## 9 Best Free Quadratic Equation Grapher Software For Windows

Here is a list of best free Quadratic equation Grapher software for Windows. Through these software, you can plot graphs of Quadratic equations on a 2DXY-plane. Not just 2D, but some software are capable to plot 3D graphs as well. Most freeware can also plot graphs using Linear, Trigonometric, Cubic, etc. equations.

Use of these graphing software is pretty easy as you just need to enter a Quadratic equation to get the resultant graph. Not just one graph, but multiple graphs can also be generated by entering multiple equations in most software. In case of multiple graphs, these software allow you to assign different color schemes for different graphs to create a distinction between graphs. After completion, you can save the graph in a software specific project format or as a PDF.

Some equation graphers can also give you solutions of various mathematical problems like Trigonometry, Linear equations, Quadratic equations, etc. Plus, various common but essential tools like Zoom in/out, Pan, Graph lines, etc. are available in most software.

### My Favorite Quadratic Equation Grapher Software For Windows:

Microsoft Mathematics is my favorite software because it can plot both 2D and 3D graphs of a Quadratic equation. It is also capable of plotting multiple graphs of different equations like Trigonometry, Cubic, etc. Plus, its property to provide step by step solution of various mathematical problems is also impressive.

You can also check out lists of best free Math Equation Editor, System Of Linear Equation Calculator, and Fibonacci Calculator software for Windows.

### Microsoft Mathematics

Microsoft Mathematics is a free quadratic equation grapher software for Windows. It is basically used to draw and solve graphs of various mathematical problems like Linear equations, Calculus, Trigonometry, Statistics, etc. It is also capable of drawing both 2D and 3D graphs of any equation. Also, its capability to give step by step solutions of input equations is really impressive.

### How to graph a quadratic equation in Microsoft Mathematics:

On the main interface of this quadratic grapher, you can view two main sections namely, Worksheet and Graphing. In its Graphing section, you can enter one or more quadratic equations to get the output graphs of all equations. Usually, the graph of quadratic equations have the parabolic shapes that you can also notice on this software. To analyze plotted points over the graph, use the Worksheet section which contains the exact values of both X and Y coordinates. Now, to view the 3D graph, again go to Graphing section and change the dimension from 2D to 3D. Always remember to press the Graph button to view the output graph. In the 3D graph, you can change the orientation of the graph through your mouse. Plus, in both 2D and 3D graphs, you can use the mouse wheel to zoom in and zoom out.

After completion, you can save the step by steps results along with graphs as Microsoft Mathematics Worksheet (.gcw) format. It lets you print the graph as well.

### GraphCalc

GraphCalc is a free and feature-rich quadratic equation grapher software for Windows. Using it, you can draw both 2D and 3D graphs of Quadratic, Linear, Cubic, Trigonometric, etc. equations. Not just graphs, but the complete solution of equations can also be obtained through it.

This software comes with a dedicated graph section where both 2D and 3D graphs are available. To draw a 2D graph, select the Graph 1 tab from the main interface. In this tab, 10 y(t) functions from y(1) to y(10) are present on the right side of the interface. Fill one or more y(t)functions with the quadratic or other supported equations to immediately get the graphs of respective equations on the XY-plane. Each y(t) function is coded with a different color, so you can quickly identify the graph of a specific equation. At any time, you can change the color of a function from the Graph options. The output graphs do not have any markings, still, you can use point cursor at any point of the graph to view its coordinates.

Similar to 2D graphs, you can easily draw 3D graphs as well by using the 3D graph tab. In this tab, you can enter the quadratic and other equations by accessing the 3D Graph menu > Options. At maximum, you can get the graphs of 6 different equations. Some other settings like the selection of colors, range, and axes can also be changed. After the setup, press OK button to view the 3D graphs. You can easily rotate and pan the graph through mouse to view it from all angles.

The resultant 2D and 3D graphs of the Quadratic equation can be saved as PDF file format.

### Graph

Graph is a free and open source Quadratic equation grapher software for Windows. It can also be used to draw graphs of various other mathematical functions or equations like Linear equations, Trigonometric functions, Cubic equations, and more.

