Графичен калкулатор

How to plot a graph

1. Enter a function. For example x^2-3. You can add a second and third function for comparison.
2. Plot the graph. The graph is drawn immediately with its axes and grid.
3. Adjust the view. Drag to move, use the wheel to zoom or set the limits by hand. The reset button restores the default view.
4. Read the zeros. The calculator lists the points where the curve crosses the x-axis.

What can you see from the graph?

The graph illustrates the function's behaviour: where it increases and decreases, where the peaks and troughs are and where the curve crosses the axes. The zeros, i.e. the crossing points with the x-axis, correspond to the solutions of the equation f(x) = 0, so the graphing calculator complements the equation solver. Plotting several functions on the same picture helps find the intersection points of the curves.

Functions you can plot

You can plot polynomials, trigonometric functions, exponential and logarithmic functions, and rational functions such as 1/x. Use the same syntax as in the other calculators: a power with a caret, functions with brackets and implicit multiplication allowed. The table of values shows the function's values at even intervals, which makes it easier to grasp its behaviour.

Example: interpreting a parabola

Let's plot f(x) = x² − 4. The graph is an upward-opening parabola whose bottom is at (0, −4). The curve crosses the x-axis where x² − 4 = 0, i.e. x = −2 and x = 2 — precisely these zeros the calculator lists. It crosses the y-axis at −4. If you add the line g(x) = x − 2 to the same picture, you see the intersection points, which correspond to the solutions of the equation x² − 4 = x − 2. Thus the graph and the equation solution support each other.

Kahden käyrän leikkaus

The intersection points of two functions are places where they take the same value. Graphically they appear as places where the curves cross; computationally they are found by setting the functions equal. A graphical examination gives a quick overview: how many intersection points there are and roughly where they lie. The exact values are best confirmed with the equation solver, but the graph tells you immediately whether there are one, two or no solutions.

Funktioperheet ja kuvaajan muunnokset

Once you know the shape of the basic graph, you can quickly derive the whole function family from it. A line, a parabola, a cubic function, the absolute value and a sine wave are the basic shapes, which are shifted and stretched by simple rules. Adding to the function, such as f(x) + 2, raises the graph upward, and f(x − 3) moves it to the right. Multiplying in front of f(x) stretches or shrinks the curve vertically.

Recognising these transformations makes plotting and interpreting graphs much easier: instead of calculating dozens of points, you grasp the basic shape and its shifts. By plotting several functions of the same family on the same picture, you see immediately how a coefficient or a constant term affects it. The graphing calculator is just right for this kind of experimentation: change the expression a little and watch how the curve moves, and the transformation rules stick in your mind.

Good to know about graphs

• The zeros are the crossing points with the x-axis and correspond to the solutions of the equation f(x) = 0.
• The y-axis crossing point is the function's value at x = 0.
• A vertical asymptote arises at the point where the denominator is zero, for example 1/x.
• If the curve is not visible, it is worth adjusting the view limits or restoring the default.

Накратко

The graphing calculator plots up to three functions on the same graph and finds the zeros. Enter the functions, adjust the view by dragging and zooming and read the zeros. The graph reveals the function's behaviour at a glance and complements equation solving by showing the intersection points visually.

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