Problem 59 - Algebra 1

Question

Sketch the graph of the exponential equation. $$y=0.9^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(y=0.9^{x}\) is a downward sloping curve, showing decay, and remains above the x-axis for all x-values.

Step 1: Behavior of the function

Evaluate the function at some key points to understand its behavior. Consider three points: \(x = -1\), \(x = 0\), and \(x = 1\). Substituting these values into the function gives \(y = 0.9^{-1}\) (for \(x=-1\)), \(y = 0.9^{0}\) (for \(x=0\)), and \(y = 0.9^{1}\) (for \(x=1\)). Thus the points to be plotted are (-1, 1.111), (0, 1), and (1, 0.9)

Step 2: Draw the plot points

Draw a graph and plot these points that were just obtained: (-1, 1.111), (0, 1), and (1, 0.9)

Step 3: Sketch the curve

Connect these points with a smooth curve. The curve will descend from left to right, being above the x-axis for all x, and it will be constantly approaching the x-axis as \(x\) increases, but will never reach it. This downward trend indicates the decay of the function.

Key Concepts

Graph Sketching

Creating a graph might seem challenging at first, but it's a fantastic way to visually understand how functions behave. When you're sketching a graph of an exponential function like \( y = 0.9^x \), it's crucial to identify some key points to guide you. These are usually points where \( x \) takes simple values, such as \(-1, 0, \) and \( 1 \).

• Start with plotting these simple points: For example, at \( x = 0 \), we find \( y = 0.9^0 = 1 \).
• Move to \( x = -1 \), giving \( y = 0.9^{-1} \approx 1.111 \).
• Then, at \( x = 1 \), \( y = 0.9^1 = 0.9 \).
• Plot these points on a graph: (-1, 1.111), (0, 1), and (1, 0.9).

Once you've got the points on the paper, connect them using a smooth curve. This curve will help you understand the general direction in which the function moves. In this case, from left to right, the curve decreases, giving you a visual of the exponential decay.

Function Behavior

Understanding the behavior of a function like \( y = 0.9^x \) is essential to grasp the bigger picture. By examining this function, you notice how it changes as \( x \) varies.

• As \( x \) increases, \( y \) gets smaller and smaller, approaching zero but never actually touching the x-axis.
• For negative values of \( x \), \( y \) increases beyond 1, showing a rise as \( x \) moves further left.
• The function is always positive, no matter what \( x \) is.

This tells us that the exponential function \( y = 0.9^x \) has no zeros and stays in the positive region of the graph. It's a neat representation of how some quantities decrease over time, yet never completely disappear.

Exponential Decay

Exponential decay is an essential concept in many fields like physics, chemistry, and finance. For the function \( y = 0.9^x \), we see exponential decay in action. "Decay" might sound scary, but it simply means how something decreases gradually over time.

• The base of the exponential function, \( 0.9 \), is less than 1, which causes this decaying behavior.
• Each increase in \( x \) results in multiplying the previous \( y \) value by 0.9, gradually reducing it.
• It never truly reaches zero, it only gets indefinitely close, which is a key characteristic of exponential decay.

This gradual decline can be observed in real life in scenarios like radioactive decay, population decreases, or even cooling processes. Remembering that exponential decay functions approach the x-axis without ever touching it can help you interpret many natural and technological processes effectively.