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# Function Calculator + Online Solver With Free Steps

The **Function Calculator **calculates the properties of a graph drawn from the equation entered in the calculator. Furthermore, the graph is drawn on a cartesian plane to represent that function **f(x)** visually**.**

This calculator takes in any kind of legitimate equation and will detect the** type** of the function (linear line, conic shape, etc.) and then provide a table showing its **properties** based on it.

There could be two graphs, with one showing a **zoomed-in view** and the other expressing the **overall trend** of the function given on the graph.

Moreover, this calculator supports **three-dimensional equations**, except that it does not give a detailed properties table for these figures and does not draw a graph to express them visually. For **higher-order polynomials**, the results show only the visual representation of the functions, and their properties are **unavailable**.

## What Is the Function Calculator?

**The Function Calculator is an online tool that determines the type of function and its properties and displays it based on the inputted polynomial in the calculator text box. Also, it draws a graph to visually represent the trend of the line on the cartesian plane. The properties differ from function to function based on the type of shape it expresses. **

The calculator consists of a single-line text box labeled “**Enter the Function, y = f(x) =,**” where you can enter any legitimate function to find its properties and type. Additionally, it is recommended to use the calculator for two-dimensional functions as it correctly represents its properties in those functions.

## How To Use the Function Calculator?

You can correctly use the **Function Calculator **by entering the desired function into the text box and submitting it. A new window will open, showing all the properties and graphs for the respective function **f(x)**.

The stepwise guidelines for the calculator’s usage are below:

### Step 1

Enter the desired equation into the** text box**.

### Step 2

Ensure that the function is written **correctly**, following the **latex format** for the calculator to calculate.

### Step 3

Press the “**Submit**” button to get the results.

### Results

A pop-up window appears showing the detailed results in the sections explained below:

**Input:**This section shows the input function written in the**mathematical equation**. You can**verify**the equation by matching your entered function with the input interpreted by the calculator and edit accordingly.**Geometric Figure:**This section shows the type of function and its geometric name. Based on its geometric shape, it also shows its properties such as y-intercept, gradient, directrix, focus, vertex, curvature, etc. You can enable the properties by clicking on the “**Show Properties**” button on the top right-hand of the section.**Plots:**Here, the graph is drawn for the functions on the**cartesian plane**. The linear line plots will have only one plot with a specific range of x-values. Furthermore, the conics are**plotted twice**, with one showing the**zoomed-in**section and the other showing the**overall trend**of the graph at a wider range of x-values.

## Solved Examples

### Example 1

Suppose a **quadratic equation**:

\[ y = x^2 + 5x + 10 \]

Find the** type of conic** which the functions represent, and calculate its **respective properties**.

### Solution

As we know, this type of quadratic equation is usually considered a **parabola**, which can be converted into the **standard parabola form**:

\[ y = a(x\,–\, h)^2 + k \]

Firstly, we convert the quadratic function into the** standard vertex form** of a parabola equation. By completing the square:

\[ y = x^2 + 2(1)\left(\frac{5}{2}x\right) + \frac{25}{4} + 10 – \frac{25}{4}\]

\[ y = \left( x + \frac{5}{2} \right)^2 + \frac{15}{4} \]

After converting to the standard form, we can find the properties of the parabola by simply comparing it to the **standard form equation**:

\[ \Rightarrow a > 0 = 1, h= -\frac{5}{2}, k = \frac{15}{4} \]

\[ \text{vertex} = (h,\, k) = \left(-\frac{5}{2},\, \frac{15}{4}\right) \]

The **Axis of Symmetry** is parallel to the y-axis, and the parabola opens upwards as a > 0. Thus the semi-axis/focal length is found by:

\[ f = \frac{1}{4a} = \frac{1}{4} \]

\[ \text{Focus :} \,\, \left(\frac{5}{2},\, \frac{15}{4} + f\right) = \left(\mathbf{\frac{5}{2},\, 4}\right) \]

The **directrix** is **perpendicular** to the Axis of Symmetry and hence a horizontal line:

\[ \text{Directrix :} \,\, y = -\frac{15}{4}-f = \mathbf{\frac{7}{2}} \]

The length of the** semi-latus rectum** equals the focal parameter:

\[ \text{Focal Parameter :} \,\, p = 2f = \mathbf{\frac{1}{2}} \]

### Example 2

Consider a **linear line** with the equation:

\[ y = 4x + 7 \]\

Find the **properties** of the line expressed by this function.

### Solution

A linear line equation is governed by the **general linear equation**:

\[ y = mx + c \]

Where **y** is the **y-coordinate**, **x** is the **x-coordinate**, **m** is the **gradient/slope** of the line, and **c **is the **y-intercept**, which is the y-coordinate at which the line **intersects** the** y-axis.**

Comparing the equation given with the general line equation, we can write the **properties** of the line as:

\[\text{y-intercept = c = } 7 \]

For the **x-intercept**, we find the value of x-coordinate when $y = 0$. Hence:

\[ 0 = 4x + 7 \]

\[ -4x = 7 \]

\[ \mathbf{x = -\frac{7}{4}} \]

The **gradient** for this is expressed as **m**, which is:

\[ \text{Gradient = m = } 4 \]

To find the **Normal Vector** of this line, we need to write the above equation in the form of $ax + by = 0$

\[ y = 4x + 7 \]

\[ -4x + x = 7 \]

Neglecting the constant value, we can take the** coefficients**, a and b, of the x and y variables as the normal coordinates (-4, 1). Now we will divide both the coordinates with the** magnitude** of the point that is described as:

\[ \text{Magnitude = } \sqrt{x^2 + y^2}\]

\[ \text{Magnitude = } \sqrt{(-4)^2 + 1^2}\]

\[ \text{Magnitude = } \sqrt{17} \]

Thus, the normal vector coordinates are $\left(-\frac{4}{\sqrt{17}}, \frac{1}{\sqrt{17}}\right)$. They are approximated in the** decimal form** as (-0.970, 0.242).

The above line is a **linear equation**. Hence, its **curvature** is equal to zero.