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Function Transformations guide

The Function Grapher visualizes mathematical functions and their transformations. You can adjust the parameters a, b, h, and k to see how stretching, shifting, and reflecting changes the shape of a parent function.

Worked examples

Transform a quadratic function

  1. Select the "Transformations" tab.
  2. Choose "Quadratic (x²)" as the Parent Function.
  3. Set a = 2, h = 1, k = -3.
  4. Observe the blue curve on the graph.
Result: y = 2(x - 1)² - 3

The parabola is stretched vertically by 2, shifted right by 1 unit, and shifted down by 3 units. The vertex moves from (0,0) to (1,-3).

Analyze a sine wave

  1. Select the "Wave" tab.
  2. Choose "Sine" as the base function.
  3. Set Amplitude (A) to 2, Frequency (B) to 2, Phase (C) to 0, and Shift (D) to 1.
Result: y = 2·sin(2x) + 1

The wave oscillates between -1 and 3 (amplitude 2, shifted up 1). It completes two full cycles in the standard 2π interval.

Find the vertex of a quadratic

  1. Select the "Quadratic" tab.
  2. Enter the coefficients for y = ax² + bx + c (e.g., a=1, b=-2, c=-3).
  3. Look at the Analysis section.
Result: Vertex: (1, -4)

The vertex formula is x = -b / 2a. Here, x = -(-2) / 2(1) = 1. Plugging x=1 back into the equation gives y = 1 - 2 - 3 = -4.

Tips

  • •A negative value for "a" reflects the graph upside down (over the x-axis).
  • •Values of "h" shift the graph horizontally. Note that (x - h) moves the graph right, not left.
  • •Use the "Evaluate at x" input to see the exact y-value at a specific point on the graph.

Frequently asked questions

What is a parent function?
The simplest form of a function family. For example, the parent function for quadratics is y = x², and for sines it is y = sin(x).
What does the parameter "a" do?
It controls the vertical stretch or compression. If |a| > 1, the graph stretches vertically. If 0 < |a| < 1, it compresses. If "a" is negative, it reflects over the x-axis.
What does the parameter "b" do?
It controls the horizontal stretch or compression. If |b| > 1, the graph compresses horizontally. If 0 < |b| < 1, it stretches horizontally.
What does the parameter "h" do?
It shifts the graph horizontally. Because the formula is f(x - h), a positive "h" moves the graph to the right, and a negative "h" moves it to the left.
What does the parameter "k" do?
It shifts the graph vertically. A positive "k" moves the graph up, and a negative "k" moves it down.
What is the vertex of a parabola?
The highest or lowest point on the graph, where the curve changes direction.
What is the axis of symmetry?
A vertical line that divides the parabola into two mirror-image halves. It passes through the vertex: x = -b / 2a.
What is the discriminant?
The part of the quadratic formula under the square root (b² - 4ac). It tells you how many real roots the equation has.
If the discriminant is negative, what does the graph look like?
The parabola does not cross the x-axis at all. It has no real roots (only complex roots).
What is the amplitude of a wave?
The maximum height of the wave from its midline. It is controlled by the "A" parameter.
What is the period of a wave?
The horizontal length of one complete cycle. It is calculated as 2π / |B|.
What is a phase shift?
A horizontal shift of the wave, controlled by the "C" parameter. It moves the starting point of the wave left or right.
What is the difference between a sine and cosine wave?
A cosine wave is just a sine wave shifted to the left by π/2 (90 degrees).
What is an exponential function?
A function where the variable is in the exponent, like y = 2^x. It grows incredibly fast and never touches the x-axis (horizontal asymptote).
What is a logarithmic function?
The inverse of an exponential function, like y = ln(x). It grows very slowly and has a vertical asymptote at x=0.
What is an asymptote?
A line that a curve approaches but never touches.
Can I graph a custom equation?
This tool focuses on transformations of standard parent functions and standard parabolas/waves. It does not support typing arbitrary equations.
How do I find where the graph crosses the x-axis?
For a quadratic, look at the "Roots (x-intercepts)" in the Analysis section. These are the points where y = 0.
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