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3
Tutorials that teach
Graph of an Exponential Equation

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Tutorial

- The Graph of y = b
^{x} - The y-intercept of y = ab
^{x} - Other Features of the Graph

**The Graph of y = b ^{x}**

Below is a sketch of the graph y = 2^{x} Notice that the base to the exponential expression is 2, and there is an implied scalar multiplier of 1 in front of this expression:

We see that, in this case, as x gets larger (approaches positive infinity), the value of y also increases (it approaches positive infinity as well). On the other hand, as x gets smaller (approaches negative infinity), the value of y gets smaller as well, but it approaches zero, rather than negative infinity.

This behavior is characteristic of exponential functions with a base larger than 1. If the base was between 0 and 1, then the behavior would be much different: y would tend towards zero as x gets larger, but tend toward infinity as x gets smaller.

Let's contrast this with the graph of an exponential equation with a larger base, such as y = 3^{x}

^{}

We have the same general behavior as in the previous graph, however things are more dramatic: as x becomes more positive, y increases, but at a faster rate than in the previous graph. This is because a larger base number is being raised to a positive exponent. Similarly, as x becomes more negative, y decreases in value (approaching zero), but at a faster rate than in the previous graph. This is we can think of y being divided by a larger number each time x decreases in value.

**The y-intercept of y = ab ^{x}**

The y-intercept to any equation is the point on the graph at which the line or curve touches or crosses the y-axis. This always occurs when x = 0. Let's return to the equation y = 2^{x} When x = 0, y evaluates to 1, because any base number raised to a power of zero is 1. That must mean this holds true for the equation y = 3^{x} The y-intercept is at the point (0, 1) because when x = 0, y = 1 for any exponential equation in the form y = b^{x}

What is the y-intercept of equations in the form y = ab^{x}? We alreay know that b^{x} evalutes to 1 when x = 0 for any base. We can deduce that the y-intercept depends on the value of "a" in this case. For the general exponential equation y = ab^{x}, the y-intercept has the coordinates (0, a)

- For equations in the form y = b
^{x}, the y–intercept has coordinates (0, 1) because b^{0}= 1 for any accepted base. - For equations in the form y = ab
^{x}, the y–intercept has coordinates (0, a), for the same reasons above.

**Other Features of the Graph of y = ab ^{x}**

If we reverse the signs of a and x in the general exponential equation, we end up with different variations of the general exponential curve. More specifically, these are reflections about either the x– or y–axes, or perhaps both, if the signs of both a and x are reversed. These patterns are illustrated in the graphs below: