Exponential Function Grapher

About the Exponential Function Grapher

Plot exponential functions y = a·bˣ and see growth or decay behavior.

How to use this calculator

1. Enter the coefficient \(a\), which sets the starting value on the graph.
2. Enter the base \(b\), which controls whether the graph grows or decays.
3. Adjust the exponent values or viewing window to see the curve shape more clearly.
4. Read the graph to compare how fast the function rises or falls as \(x\) changes.

The formula explained

The formula \(y=a\cdot b^x\) computes the value of an exponential function for each input \(x\). The graph shows how the output changes by repeated multiplication rather than repeated addition.

• a = the initial value, or the y-intercept when \(x=0\)
• b = the growth factor if \(b>1\), or the decay factor if \(0<b<1\)
• x = the input, usually the horizontal-axis value
• y = the output, or the vertical-axis value

Step by step method

1. Start with the function in the form \(y=a\cdot b^x\).
2. Choose a few \(x\)-values, such as 0, 1, and 2, and compute \(y\) each time by substituting into the formula.
3. Plot the points on the coordinate plane, then connect them with a smooth curve.
4. Check whether the graph increases for growth, or decreases for decay, based on the value of \(b\).

Worked example

Problem. Graph the exponential function \(y=3\cdot 2^x\).

1. When \(x=0\), \(y=3\cdot 2^0=3\). So the graph crosses the y-axis at \((0,3)\).
2. When \(x=1\), \(y=3\cdot 2^1=6\). When \(x=2\), \(y=3\cdot 2^2=12\).
3. Plot \((0,3)\), \((1,6)\), and \((2,12)\), then draw a smooth increasing curve through them.

Tips and common mistakes

• If \(0<b<1\), the function shows decay, not growth, even if \(a\) is positive.
• Do not treat exponential change like linear change, because the output is multiplied by the base each step, not increased by a fixed amount.

Frequently asked questions