In this software, you can draw multiple graphs on the same XY-plane by entering more than one Quadratic equations. Not just one type of equation, but two or more absolutely different equations like Linear, Trigonometric, and Quadratic equations on the same XY plane can also be drawn. And, you don’t need to select different equation types before entering the equations because it can self-identify the differences between equations by analyzing their structure. In case of multiple equations, you can assign a color scheme for each equation to quickly identify a particular graph. Besides color coding, there are some other handy features in it like Shading (to shade different parts of the graph), Series (to add colored dot on the graph), Label (to add text), Relation (to add relations between equations), etc.

Now, to plot graphs for Quadratic equations, go to Functions menu and select Insert function. In Insert Function, there is one Function equation field where you need to enter one equation at a time. Color coding option is also available in this menu, so select a color for the entered equation and click on OK button to view the graph on XY plane. Repeat the same process again to plot multiple graphs by entering more equations. After completion, you can save the final graph in various formats like SVG, BMP, JPG, PNG, etc.

### ZGrapher

ZGrapher is a feature-rich graph generating utility software for Windows. Through it, you can generate graphs for various mathematical problems like Quadratic equations, Linear equations, Derivation, Integration, Regression, Tangents, and more. Like other similar software, you need to enter the Quadratic equation to this software to get the output graph. The graph plotted by this software is very well marked. Still, you get some additional tools to add labels and points to the graph as well.

This freeware also lets you plot multiple graphs on its XY-plane by entering multiple Quadratic equations. Plus, you can use different colors to differentiate each graph from each other from its Graph properties.

### How to plot a graph using ZGrapher:

• Now, add Quadratic Equation to it and press OK to get the graph on the XY-plane.
• You can repeat the second point to plot more graphs on the XY-plane.

After completion, you can save the graph as an Image (BMP, GIF, and PCX) or as ZGrapher (.zgd) file.

### Equation Analyzer

Equation Analyzer is a free quadratic equation grapher app for Windows 10. It is a very simple app that can plot a graph for any linear or quadratic equation.

When you launch this app, you can view only two main options namely, Linear Equations and Quadratic Equations. To plot a graph of a Quadratic equation, select the Quadratic Equation option and enter the equation. After that, press Calculate to view the resultant graph. Besides graph, it also clearly defines the meaning of graph by showing various graph parameters.

Graph Parameters provided by this software are:

• Even or Odd: To show whether the result is even, odd, not even, or not odd.
• Bounded: To let you know whether the graph is bounded or not.
• Continuous: To show function is continuous or not.
• Parabola: It shows whether parabola opens upwards or downwards.
• Minimum or maximum point: It shows minimum or maximum values of X and Y coordinates.

Similar to the Quadratic equation, you can plot the graph of linear equation by going to Linear equations option.

### GeoGebra graphing Calculator

GeoGebra graphing Calculator is another free quadratic equation grapher app for Windows 10. This app can plot graphs using Quadratic, Linear, Trigonometric, and various other mathematical equations. Plus, it also allows you to enter multiple equations to plot their graphs on the same XY-Plane. To create a difference between graphs, this app automatically selects a different color for each graph.

The process of generating the graph is simple as you just need to enter equations in its Algebra section to get the respective graph. The unique property of this app is that it shows the graph in real time. That means as soon as you enter a single digit of the equation, it starts plotting the graph from that instance.

This app also comes with a lot of tools that you can use to further add various elements on the graph. Some of the useful tools of this software are:

• Basic Tools: These tools include move, point, slider, intersect, etc.
• Edit: It includes tools, like select objects, move graphics view, delete, show/hide label, and show/hide an object.
• Media: Using these tools, you can add an image and any text to the graph.
• Measure: This tool lets you measure an angle between two line segments, distance between two points of a line segment, and the area of a shape such as a circle.

After completion, you can save the graph as a project and also share it with others through Email and Onedrive.

