Function Graphing Calculator

What Is a Function Graph?

A function is a rule. Give it an x, and it gives you one y. The rule y = 2x + 3 says: take x, double it, add 3. The graph draws every pair at once — a straight line that rises as x grows.

A function graph turns a rule into a picture. You can see the slope of a line, the vertex of a parabola, and the wave of a sine curve. Numbers in a table are hard to read. A picture shows the idea in one look.

Every function has a shape. A line is straight. A quadratic is a parabola. A sine wave repeats. An exponential curve grows faster and faster. Once you know the shapes, you can read a new function by its graph before you do any algebra.

How to Graph a Function

1. Type the function. Use the Y= key and type the rule, for example y = x² − 4. Use x as the variable.
2. Press GRAPH. The curve appears. If you cannot see it, press ZOOM and pick "Zoom Fit".
3. Use TRACE. Move along the curve with the arrow keys. The screen shows the x and y values of each point.
4. Open the TABLE. See exact values of x and y side by side, with a step size you pick.

To solve an equation, graph both sides. For x² = 4, graph y = x² and y = 4. Where the two curves cross, x is the answer. The TI-84 calculator online has the same steps, with the full TI-84 style keypad.

Functions You Can Graph

• Linear functions — y = mx + b. A straight line. The slope m sets the steepness, and b sets where it crosses the y-axis.
• Quadratic functions — y = ax² + bx + c. A parabola. The vertex is the highest or lowest point.
• Cubic functions — y = x³. An S-shaped curve that rises on both ends in opposite directions.
• Trigonometric functions — sin(x), cos(x), tan(x). Waves that repeat. Change the number in front to change the height.
• Exponential functions — y = 2 to the power of x. Slow growth at first, then very fast.
• Logarithmic functions — y = log(x). Fast growth at first, then very slow.
• Square root functions — y = the square root of x. Starts at zero and rises slowly.
• Absolute value functions — y = the absolute value of x. A V shape with a sharp point at zero.

For curves with r and theta, use the polar graphing calculator. For surfaces with two inputs, use the 3D graphing calculator.

Reading a Function Graph

A graph answers four questions fast:

• Where does it cross zero? Look for the x-intercepts. Those are the roots of the equation.
• Where does it turn? The vertex of a parabola and the peak of a wave are the maximum or minimum.
• Is it rising or falling? A line that goes up to the right has a positive slope. One that goes down has a negative slope.
• Where do two graphs meet? Graph both and find the intersection. That point solves both equations at once.

Practice with a simple function first. Graph y = x² − 4 and find its roots, its vertex, and its y-intercept. Then change one number — make it y = x² − 9 — and watch the graph move. Changing one number at a time is the fastest way to learn what each part of a function does.

Specialized Graphing Tools

Need a different view? These tools work the same way and are all free.

Limits, Inequalities and Piecewise Functions: What This Calculator Can Do

This is a function grapher with the keys of a TI-84, so it is best at plotting y = f(x). Here is what that means for three common topics.

• Limits: graph the function, then open the table (2ND + GRAPH) and set the start value close to the point with TBLSET. Read the values from both sides to see where the function is heading. A hole or jump is easy to spot on the graph with TRACE.
• Inequalities: graph the boundary line or curve, then test a point on the home screen to decide which side to shade on paper. This calculator does not shade regions.
• Piecewise functions: the keypad has no comparison keys, so a piecewise rule cannot be typed in one line. Graph each piece on its own and use WINDOW to limit the x-range you look at.
• Coordinate plane: graph lines such as 2*x+1 and use TRACE to read the coordinates of points on them.

For a full walkthrough, read how to use a graphing calculator.

Frequently Asked Questions