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Graphing Calculator

Draw up to four graphs, then zoom and trace them. Find where a graph crosses zero, where two graphs meet, and its lowest or highest point.

Level: Grades 8–12

  • Common Core standard HSA.REI.D.11Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • Common Core standard HSF.IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
  • Common Core standard 8.F.A.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Printable worksheet

Graphing Calculator

Functions
Angle unit

Window

Graph of Y1 = x^2 (violet solid line), Y2 = 2x + 3 (orange dashed line), for x from −10 to 10 and y from −10 to 10, in radians. The table of values below lists points on each graph.

Drag to move. Hold Ctrl (⌘ on a Mac) and scroll, pinch on a trackpad, or use the buttons to zoom. Click the graph or focus it and use the arrow keys to trace.

  • Y1 = x^2
  • Y2 = 2x + 3

Value at x

  • Y1(2) = 4
  • Y2(2) = 7

Find on an interval

  • x ≈ −1, y ≈ 1
  • x ≈ 3, y ≈ 9

Approximate: each value is found numerically and rounded to the digits it can be trusted to (at most 10 significant digits; fewer where a graph crosses very flatly or only touches).

Table of values

xY1 = x^2Y2 = 2x + 3
−39−3
−24−1
−111
003
115
247
399
41611
52513
63615
74917

How to solve this on a calculator

The keys to press to get this page's worked-example answer on an ordinary calculator, with what the display should show after each step. A named-model tab follows that calculator's official guidebook.

Where a parabola meets a line

Problem
Find where y = x² meets y = 2x + 3 by graphing both and using Intersect.
Formula
The graphs meet where x² = 2x + 3; y is either function's value there
Example values
Y1 = x² (the parabola); Y2 = 2x + 3 (the line). These are the fixed values from the worked example on this page, so the keys stay the same whatever you type into the calculator above.
Why a graphing calculator
Intersect solves Y1 = Y2 straight from the picture, which works even for equations with no algebraic method.
The whole sequence
Y=Y1 = x^2Y=Y2 = 2x+3Windowx −10 to 10, y −10 to 10IntersectY1 and Y2, from −5 to 0Valuex = −1IntersectY1 and Y2, from 0 to 5Valuex = 3

Step by step

  1. Y=Y1 = x^2

    Press Y= and enter the parabola as Y1 = x^2.

    Display: Y1=x^2

  2. Y=Y2 = 2x+3

    Enter the line as Y2 = 2x + 3. Implied multiplication works: 2x means 2 × x.

    Display: Y2=2x+3

  3. Windowx −10 to 10, y −10 to 10

    Use the standard window. The line crosses the parabola twice.

    Display: Window: x -10 to 10, y -10 to 10

  4. IntersectY1 and Y2, from −5 to 0

    Choose Intersect for Y1 and Y2 with bounds −5 and 0, around the left crossing.

    Display: Intersection: x≈-1, y≈1

  5. Valuex = −1

    Check with Value at x = −1: Y1 = (−1)² = 1, the same y as the line, 2(−1) + 3 = 1.

    Display: x=-1, Y1=1

  6. IntersectY1 and Y2, from 0 to 5

    Intersect again with bounds 0 and 5 for the right crossing.

    Display: Intersection: x≈3, y≈9

  7. Valuex = 3

    Value at x = 3: 3² = 9, and 2(3) + 3 = 9 too.

    Display: x=3, Y1=9

If your answer looks wrong

  • If Intersect reports that the graphs don't meet, the bounds miss the crossing: widen them, or look at the graph to see roughly where it is.
  • Bounds of −5 and 5 contain both crossings and return only the left one, x≈-1. Use one pair of bounds per crossing.
  • Entering 2x+3 as Y1 and x^2 as Y2 finds the same points: Intersect doesn't depend on the order.
  • Zero, Intersect and Minimum are numeric searches: the calculator shows ≈ and rounds to the digits it can trust, so a real calculator may show something like 2.9999999998 instead of 3.