### WeissCalc

WeissCalc is a free and lightweight Quadratic equation grapher software for Windows. Using it, you can draw 2D graphs of both linear and quadratic equations. It is also capable of plotting multiple graphs for different linear and quadratics equation at a time. At maximum, you can plot graphs for 3 linear and 3 quadratic equations. It has a dedicated Graphing section into which you need to enter quadratic equations to get the respective graph. The graphs produced by this software are pretty accurate in terms of shapes, but they lack important markings (X and Y axis plot values). Plus, it is really hard to identify the graphs for different equations as all graphs use same color scheme.

Apart from graphs, you can also use this freeware as a basic calculator, quadratic equation solver, prime number checker, and Trigonometry function solver.

Quadratic Solver is another lightweight quadratic equation grapher software for Windows. It is a straightforward software that plots a graph according to the input quadric equation. Not just graph, it can also be used to get the solution of the equation as well as to find the value of Y by entering the value of X in the same equation.

To plot a graph, first, you need to enter a Quadratic equation to this software. After that, press the Plot button to view the graph. In the output graph, you can notice that it does not have any markings on both axes which is the main drawback of this software. Overall, it is a simple and easy software to get the quadratic equation graph.

### Console Grapher

Console Grapher is a free and portable console-based Quadratic equation grapher software for Windows. It is a very lightweight software that you can carry around in a portable storage device and use on the go.

This software is capable of plotting graphs of various equation types like Quadratic, Linear, Cubic, Trigonometric equations, etc. When you launch this software, a console window will open up where you can enter one equation at a time. After writing the equation press Enter to view the resultant graph. In the output graph, you can notice that it contains very few points (#) to represent the shape of the graph, usually parabola in case of Quadratic Equations. Plus, the lack of markings is another letdown of this software. Through its output graph, you can get a hint of the overall shape of the graph but nothing more.

Overall, it a quite simple but also a very basic Quadratic grapher software with almost no additional features.

This is a program that can plot graphs for quadratic functions. The formula of a quadratic equation is f(x)= ax2+bx+c , where a, b and c are constant.

This program  employs a picture box as the plot area and three text boxes to obtain the values of the coefficients a, b, c of the quadratic equation from the users. We also need to modify the scale factor in the properties windows of the picture box. We are using a scale of 0.5 cm to represent 1 unit .
Besides, we need to make some transformation as the coordinates in VB start from top left but we want it to start from the middle. We can use the Pset method to draw the graph using a very small increment. Pset is a method that draws a dot on the screen, the syntax is

Pset(x,y), color

Where (x,y) is the coordinates of the dot and color is the color of the dot. Using the For Next loop together with Pset, we can draw a line on the screen.

### The Code

Private Sub cmd_draw_Click()
Dim a, b, c As Integer
Dim w, v As Single
a = Val(txt_a.Text)
b = Val(txt_b.Text)
c = Val(txt_c.Text)

'Using a scale of 0.5 cm to represent i unit to draw the graph
' Need to make some transformation as the coordinates in VB start from top left
For w = 0 To 10 Step 0.001

v = a * (5 - w) ^ 2 - b * (5 - w) + c
pic_graph.PSet (w, 5 - v)

Next w

End Sub

Private Sub Command1_Click()
pic_graph.Cls

txt_a.Text = ""
txt_b.Text = ""
txt_c.Text = ""

End Sub

Sours: https://www.vbtutor.net/VB_Sample/QGraphplotter.htm

Quadratic function has the form $f(x) = ax^2 + bx + c$ where a, b and c are numbers

You can sketch quadratic function in 4 steps. I will explain these steps in following examples.

Example 1:

Sketch the graph of the quadratic function

$${\color{blue}{ f(x) = x^2+2x-3 }}$$

Solution:

In this case we have $a=1, b=2$ and $c=-3$

STEP 1: Find the vertex.

To find x - coordinate of the vertex we use formula:

$$x=-\frac{b}{2a}$$

So, we substitute $1$ in for $a$ and $2$ in for $b$ to get

$$x=-\frac{b}{2a} = -\frac{2}{2\cdot1} = -1$$

To find y - coordinate plug in $x=-1$ into the original equation:

$$y = f(-1) = (-1)^2 + 2\cdot(-1) - 3 = 1 - 2 - 3 = -4$$

So, the vertex of the parabola is ${\color{red}{ (-1,-4) }}$

STEP 2: Find the y-intercept.

