Problem 24 - Algebra 2

Question

Graph each function. $$ y=(0.75)^{x} $$

Step-by-Step Solution

Answer

The graph of the function \(y=(0.75)^{x}\) is a decreasing curve, starting from a high point on the left, passing through the point (0,1), and approaching but never crossing the x-axis as x increases.

Step 1: Identify the base of the exponential function

In this case, the base of the exponential function is \(0.75\).

Step 2: Identify key points for the graph

Recall that all exponential functions pass through the point (0,1). When \(x=0\), \(y=(0.75)^{0}=1\). So, the point (0,1) is on the graph.

Step 3: Draw the graph

Plot the point (0,1) and knowing that the function is decreasing (since the base 0.75 is less than 1), draw a curve that starts from a high point on the left, passes through (0,1), and approaches but never crosses the x-axis.

Key Concepts

Graphing Exponential Functions

Graphing an exponential function involves plotting its behavior on a graph to understand how it changes with respect to the variable, typically x. An exponential function can be written in the form \( y = a^x \), where \( a \) is the base. A crucial step in graphing this function is to recognize the characteristics based on the base. If the base \( a \) is greater than 1, the graph will show exponential growth; it will rise sharply as x increases. Conversely, if the base is between 0 and 1, as in our function \( y = (0.75)^x \), it demonstrates exponential decay. This means as \( x \) increases, \( y \) decreases, creating a downward curve.

When graphing, ensure you identify and plot key points which serve as anchors. You start by locating the point \( (0, 1) \), since any exponential function raised to the power of zero equals 1. Then, plot additional points by calculating values of \( y \) for chosen x-values to decide how the curve will further shape. Connect these points smoothly, and illustrate the direction of the curve to visualize how the function behaves across x-values.

Exponential Decay

Exponential decay is a pattern observed in functions where the quantity decreases at a rate proportional to its current value. It is a prevalent model for processes like cooling temperatures or radioactive decay. In our function \( y = (0.75)^x \), the base 0.75 informs us of this decay. As \( x \) increases, \( y \) reduces by multiplying by 0.75 repeatedly.

This consistent decrease means the function will approach the x-axis but never actually touch it, forming an asymptote. When graphing, observe how steeply the values of y drop, especially for positive x, since this will depict how quickly the quantity fades away. At negative values of x, the function will rise, as a diminishing base raised to a negative power indicates increasing y-values. By conceptualizing this decay, you can predict a smooth curve diminishing across the positive x-range.

Key Points on Graphs

Identifying key points on graphs of exponential functions strengthens understanding and accuracy in plotting curves. These points serve as essential references in defining the graph's shape. The basic exponential function \( y = a^x \) always includes the point \( (0,1) \) because any number to the zero power is 1. In \( y = (0.75)^x \), this point is confirmed as foundational.

Additionally, consider some familiar easy-to-calculate points. For instance, when \( x = 1 \), \( y = 0.75^1 \), yielding (1, 0.75). Another helpful point is \( x = -1 \), resulting in \( y = (0.75)^{-1} = \frac{1}{0.75} \approx 1.33 \). Calculating these values allows you to identify how quickly the function grows or decays, giving you a picture of the exponential curve. By connecting these points with a smooth line, you'll accurately map out the exponential decay signature to display understanding clearly on a graph.