Function Graph Calculator

1. Choose a function family: linear, quadratic, exponential, absolute value, or square root.
2. Enter the numbers for that family. Only the boxes you need are shown.
3. Set the x min and x max to choose how much of the graph you see. The y-axis fits itself.
4. Read the key feature, the graph, the cards, and the table of values. Press Copy report or Save to keep them.

The graph is drawn to scale, and the dark dots mark the intercepts and the vertex or start point when they fall inside the view.

A function is a rule that turns each input into exactly one output. We usually call the input x and the output f(x), or y. The graph is every point (x, f(x)) drawn on a grid.

The key word is one. An input can never give two different outputs. Two inputs may share an output, though. In y = x^2, both 2 and -2 give 4, and that is fine.

The Vertical Line Test

You can check a graph with a vertical line. Slide it from left to right across the picture. If it ever hits the graph in two places, the graph is not a function. A circle fails this test. A parabola that opens up or down passes it.

The domain is the set of x-values you may use. The range is the set of y-values the function actually produces. The calculator gives both in words and in interval notation.

In interval notation, a square bracket [ means the end number is included. A round bracket ( means it is not. We write infinity as inf, and infinity always gets a round bracket.

Only the square root family limits the domain here. You cannot take the square root of a negative number and get a real answer.

Each family starts from a simple parent function. The other numbers move, stretch, or flip that parent shape.

A line has the same slope everywhere. The other four families bend, so their steepness changes as x changes.

The letters a, h, and k change a parent graph in predictable ways. Once you know them, you can sketch a graph without plotting many points.

• Shift right or left: replacing x with x - h moves the graph h units right. So |x - 2| sits 2 units right of |x|, and sqrt(x + 4) sits 4 units left of sqrt(x).
• Shift up or down: adding k moves the whole graph up k units. A negative k moves it down.
• Stretch or shrink: multiplying by a with a size above 1 makes the graph steeper. A size between 0 and 1 makes it flatter.
• Reflect: a negative a flips the graph over the x-axis. That is why y = -x^2 + 4 opens downward.

Watch the sign inside the brackets. The shift goes the opposite way from the sign you see, which trips up many students.

The y-intercept is where the graph crosses the y-axis. Find it by putting x = 0 into the function. The x-intercepts are where the graph crosses the x-axis. Find them by setting f(x) = 0 and solving.

Worked Example: y = x^2 - 4x + 3

1. y-intercept: f(0) = 0 - 0 + 3 = 3, so the point is (0, 3).
2. Discriminant: b^2 - 4ac = 16 - 12 = 4. It is positive, so there are two x-intercepts.
3. x-intercepts: x = (4 - 2) / 2 = 1 and x = (4 + 2) / 2 = 3, so (1, 0) and (3, 0).
4. Vertex: h = 4 / 2 = 2 and k = 4 - 8 + 3 = -1, so the vertex is (2, -1).

Now try y = x^2 + 1. Its discriminant is 0 - 4 = -4, which is negative. The parabola sits above the x-axis, so it has no x-intercepts. Need more help with lines? See our guide on how to find the slope-intercept form.

1. Shifting the Wrong Way

In |x - 2| the graph moves right, not left. The inside sign is the opposite of the direction. Check by finding the x that makes the inside equal 0.

2. Squaring a Negative Without Brackets

For y = x^2, the value at x = -3 is (-3)^2 = 9. Typing -3^2 on many calculators gives -9 instead, because it squares 3 first.

3. Forgetting the Square Root Domain

A table for sqrt(x + 4) has no value at x = -10. That point is not on the graph at all, so do not plot a 0 there.

4. Expecting an Exponential to Cross y = 0

The curve y = 2^x gets very close to the x-axis on the left but never touches it. It has no x-intercept.

The domain is every x you are allowed to put in. The range is every y that comes out. Look for inputs that break the rule, such as a negative number under a square root. For y = sqrt(x + 4), the domain is x >= -4 and the range is y >= 0.

Set the function equal to 0 and solve. The discriminant b^2 - 4ac tells you how many answers exist. If it is positive there are two x-intercepts, if it is 0 there is one, and if it is negative there are none. For x^2 - 4x + 3 the x-intercepts are 1 and 3.

The vertex is the turning point of the parabola. Its x-value is -b / 2a. Put that x back into the function to get y. For y = x^2 - 4x + 3, x = 4 / 2 = 2 and y = 4 - 8 + 3 = -1, so the vertex is (2, -1).

Draw any vertical line through a graph. If a line ever crosses the graph more than once, the graph is not a function, because one x would have two y-values. A circle fails the test, but every graph this calculator draws passes it.

The square root of a negative number is not a real number. So y = sqrt(x - h) + k only exists when x is h or larger. The graph starts at (h, k) and moves right. For y = sqrt(x + 4), it starts at (-4, 0).

A base of 1 gives 1^x = 1 for every x, which is just a flat line, not growth or decay. A negative base breaks at many inputs. For example, (-2)^0.5 is not a real number. So the base must be positive and not equal to 1.

No. For y = a * b^x, the value b^x is always positive, so y never reaches 0. The graph gets closer and closer to the line y = 0, called a horizontal asymptote, but has no x-intercept.

• OpenStax. College Algebra 2e, 3.1 Functions and Function Notation (definition of a function, vertical line test).
• OpenStax. College Algebra 2e, 3.2 Domain and Range (interval notation, domains of root functions).
• OpenStax. College Algebra 2e, 3.5 Transformation of Functions (shifts, stretches, reflections).
• OpenStax. College Algebra 2e, 3.6 Absolute Value Functions (vertex and intercepts of absolute value graphs).
• Dawkins, P. The Definition of a Function. Paul's Online Math Notes, Lamar University (functions, domain and range).
• Weisstein, E. W. Function. MathWorld, Wolfram Research (formal definition of a function).