To find y - intercept plug in $x=0$ into the original equation:

$$f(0) = (0)^2 + 2\cdot(0) - 3 = 0 - 0 - 3 = -3$$

So, the y-intercept of the parabola is ${\color{blue}{ y = -3 }}$

STEP 3: Find the x-intercept.

To find x - intercept solve quadratic equation $f(x)=0$ in our case we have:

$$x^2+2x-3 = 0$$

Solutions for this equation are:

$${\color{blue}{ x_1 = -3 }} ~~~\text{and}~~~ {\color{blue}{ x_2 = 1 }}$$

( to learn how to solve quadratic equation use quadratic equation solver )

STEP 4: plot the parabola.

Example 2:

Sketch the graph of the quadratic function

$${\color{blue}{ f(x) = -x^2+2x-2 }}$$

Solution:

Here we have $a=-1, b=2$ and $c=-2$

The x-coordinate of the vertex is:

$${\color{blue}{ x = -\frac{b}{2a} }} = -\frac{2}{2\cdot(-1)}= 1$$

The y-coordinate of the vertex is:

$$y = f(1) = -1^2+2\cdot1-2 = -1 + 2 - 2 = -1$$

The y-intercept is:

$$y = f(0) = -0^2+2\cdot0-2 = -0 + 0 - 2 = -2$$

In this case x-intercept doesn't exist since equation $-x^2+2x-2=0$ does not has the solutions (use quadratic equation solver to check ). So, in this case we will plot the graph using only two points

How to Graph Parabolas

### Parts of a Parabola

The graph of a quadratic function is a parabola, and its parts provide valuable information about the function.

### Learning Objectives

Describe the parts and features of parabolas

### Key Takeaways

#### Key Points

• The graph of a quadratic function is a U-shaped curve called a parabola.
• The sign on the coefficient $a$ of the quadratic function affects whether the graph opens up or down. If $a<0$, the graph makes a frown (opens down) and if $a>0$ then the graph makes a smile (opens up).
• The extreme point ( maximum or minimum ) of a parabola is called the vertex, and the axis of symmetry is a vertical line that passes through the vertex.
• The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function.

#### Key Terms

• vertex: The point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function.
• axis of symmetry: A vertical line drawn through the vertex of a parabola around which the parabola is symmetric.
• zeros: In a given function, the values of $x$ at which $y=0$, also called roots.

Recall that a quadratic function has the form

$\displaystyle f(x)=ax^{2}+bx+c$.

where $a$, $b$, and $c$ are constants, and $a\neq 0$.

The graph of a quadratic function is a U-shaped curve called a parabola.  This shape is shown below.

Parabola : The graph of a quadratic function is a parabola.

In graphs of quadratic functions, the sign on the coefficient $a$ affects whether the graph opens up or down. If $a<0$, the graph makes a frown (opens down) and if $a>0$ then the graph makes a smile (opens up). This is shown below.

Direction of Parabolas: The sign on the coefficient $a$ determines the direction of the parabola.

### Features of Parabolas

Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane.

### Vertex

One important feature of the parabola is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph.

### Axis of Symmetry

Parabolas also have an axis of symmetry, which is parallel to the y-axis. The axis of symmetry is a vertical line drawn through the vertex.

### $y$-intercept

The y-intercept is the point at which the parabola crosses the y-axis. There cannot be more than one such point, for the graph of a quadratic function. If there were, the curve would not be a function, as there would be two $y$ values for one $x$ value, at zero.

### $x$-intercepts

The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of $x$ at which $y=0$. There may be zero, one, or two $x$-intercepts. The number of $x$-intercepts varies depending upon the location of the graph (see the diagram below).

Possible $x$-intercepts: A parabola can have no x-intercepts, one x-intercept, or two x-intercepts

Recall that if the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the function. These are the same roots that are observable as the $x$-intercepts of the parabola.

