Problem 4 - Algebra 1: Concepts and Skills

Question

Graph the function \(y=3\left(\frac{1}{4}\right)^{x}\)

Step-by-Step Solution

Answer

The graph of the function \(y=3\left(\frac{1}{4}\right)^{x}\) is a decreasing curve that starts from the point (0,3) and approaches the x-axis as x increases, forming an asymptote at \(y=0\). It passes through the point (1,0.75).

Step 1: Identify the Function's Properties

Recognize the mathematical function as an exponential function of the base \( \frac{1}{4} \) and the rate of 3. This means that the graph will be a rapidly decreasing curve because the base of the exponent is less than 1. Additionally, the multiplier of 3 indicates the curve will be three times as steep when compared to the graph of \( y=\left(\frac{1}{4}\right)^x \).

Step 2: Identify Key Points

This function passes through the point (0,3) because any non-zero number raised to the power 0 is 1 multiplying by 3 gives 3. It would also pass through the point (1,0.75) because \( 3\left(\frac{1}{4}\right)^1 = 3*0.25 = 0.75 \). These two points provide initial guidance for sketching the rough graph.

Step 3: Sketch the Curve

Plot the points (0,3) and (1,0.75) on the graph. Then, sketch in a smooth curve that descends rapidly from the point (0,3) and passes through the point (1,0.75). The curve approaches the x-axis as x increases, but never crosses the x-axis, representing the asymptote at \(y=0\).

Key Concepts

Exponential Function Properties

Exponential functions are a type of mathematical function where the variable is an exponent. A basic form of an exponential function is represented as
\( f(x) = ab^x \), where
\( a \) is the initial value (or y-intercept),
\( b \) is the base, and
\( x \) is the exponent. The properties of exponential functions are distinctive:

• If
  \( b > 1 \), the function continuously increases, resulting in a growth curve.
• If
  \( 0 < b < 1 \), the function continuously decreases, producing a decay curve.
• The y-intercept of the graph is at the point
  \( (0, a) \).
• The function never touches the x-axis; hence, the x-axis acts as a horizontal asymptote.
• The rate at which the function grows or decays is determined by the base
  \( b \).

In the context of the given exercise, the properties of the function \( y=3\left(\frac{1}{4}\right)^{x} \) indicate that it is a decreasing exponential function due to its base \( \frac{1}{4} \) being less than 1. The multiplier 3 affects the steepness and position of the curve but does not alter the basic shape.

Plotting Points on a Graph

To accurately graph functions, it's essential to determine specific points through which the graph will pass. To plot points, you calculate the y-value for given x-values using the function's formula. Start by selecting a range of x-values that make sense for the function you're dealing with. For exponential functions, it's often useful to include
\( x = 0 \) because it helps to define the y-intercept.
With the exercise's function \( y=3\left(\frac{1}{4}\right)^{x} \), we know it passes through \( (0,3) \) since any number to the power of zero equals one, and \( (1,0.75) \) because \( 3\left(\frac{1}{4}\right) \) equals 0.75.After calculating a few key points based on the x-values you've selected, plot these points on a coordinate graph, then draw a smooth curve through these plots to form the shape of the graph. Always check if you have plotted a reasonable range of x-values for the specific function to cover its notable characteristics.

Asymptotes of Exponential Functions

Asymptotes are lines that a graph approaches but never actually touches or crosses. For exponential functions, horizontal asymOptotes are common and can usually be identified by looking at the function's behavior as \( x \) approaches infinity. For the function \( y=3\left(\frac{1}{4}\right)^{x} \), as \( x \) increases, the value of \( \left(\frac{1}{4}\right)^{x} \) gets closer and closer to zero, but never reaches it. This results in a horizontal asymptote along the x-axis at \( y=0 \). In summary, no matter how large \( x \) becomes, the value of \( y \) will approach, but not equal or fall below, the asymptote's y-value.Identifying and understanding asymptotes are crucial when sketching graphs, as they guide the overall shape and direction of the curve, ensuring accuracy in the visual representation of the function.