1}{3},\frac{4}{3}} \right)\]
Note that the parabola will open downward (since \(a\) is negative), but the vertex has a positive \(y\)-coordinate.This means that the parabola crosses the \(x\)-axis. In other words, the quadratic function as real zeroes. Let us calculate these zeroes:
\[x = \frac{{ - \left( { - 2} \right) \pm \sqrt {16}}}{{ - 6}} = - 1,\frac{1}{3}\]
The parabola will cross the \(x\)-axis at these \(x\) values.
Finally, using all this information, we plot the quadratic graph:
| \(\therefore\) The graph of the function is shown. |
| Example 2 |
Ms. Emma asked the students to plot the graph of the function shown on the blackboard.
Can you help the students to plot the graph of this equation?
Solution
We have:
\[\begin{array}{l}a = - 1,\,\,\,b = 5,\,\,\,c = - 4\\ \Rightarrow \,\,\,D = {b^2} - 4ac =9\end{array}\]
Since \(a\) is negative, the parabola will open downward.
The coordinates of the vertex are
\[V \equiv \left( { - \frac{b}{{2a}}, -\frac{D}{{4a}}} \right) = \left( {\frac{5}{2},\frac{9}{4}} \right)\]
We can calculate the roots using the quadratic formula or through factorization; we use the latter approach in this case:
\[\begin{array}{l}f\left( x \right) = 5x - 4 - {x^2} =0\\ \Rightarrow \,\,\,{x^2} - 5x + 4 =0\\ \Rightarrow \,\,\,\left( {x - 1}\right)\left( {x - 4} \right) = 0\\ \Rightarrow \,\,\,x = 1,4\end{array}\]
Finally, using all this information, we plot the quadratic graph:
| \(\therefore\) The graph of the function is shown. |
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
We hope you enjoyed learning about the graphing quadratic functions with the simulations and practice questions. Now you will be able to easily solve problems related to the graphing quadratic functions.
About Cuemath
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FAQs on Graphing Quadratic Functions
1. What is a quadratic function?
A function of the form \(f(x)=ax^2+bx+c\), where \(a \neq 0\) is called a quadratic function.
2. How do you graph a quadratic function?
Follow the steps shown below to graph a quadratic function:
Step-1: \({x^2} \to a{x^2}\): This will imply a vertical scaling of the original parabola. If \(a\) is negative, the parabola will also flip its mouth from the positive to the negative side. The magnitude of the scaling depends upon the magnitude of \(a\).
Step-2: \(a{x^2} \to a{\left( {x + \dfrac{b}{{2a}}}\right)^2}\): This is a horizontal shift of magnitude \(\left| {\dfrac{b}{{2a}}} \right|\) units. The direction of the shift will be decided by the sign of \(\dfrac{b}{{2a}}\). The new vertex of the parabola will be at \(\left( { -\dfrac{b}{{2a}},0} \right)\).
Step-3: \(a{\left( {x + \dfrac{b}{{2a}}} \right)^2} \to a{\left( {x + \dfrac{b}{{2a}}} \right)^2} - \dfrac{D}{{4a}}\): This transformation is a vertical shift of magnitude \(\left| {\dfrac{D}{{4a}}} \right|\) units. The direction of the shift will be decided by the sign of \(\dfrac{D}{{4a}}\). The final vertex of the parabola will be at \(\left( { - \dfrac{b}{{2a}},- \dfrac{D}{{4a}}} \right)\).
3. How do you tell if a graph is a quadratic function?
If the curve of the graph is in the form of a parabola, then the graph is a quadratic function.
4. How do you find the quadratic function?
If the function is of the form \(f(x)=ax^2+bx+c\), where \(a \neq 0\), then the function is a quadratic function.
5. How do you graph a quadratic function in standard form?
Follow the steps shown below to graph a quadratic function in the standard form:
Step-1: Convert the standard form of a quadratic function to the vertex form.
Step-2: \({x^2} \to a{x^2}\): This will imply a vertical scaling of the original parabola. If \(a\) is negative, the parabola will also flip its mouth from the positive to the negative side. The magnitude of the scaling depends upon the magnitude of \(a\).
Step-3: \(a{x^2} \to a{\left( {x + \dfrac{b}{{2a}}}\right)^2}\): This is a horizontal shift of magnitude \(\left| {\dfrac{b}{{2a}}} \right|\) units. The direction of the shift will be decided by the sign of \(\dfrac{b}{{2a}}\). The new vertex of the parabola will be at \(\left( { -\dfrac{b}{{2a}},0} \right)\).
Step-4: \(a{\left( {x + \dfrac{b}{{2a}}} \right)^2} \to a{\left( {x + \dfrac{b}{{2a}}} \right)^2} - \dfrac{D}{{4a}}\): This transformation is a vertical shift of magnitude \(\left| {\dfrac{D}{{4a}}} \right|\) units. The direction of the shift will be decided by the sign of \(\dfrac{D}{{4a}}\). The final vertex of the parabola will be at \(\left( { - \dfrac{b}{{2a}},- \dfrac{D}{{4a}}} \right)\).
6. What are quadratic graphs called?
The quadratic graphs are called parabolas.
7. How do you graph a quadratic function in vertex form?
Follow the steps shown below to graph a quadratic function in vertex form:
Step-1: \({x^2} \to a{x^2}\): This will imply a vertical scaling of the original parabola. If \(a\) is negative, the parabola will also flip its mouth from the positive to the negative side. The magnitude of the scaling depends upon the magnitude of \(a\).
Step-2: \(a{x^2} \to a{\left( {x + \dfrac{b}{{2a}}}\right)^2}\): This is a horizontal shift of magnitude \(\left| {\dfrac{b}{{2a}}} \right|\) units. The direction of the shift will be decided by the sign of \(\dfrac{b}{{2a}}\). The new vertex of the parabola will be at \(\left( { -\dfrac{b}{{2a}},0} \right)\).
Step-3: \(a{\left( {x + \dfrac{b}{{2a}}} \right)^2} \to a{\left( {x + \dfrac{b}{{2a}}} \right)^2} - \dfrac{D}{{4a}}\): This transformation is a vertical shift of magnitude \(\left| {\dfrac{D}{{4a}}} \right|\) units. The direction of the shift will be decided by the sign of \(\dfrac{D}{{4a}}\). The final vertex of the parabola will be at \(\left( { - \dfrac{b}{{2a}},- \dfrac{D}{{4a}}} \right)\).
8. What is a real-life example of quadratic functions?
When a ball is thrown in the air, its path is modeled by a quadratic function.