Notice that, for parabolas with two $x$-intercepts, the vertex always falls between the roots. Due to the fact that parabolas are symmetric, the $x$-coordinate of the vertex is exactly in the middle of the $x$-coordinates of the two roots.

### A Graphical Interpretation of Quadratic Solutions

The roots of a quadratic function can be found algebraically or graphically.

### Learning Objectives

Describe the solutions to a quadratic equation as the points where the parabola crosses the x-axis

### Key Takeaways

#### Key Points

• The roots of a quadratic function can be found algebraically with the quadratic formula, and graphically by making observations about its parabola.
• The solutions, or roots, of a given quadratic equation are the same as the zeros, or $x$-intercepts, of the graph of the corresponding quadratic function.

#### Key Terms

• zeros: In a given function, the values of $x$ at which $y = 0$, also called roots.

Recall how the roots of quadratic functions can be found algebraically, using the quadratic formula $(x=\frac{-b \pm \sqrt {b^2-4ac}}{2a})$. The roots of a quadratic function can also be found graphically by making observations about its graph. These are two different methods that can be used to reach the same values, and we will now see how they are related.

Consider the quadratic function that is graphed below. Let’s solve for its roots both graphically and algebraically.

Graph of the quadratic function $f(x) = x^2 – x – 2$: Graph showing the parabola on the Cartesian plane, including the points where it crosses the x-axis.

Notice that the parabola intersects the $x$-axis at two points: $(-1, 0)$ and $(2, 0)$. Recall that the $x$-intercepts of a parabola indicate the roots, or zeros, of the quadratic function. Therefore, there are roots at $x = -1$ and $x = 2$.

Now, let’s solve for the roots of $f(x) = x^2 - x- 2$ algebraically with the quadratic formula.

Recall that the quadratic equation sets the quadratic expression equal to zero instead of $f(x)$:

$0 = x^2 - x - 2$

Now the quadratic formula can be applied to find the $x$-values for which this statement is true. For the given equation, we have the following coefficients: $a = 1$, $b = -1$, and $c = -2$.

Substitute these values in the quadratic formula:

$x = \dfrac{-(-1) \pm \sqrt {(-1)^2-4(1)(-2)}}{2(1)}$

Simplifying, we have:

$x = \dfrac{1 \pm \sqrt {9}}{2} \\$

and

$x = \dfrac{1 \pm 3}{2}$

We now have two possible values for x: $\frac{1+3}{2}$ and $\frac{1-3}{2}$.

These reduce to $x = 2$ and $x = - 1$, respectively. Notice that these are the same values that when found when we solved for roots graphically.

### Example

Find the roots of the quadratic function $f(x) = x^2 - 4x + 4$. Solve graphically and algebraically.

The graph of $f(x) = x^2 – 4x + 4$.: The graph of the above function, with the vertex labeled at $(2, 1)$.

Looking at the graph of the function, we notice that it does not intersect the $x$-axis. Therefore, it has no real roots.

We can verify this algebraically. First, identify the values for the coefficients: $a = 1$, $b = - 4$, and $c = 5$.

Substituting these into the quadratic formula, we have:

$x=\dfrac{-(-4) \pm \sqrt {(-4)^2-4(1)(5)}}{2(1)}$

Simplifying, we have:

$x=\dfrac{4 \pm \sqrt {16-20}}{2} \\ x=\dfrac{4 \pm \sqrt {-4}}{2}$

Notice that we have $\sqrt{-4}$ in the formula, which is not a real number. Therefore, there are no real roots for the given quadratic function. We have arrived at the same conclusion that we reached graphically.

### Graphing Quadratic Equations in Vertex Form

The vertex form of a quadratic function lets its vertex be found easily.

### Learning Objectives

Explain the meanings of the constants $a$, $h$, and $k$ for a quadratic equation in vertex form

### Key Takeaways

#### Key Points

• An important form of a quadratic function is vertex form: $f(x) = a(x-h)^2 + k$
• When written in vertex form, it is easy to see the vertex of the parabola at $(h, k)$.
• It is easy to convert from vertex form to standard form.
• It is more difficult, but still possible, to convert from standard form to vertex form. The process involves a technique called completing the square.

