Problem 26 - Algebra 2

Question

Graph each function. $$ y=(0.25)^{x} $$

Step-by-Step Solution

Answer

The graph of the function \(y = (0.25)^x\) starts at the y-intercept (0,1) and decreases, approaching but never reaching the x-axis (the line y = 0).

Step 1: Identify the y-intercept

The y-intercept of a function is the point where the line cuts the y-axis. For any exponential function in the form \(y = a^{x}\), when x=0, y equals 1 because any non-zero number raised to the power of 0 equals 1. So, for our function \(y=(0.25)^{x}\), when x=0, \(y=(0.25)^0=1\). Hence, the y-intercept is at (0,1).

Step 2: Identify the horizontal asymptote

Horizontal asymptotes are lines that the graph approaches but never touches. For an exponential function in the form \(y = a^{x}\), as x tends to positive infinity, y tends to 0. Hence, the line y=0 is a horizontal asymptote.

Step 3: Plot the key points

The y-intercept is (0,1) and the horizontal asymptote is y=0. The graph increases from left to right and is getting closer and closer to the x-axis but never touches it. The graph passes through the y-intercept and tends to the x-axis as x tends to infinity.

Step 4: Draw the graph

Using the y-intercept and horizontal asymptote, sketch the graph. Remember that the graph will never drop below the x-axis, so the area under the x-axis (where y < 0) should be left blank.

Key Concepts

Graphing

Graphing exponential functions is a fun and insightful way to understand their behavior in a visual format. Let's break down the process:- **First**, identify the equation of the exponential function you are dealing with, like \( y = (0.25)^x \).- **Next**, compute important features like the **y-intercept** and the **horizontal asymptote**.When graphing this function, plot its y-intercept first, which is the point where the graph crosses the y-axis. Since our function is of the form \( y = a^x \), the graph will cross at the y-intercept of (0, 1) by default when x equals zero because any non-zero number to the power of zero is one.As you draw the graph, notice how it behaves. It starts from the y-axis and approaches the x-axis as it extends to the right. These curves are smooth and reflective of the exponential decay in this example (since 0 < 0.25 < 1). Recall, the graph never actually touches the x-axis but gets very close, which introduces the concept of asymptotes that we'll explore next. Moreover, make sure your plot showcases:

• A continuous curve without breaks.
• A smooth transition passing through the y-intercept.
• Approach towards the x-axis as x increases, indicating decay.

Incorporating these elements will result in an accurate graph of the function.

Horizontal Asymptote

A horizontal asymptote in exponential functions tells us about the behavior of the graph as x approaches positive or negative infinity. For the function \( y = (0.25)^x \), this is a line that the function comes infinitesimally close to but never actually reaches or crosses.In our case, as x increases, \( y = (0.25)^x \) tends towards zero but will not become negative or reach zero. This creates a horizontal asymptote at \( y = 0 \).Here are some key points about horizontal asymptotes:

• They do not affect the y-intercept or the initial behavior of the graph, rather they define the behavior at extreme x-values.
• The asymptote with \( y = (0.25)^x \) is horizontal because the exponential base is positive and smaller than 1, indicating decay as x grows.

To sketch this on your graph, you can draw a dashed line along the x-axis to represent the asymptote, guiding your eye to how the graph approaches this line without crossing it.

Y-Intercept

The y-intercept is a key point on the graph of a function, providing an easy starting location for plotting. For exponential functions like our example \( y = (0.25)^x \), finding the y-intercept is straightforward and involves evaluating the function at x = 0.When substituting x = 0 into the equation, it simplifies to \( y = (0.25)^0 = 1 \). Thus, the graph crosses the y-axis at the point (0, 1).Here are some essential aspects regarding y-intercepts:

• It is always calculated by setting x to zero in the exponential function.
• The y-intercept for any exponential function \( y = a^x \) with a constant base "a" will always be at y = 1, assuming no vertical shifts.
• This starting point is foundational for plotting the rest of the function as it helps dictate the initial slope and direction of the curve.

Plotting this point is crucial, as it contributes to the initial anchoring of the graph, accentuating the characteristic exponential growth or decay pattern of the whole function.