How this is checked: an automated test presses these keys on this site's graphing calculator, checks every display along the way, and confirms the answer matches the calculator on this page (last checked 2026-10-19). It has not been checked on any particular brand or model, and real calculators label some keys differently.

What this calculator does

Type a rule like y = x² and see its graph. A function is a rule that turns each x into one y, written y = f(x). The graph shows its shape: where it crosses the x-axis, where two graphs meet, where it turns, and where it has no value. You can type up to four functions, such as x^2 − 2, 2x + 3 or sin(x). They're drawn together in different colours and line styles.

Drag the graph to move around. Use the mouse wheel or the zoom buttons to look closer. Trace moves a cursor along a curve with the arrow keys and shows its coordinates (its x and y).

The tools can also find a function's value at any x and make a table of values. They find zeros (where the graph crosses zero), where two graphs meet, and the lowest or highest point in a range. These answers come from a step-by-step computer search, not exact algebra. So they're marked as approximate and rounded to the digits you can trust.

Where it's used

Why this math matters outside the classroom:

  • Solving equations graphically. Algebra students solve x² = 2x + 3 by graphing both sides and reading the intersections at x = −1 and x = 3.
  • Break-even points. A small business graphs revenue 12x and cost 500 + 7x and finds they meet at 100 units.
  • Projectile height. Physics students graph h = −4.9t² + 20t + 1.5 and find the maximum height of about 21.9 m at about 2.04 s.
  • Checking homework. Students confirm that x² − 2 has zeros near ±1.414213562, matching ±√2.
  • Comparing growth. Savers graph 1000 × 1.05^x and 1000 + 80x to see when compound growth overtakes a fixed yearly amount.

Grade level and standards: Grades 8–12; Common Core State Standards for Mathematics HSA.REI.D.11, HSF.IF.C.7, 8.F.A.1. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Type a function that uses x after Y1 =, for example x^2 − 2. Add more in Y2 to Y4. You can leave out the × sign: 2x means 2 × x, and sinx means sin(x).
  2. Choose radians or degrees for sin, cos and tan. Radians are the usual choice for graphing. Degrees make sin(x) repeat every 360.
  3. Move the view: drag the graph, or zoom with the wheel or the + and − buttons. Standard shows −10 to 10 on both axes. You can also type exact window edges.
  4. Trace: click the graph (or tab to it) and press the left and right arrow keys to step along x. Up and down switch between functions.
  5. Use Value to get y at an exact x. Use Zeros, Intersections and Min/Max to search a range of x, and the table of values for a list of points.

What the inputs mean

Y1 =
Use x as the variable, with + − × ÷ ^ and brackets. You can also use sin, cos, tan and their inverses, log, ln, √ (sqrt), ∛ (cbrt), abs, exp, π and e. If you start with "y =", it's ignored.
Y2 =
Keep the example or type a second function. The intersections are the points where Y1 and Y2 have the same value.
Angle unit
This only matters for sin, cos and tan. In degrees, sin(x) repeats every 360. In radians, it repeats every 2π ≈ 6.283.
Interval start (x)
The searches look at x values from here to the interval end.
Evaluate at x
Y1 and Y2 are worked out exactly at this x, as far as computer arithmetic allows.

Formula and method

Point on the graph: (x, f(x))
Zero of f: a value of x with f(x) = 0
Intersection of f and g: a zero of f(x) − g(x), at the point (x, f(x))
Minimum on [a, b]: the lowest f(x) for a ≤ x ≤ b, possibly at a or b

In the formulas, f and g are functions, and f(x) is the y value f gives for a particular x. [a, b] means the range of x from a to b.

To draw a curve, the calculator works out f at several hundred x values across the window and joins the dots. Where the curve bends sharply or jumps, it works out more points in between. Some functions have no value at some x, like √x for negative x. Others shoot off to infinity, like 1/x at 0 or tan x at 90°. There, the line is left broken instead of being joined by a false vertical line.

Zeros and intersections come from a search. The calculator checks the range in 2,000 steps, looking for places where f(x) (or f(x) − g(x)) switches between positive and negative. It then zooms in on each one with Brent's method, a standard technique that keeps halving the gap and makes smart guesses to get there faster.