#### Key Terms

• constant: An identifier that is bound to an invariant value.
• vertex: A point on the curve with a local minimum or maximum of curvature.
• quadratic: A polynomial of degree two.

Quadratic equations may take various forms. You have already seen the standard form:

$f(x)=a{ x }^{ 2 }+bx+c$

Another common form is called vertex form, because when a quadratic is written in this form, it is very easy to tell where its vertex is located. The vertex form is given by:

$f(x)=a(x-h)^2+k$

The vertex is $(h,k).$ Note that if the form were $f(x)=a(x+h)^2+k$, the vertex would be $(-h,k).$ The coefficient $a$ as before controls whether the parabola opens upward or downward, as well as the speed of increase or decrease of the parabola.

### Converting From Vertex Form to Standard Form

If you want to convert a quadratic in vertex form to one in standard form, simply multiply out the square and combine like terms. For example, the quadratic

$y=(x-2)^2+1$

Can be rewritten as follows:

\begin{align} y&=(x-2)(x-2)+1 \\ &=x^2-2x-2x+4+1 \\ &=x^2-4x+5 \end{align}

### Converting From Standard Form to Vertex Form

It is more difficult to convert from standard form to vertex form. The process is called “completing the square.”

### Conversion When $a=1$

Consider the following example: suppose you want to write $y=x^2+4x+6$ in vertex form. Note that the coefficient on $x^2$ (the one we call $a$) is $1$. When this is the case, we look at the coefficient on $x$ (the one we call $b$) and take half of it. Then we square that number. Thus for this example, we divide $4$ by $2$ to obtain $2$ and then square it to obtain $4$.

We then both add and subtract this number as follows:

$y=(x^2+4x+4)+6-4$

Note that we both added and subtracted 4, so we didn’t actually change our function. Now the expression in the parentheses is a square; we can write $y=(x+2)^2+2.$ Our equation is now in vertex form and we can see that the vertex is $(-2,2).$

### Conversion When $a \neq 1$

It is slightly more complicated to convert standard form to vertex form when the coefficient $a$ is not equal to $1$. We can still use the technique, but must be careful to first factor out the $a$ as in the following example:

Consider $y=2x^2+12x+5.$ We factor out the coefficient $2$ from the first two terms, writing this as:

$y=2(x^2+6x) + 5$

We then complete the square within the parentheses. Note that half of $6$ is $3$ and $3^2=9$. So we add and subtract $9$ within the parentheses, obtaining:

$y=2(x^2+6x+9-9)+5$

We can then finish the calculation as follows:

\begin{align} y&=2((x+3)^2-9)+5 \\ &=2(x+3)^2-18+5 \\ &=(x+3)^2-13 \end{align}

So the vertex of this parabola is $(-3,-13).$

### Graphing Quadratic Equations In Standard Form

A quadratic function is a polynomial function of the form $y=ax^2+bx+c$.

### Learning Objectives

Explain the meanings of the constants $a$, $b$, and $c$ for a quadratic equation in standard form

### Key Takeaways

#### Key Points

• The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the $y$-axis.
• The coefficients $a, b,$ and $c$ in the equation $y=ax^2+bx+c$ control various facets of what the parabola looks like when graphed.

#### Key Terms

• vertex: The maximum or minimum of a quadratic function.
• parabola: The shape formed by the graph of a quadratic function.
• quadratic: A polynomial of degree two.

A quadratic function in the form

$f(x)=a{ x }^{ 2 }+bx+x$

is in standard form.

Regardless of the format, the graph of a quadratic function is a parabola.

The graph of $y=x^2-4x+3$ : The graph of any quadratic equation is always a parabola.

### Coefficients and Graphs of Quadratic Function

Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function’s graph.

### Coefficient of $x^2$, $a$

The coefficient $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex. A larger, positive $a$ makes the function increase faster and the graph appear thinner.

The coefficient $a$ controls the speed of increase of the parabola.: The black curve is $y=4x^2$ while the blue curve is $y=3x^2.$ The black curve appears thinner because its coefficient $a$ is bigger than that of the blue curve.

Whether the parabola opens upward or downward is also controlled by $a$. If the coefficient $a>0$, the parabola opens upward, and if the coefficient $a<0$, the parabola opens downward.

Quadratics either open upward or downward: The blue parabola is the graph of $y=3x^2.$ It opens upward since $a=3>0.$ The black parabola is the graph of $y=-3x^2.$ It opens downward since $a=-3<0.$.

### The Axis of Symmetry

The coefficients $b$ and $a$ together control the axis of symmetry of the parabola and the $x$-coordinate of the vertex. The axis of symmetry for a parabola is given by:

$x=-\dfrac{b}{2a}$

For example, consider the parabola $y=2x^2-4x+4$ shown below. Because $a=2$ and $b=-4,$ the axis of symmetry is:

$x=-\frac{-4}{2\cdot 2} = 1$

The vertex also has $x$ coordinate $1$.

The graph of $y=2x^2-4x+4.$: The axis of symmetry is a vertical line parallel to the y-axis at  $x=1$.

### The $y$-intercept of the Parabola

The coefficient $c$ controls the height of the parabola. More specifically, it is the point where the parabola intercepts the y-axis. The point $(0,c)$ is the $y$ intercept of the parabola. Note that the parabola above has $c=4$ and it intercepts the $y$-axis at the point $(0,4).$

A Quadratic Equation in Standard Form
(a, b, and c can have any value, except that a can't be 0.)

Here is an example:

### Graphing

You can graph a Quadratic Equation using the Function Grapher, but to really understand what is going on, you can make the graph yourself. Read On!

f(x) = x2

And its graph is simple too:

This is the curve f(x) = x2
It is a parabola.

Now let us see what happens when we introduce the "a" value:

f(x) = ax2

• Larger values of a squash the curve inwards
• Smaller values of a expand it outwards
• And negative values of a flip it upside down

### Play With It

Now is a good time to play with the
"Quadratic Equation Explorer" so you can
see what different values of a, b and c do.

Before graphing we rearrange the equation, from this:

f(x) = ax2 + bx + c

To this:

f(x) = a(x-h)2 + k

Where:

In other words, calculate h (= −b/2a), then find k by calculating the whole equation for x=h

### But Why?

The wonderful thing about this new form is that h and k show us the very lowest (or very highest) point, called the vertex:

And also the curve is symmetrical (mirror image) about the axis that passes through x=h, making it easy to graph

### So ...

• h shows us how far left (or right) the curve has been shifted from x=0
• k shows us how far up (or down) the curve has been shifted from y=0

Lets see an example of how to do this:

### Example: Plot f(x) = 2x2 − 12x + 16

First, let's note down:

• a = 2,
• b = −12, and
• c = 16

Now, what do we know?

• a is positive, so it is an "upwards" graph ("U" shaped)
• a is 2, so it is a little "squashed" compared to the x2 graph

Next, let's calculate h:

h = −b/2a = −(−12)/(2x2) = 3

And next we can calculate k (using h=3):

k = f(3) = 2(3)2 − 12·3 + 16 = 18−36+16 = −2

So now we can plot the graph (with real understanding!):

We also know: the vertex is (3,−2), and the axis is x=3

### From A Graph to The Equation

What if we have a graph, and want to find an equation?

### Example: you have just plotted some interesting data, and it looks Quadratic:

Just knowing those two points we can come up with an equation.

Firstly, we know h and k (at the vertex):

(h, k) = (1, 1)

So let's put that into this form of the equation:

f(x) = a(x-h)2 + k

f(x) = a(x−1)2 + 1

Then we calculate "a":

We know the point (0, 1.5) so:f(0) = 1.5

And a(x−1)2 + 1 at x=0 is:f(0) = a(0−1)2 + 1

They are both f(0) so make them equal: a(0−1)2 + 1 = 1.5

Simplify:a + 1 = 1.5

a = 0.5

And so here is the resulting Quadratic Equation:

f(x) = 0.5(x−1)2 + 1

Note: This may not be the correct equation for the data, but it’s a good model and the best we can come up with.

How to Graph Parabolas

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