Graphing Calculator
Graph up to three functions of x online, with zoom by range, roots, y-intercepts, intersection points and a table of values. Free, with no app or sign-up.
Graph
Key points
| Item | Value |
|---|
Table of values
| x | f(x) | g(x) | h(x) |
|---|
Quick Answer
Type a function such as x^2 − 4 and press Graph. The calculator draws it and lists the key points. For example, x^2 − 4 crosses the x-axis at x = −2 and x = 2 and the y-axis at −4. With g(x) = 2x + 1 added, the two graphs meet at x = 1 − √6 ≈ -1.449490 and x = 1 + √6 ≈ 3.449490.
Embed this calculator
Adjust the size if needed, then copy the code below into your page or blog. The calculator will load in an iframe and stay updated automatically.
Table of Contents
- Graphing Calculator
- Quick Answer
- Related Calculators
- How This Graphing Calculator Works
- How to Use the Graphing Calculator
- How to Type Functions
- Key Terms
- Worked Example
- Real-Life Examples
- Online Graphing Calculator vs Handheld Graphing Calculator
- Common Graphs and Their Shapes
- Reading a Graph
- Zooming and Choosing a Window
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Graphing Calculator Works
Type up to three functions of x, choose the x range and, if you like, the y range. The calculator works out each function at hundreds of points across the screen and joins them into a curve. Leave the y range blank and it picks one that fits the graphs. It also lists where each graph crosses the axes and where the graphs meet, plus a table of values.
Quick explanation: a graph shows every pair (x, y) where y = f(x).
- Roots: where a graph crosses the x-axis (y = 0).
- Y-intercept: where it crosses the y-axis (x = 0).
- Intersections: where two graphs have the same y at the same x.
How to Use the Graphing Calculator
- Type a function of x in the f(x) box, such as x^2 - 4.
- Add up to two more functions to compare, if you want.
- Set the x range, and the y range or leave it blank for auto.
- Press Calculate to draw the graph.
- Read the roots, intercepts, intersections and table of values.
How to Type Functions
- Powers: x^2, x^3, 2^x. Use brackets for a power that is a sum, such as e^(2x + 1).
- Multiplying: 3x, 2(x + 1) and x*sin(x) all work.
- Fractions: 1/x or (x + 1)/(x − 2). Use brackets around the top and bottom.
- Roots and logs: sqrt(x), ln(x), log(x) for base 10, abs(x).
- Trig: sin(x), cos(x), tan(x) in radians. Use pi for π.
You can also type y = or f(x) = at the start; the calculator ignores it.
Key Terms
- Function: a rule that gives one y value for each x value, written y = f(x).
- Root (zero): an x value where f(x) = 0, so the graph crosses or touches the x-axis.
- Y-intercept: the value f(0), where the graph crosses the y-axis.
- Intersection: a point where two graphs have the same x and y.
- Window: the x and y ranges shown on the graph.
Worked Example
Graph f(x) = x^2 − 4 and g(x) = 2x + 1 from x = −10 to 10.
- Roots of f: x^2 − 4 = 0, so x = −2 and x = 2. The y-intercept is f(0) = −4.
- Root of g: 2x + 1 = 0, so x = −0.5. The y-intercept is 1.
- Intersections: x^2 − 4 = 2x + 1 gives x^2 − 2x − 5 = 0, so x = 1 ± √6.
- That is x ≈ -1.449490 (y ≈ -1.898979) and x ≈ 3.449490 (y ≈ 7.898979).
Real-Life Examples
- Homework checks: x^2 − 4 crosses the x-axis at x = −2 and x = 2, matching the factored form (x − 2)(x + 2).
- Solving equations: graph both sides of x^2 − 4 = 2x + 1; they meet at x ≈ -1.4495 and x ≈ 3.4495.
- Trig: sin(x) from −10 to 10 crosses zero 7 times, at multiples of π such as 3.1416.
- Comparing growth: plot x^2 and 2^x to see that 2^x overtakes x^2 for good after x = 4.
Online Graphing Calculator vs Handheld Graphing Calculator
- Online: free, works on any phone or computer, and is easy to read on a big screen.
- Handheld: required or allowed in some exams, works without the internet, and has built-in statistics and programs.
- For learning and checking work, an online graph is quicker. For an exam, check which calculators your exam board allows.
Common Graphs and Their Shapes
| Function | Shape | Roots | Y-intercept |
|---|---|---|---|
| y = 2x + 1 | Straight line, slope 2 | x = −0.5 | 1 |
| y = x^2 − 4 | U-shaped parabola | x = −2, 2 | −4 |
| y = x^3 − x | S-shaped cubic | x = −1, 0, 1 | 0 |
| y = 2^x | Exponential growth | None | 1 |
| y = ln(x) | Slow growth, x > 0 only | x = 1 | None |
| y = sin(x) | Wave, period 2π | x = 0, ±π, ±2π, … | 0 |
| y = 1/x | Two branches (hyperbola) | None | None |
Reading a Graph
- Lines (y = mx + b) are straight. m is the slope and b is where the line crosses the y-axis.
- Parabolas (y = ax^2 + bx + c) open up when a is positive and down when a is negative.
- Exponentials such as 2^x grow faster and faster; e^(−x) shrinks towards zero.
- Sine and cosine repeat every 2π, about 6.283, and stay between −1 and 1.
- Gaps and jumps appear where a function is undefined, such as 1/x at x = 0 or tan(x) at π/2.
Zooming and Choosing a Window
The x range and y range set the viewing window. To zoom in on a crossing point, narrow the x range around it, for example from 3 to 4, and leave the y range blank. To compare graphs that grow at very different speeds, set the y range yourself so one curve does not squash the others.
If a graph looks like a flat line, the y range is probably too wide; if it disappears, it may be outside the window or undefined there (for example, sqrt(x) has no real value for negative x). Each time you change a range, press Graph again.
Common Mistakes to Avoid
- Forgetting brackets. sin x + 1 is read as sin(x) + 1; write sin(x + 1) when you mean that.
- Using degrees. Trig works in radians, so sin(90) is not 1. Use sin(pi/2).
- Writing powers without ^. Type x^2, not x2.
- A window that is too wide. Narrow the x range to see detail near a point.
- Reading a vertical jump as part of the curve. Graphs like tan(x) and 1/x break where they are undefined.
Frequently Asked Questions
Type it using x as the variable, such as x^2 - 4, choose an x range and press Calculate.
Yes. Enter up to three functions; each gets its own colour, and the calculator lists where they meet.
Enter both functions. The key points table lists each intersection as (x, y).
Make the x range smaller around the part you want, and leave the y range blank for auto.
Radians. To use degrees, convert first: sin(x*pi/180).
The function is undefined there, for example sqrt(x) for negative x or 1/x at x = 0.
Yes. You can type y = 2x + 1; the y = part is ignored.
Not directly. This tool graphs y as a function of x. A circle can be drawn as two halves: sqrt(9 - x^2) and -sqrt(9 - x^2).
The calculator looks for places where the graph changes sign, then narrows each one down by bisection to about six decimal places.
A graph that only touches the axis, such as x^2 at 0, does not change sign and may not be listed. Zoom in to check.
Yes. It runs in your browser with no app or account.
Usually not. Exams that allow graphing calculators mean approved handheld models.
References & Sources
| Source | Used For | Link |
|---|---|---|
| OpenStax, College Algebra 2e, Section 3.3: Graphs of Functions | How to read and plot graphs of functions, roots and intercepts. | https://openstax.org/books/college-algebra-2e/pages/3-introduction-to-functions |
| OpenStax, Precalculus 2e, Graphs of the Sine and Cosine Functions | Period and shape of trigonometric graphs. | https://openstax.org/books/precalculus-2e/pages/6-1-graphs-of-the-sine-and-cosine-functions |