Exponential Parent Function Graph
The exponential parent function is a basic function that serves as the foundation for all other exponential functions
The exponential parent function is a basic function that serves as the foundation for all other exponential functions. It is represented by the equation:
f(x) = b^x
where b is the base and x is the exponent.
The graph of the exponential parent function depends on the value of the base (b). We will explore two cases: when b is greater than 1 (b > 1) and when b is between 0 and 1 (0 < b < 1).
Case 1: b > 1
When the base (b) is greater than 1, the exponential function grows exponentially as x increases. The graph of the exponential parent function will have a positive y-intercept and will be increasing from left to right. It will resemble an upward curve that becomes steeper as x increases.
For example, let’s consider the exponential parent function with the base b = 2:
f(x) = 2^x
If we substitute different values of x, we can find the corresponding y-values and plot the points to graph the function. Here are a few points:
f(0) = 2^0 = 1
f(1) = 2^1 = 2
f(2) = 2^2 = 4
f(3) = 2^3 = 8
Plotting these points and connecting them with a smooth curve will result in an increasing graph that gets steeper as x increases. The y-intercept is (0, 1).
Case 2: 0 < b < 1 When the base (b) is between 0 and 1, the exponential function decays exponentially as x increases. The graph of the exponential parent function will have a positive y-intercept and will be decreasing from left to right. It will resemble a downward curve that gets closer and closer to the x-axis as x increases. For example, let's consider the exponential parent function with the base b = 0.5 (or 1/2): f(x) = (1/2)^x Using the same process as before, we can find some points to graph the function: f(0) = (1/2)^0 = 1 f(1) = (1/2)^1 = 1/2 f(2) = (1/2)^2 = 1/4 f(3) = (1/2)^3 = 1/8 Plotting these points and connecting them results in a decreasing graph that gets closer to the x-axis as x increases. The y-intercept is (0, 1). It is worth noting that the exponential parent function does not intersect the x-axis. As x approaches negative infinity, the function approaches 0 but never reaches it. This is because the base b, when raised to any power, is always positive except when b equals 0.
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