Problem 18 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$f(x)=5^{x}$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 5^x\) is an upwardly increasing curve that passes through the points (0, 1) and (1, 5). As \(x\) increases, the curve rises very quickly. As \(x\) decreases, the curve gets closer and closer to \(y=0\), without touching it, giving a horizontal asymptote at \(y=0\).

Step 1 - Understand the Basic Shape of the Graph

The shape of the graph depends on the base of the exponential function. As \(5 > 1\), the function \(f(x) =5^x\) will have a basic exponential growth pattern that rises from left to right.

Step 2 - Identify Key Points

Identify key points of the function to help with the sketch. Since it's an exponential function, the y-intercept (where the graph crosses the y-axis) will be at (0, 1). This is because any number to the power of 0 is 1, so when \(x=0\), \(y=5^{0}\), hence \(y=1\). Another key point is (1, 5), when \(x=1\), \(y=5^{1}\), therefore, \(y=5\).

Step 3 - Sketch the Graph

Join the points together in a smooth curve which starts from the y-intercept. As you move to the right (i.e. as \(x\) increases), the curve rises quickly. As you move to the left (i.e. as \(x\) decreases), the curve approaches \(y=0\) but never touches or crosses it, giving it a horizontal asymptote at \(y=0\). Keep in mind that the graph never touches or crosses the x-axis.

Key Concepts

Exponential Growth

Exponential functions, such as \(f(x) = 5^x\), reflect a type of growth that increases rapidly as the input value gets larger. This is known as exponential growth. To understand this concept, imagine a situation where something is repeatedly multiplied by the same number—like bacteria doubling every hour. In the mathematical function \(f(x) = 5^x\), the base, which is 5 in this case, is the constant factor by which the function's value is multiplied as \(x\) increases.

Therefore, as you move along the graph from left to right, every step you take along the x-axis results in the y-value being multiplied by 5 again, making the graph climb steeply upwards. This characteristic steep rise demonstrates the 'explosive' increase typical of exponential growth models in real-world scenarios, such as population growth, investments, and certain contagions.

Y-Intercept

The y-intercept is a fundamental component of graphing exponential functions. It represents the point where the graph of a function crosses the y-axis. For the function \(f(x) = 5^x\), as with any exponential function \(b^x\) where \(b\) is a positive number, the y-intercept is always at \((0,1)\).

• This is because any non-zero number raised to the power of 0 equals 1. So, if you substitute \(x\) with 0 in the equation, \(f(0) = 5^0 = 1\), showing that the y-intercept occurs at \(y=1\).
• Understanding where the y-intercept lies on a graph is crucial for sketching exponential functions since it serves as a starting point from which the curve emerges.

Horizontal Asymptote

Another key concept in graphing exponential functions is the horizontal asymptote. An asymptote is a line that a graph gets closer and closer to, but never actually touches or crosses. For exponential growth functions where the base is greater than 1, the horizontal asymptote is always \(y=0\).

The graph of \(f(x) = 5^x\) illustrates this as it approaches the x-axis infinitely as \(x\) decreases without touching it. This happens because, as \(x\) becomes more negative, \(5^x\) gets closer to zero—but since a positive number raised to any power can never be zero or negative, the graph will never meet the x-axis. Recognizing the location of a horizontal asymptote helps in sketching and understanding the behavior of an exponential function, especially as it stretches towards infinity.