Some sign switches happen at a pole, where the graph shoots to infinity. Those are rejected, because f isn't close to 0 there. Points where the graph only touches the axis, like x² at 0, are found by looking for the smallest size of f(x) instead. They're less precise.

The minimum and maximum come from the same search, followed by Brent's method for finding a lowest point. It pins down the turning point to about 7 digits. That's the limit for any method that compares y values, because near a turning point y barely changes as x moves.

Worked example

Where does the U-shaped curve (parabola) y = x² meet the line y = 2x + 3?

  1. Enter Y1 = x^2 and Y2 = 2x + 3 and use the standard window, −10 to 10 on both axes.
  2. The line crosses the parabola twice. The graphs meet where x² = 2x + 3, which is where x² − 2x − 3 = 0.
  3. Searching from −10 to 10 for sign changes of x² − (2x + 3) finds x ≈ −1 and x ≈ 3.
  4. At x = −1, y = (−1)² = 1. At x = 3, y = 3² = 9. Both points also fit y = 2x + 3: 2(−1) + 3 = 1 and 2(3) + 3 = 9.
  5. Algebra agrees: x² − 2x − 3 = (x − 3)(x + 1), so x = 3 or x = −1.

Result: The graphs meet at (−1, 1) and (3, 9). The search reports them as x ≈ −1, y ≈ 1 and x ≈ 3, y ≈ 9.

Understanding the result

A zero is where the graph crosses or touches the x-axis. For a polynomial (like x² − 2x − 3), it's a root, a solution of f(x) = 0. An intersection is a solution of f(x) = g(x). That's how graphing solves equations that have no neat algebra method, such as 2^x = x + 3.

Searched answers are marked ≈ and rounded to the digits you can trust. A clean crossing is good to 10 digits. A flat crossing, such as (x − 1)³ at 1, or a point where the graph only touches the axis, shows fewer digits.

The searches look only inside the range you give. If a zero is missing, widen the range or look at the graph. The search can miss two zeros that are closer together than one search step.

Assumptions and limitations

  • Real numbers only. Points where f has no real value, such as √x for negative x, are left blank.
  • Leaving out the × sign ranks the same as × and ÷, read left to right, as on the scientific calculator.
  • Degree mode uses exact values at multiples of 15°, so tan(90) has no value instead of being a huge number.
  • Functions of x only. It doesn't draw parametric, polar or implicit equations, inequalities or statistics plots.
  • Zeros and intersections are close estimates, not exact answers in symbols. The quadratic equation calculator gives exact roots of quadratics.
  • Functions that wiggle extremely fast, such as sin(1/x) near 0, can't be drawn accurately at any zoom.

Frequently asked questions

Why is the line broken on graphs like 1/x or tan(x)?

Because the function has no value at the gap. 1/x has no value at x = 0, and it shoots to infinity on either side. Joining the last point on the left to the first point on the right would draw a vertical line that isn't part of the graph. The calculator spots these breaks (called poles) and leaves the gap.

How accurate are the zeros and intersections?

Each is rounded to the number of digits you can trust, at most 10. For an ordinary crossing, that's all 10. Where a graph crosses very flatly or only touches the axis, tiny computer rounding errors limit the precision. Then fewer digits are shown.

Should I use degrees or radians?

Use radians for most graphing. Calculus and physics formulas assume radians, and sin(x) then repeats every 2π. Choose degrees when x is an angle in degrees. For example, you'll see that sin(x) = 0.5 at 30 and 150.

Can I find where a graph meets a horizontal line?

Yes. Put the line in another slot, such as Y2 = 0.5, and find the intersections. That solves Y1 = 0.5.

How this calculator is checked

Method: Functions are parsed by the site's own expression parser with x bound as a variable. Zeros use a sign-change scan plus Brent's method; extrema use Brent's minimisation; precision is estimated from the slope at the root against floating-point noise. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-19. Found a problem? Tell us.

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