Problem 63 - Algebra 2

Question

Sketch the graph of the function \(y=2 \cdot\left(\frac{1}{4}\right)^{x}\)

Step-by-Step Solution

Answer

The graph of the function \(y=2 \cdot\left(\frac{1}{4}\right)^{x}\) starts at the y-intercept (0, 2), passes through the points (1, \(\frac{1}{2}\)) and (-1, 8) and decreases as x increases, representing an exponential decay.

Step 1: Identify the y-intercept

The y-intercept of the function is the value of y when x=0. Substituting x=0 into the equation, we get \(y=2 \cdot\left(\frac{1}{4}\right)^{0} = 2 \cdot 1 = 2\). So, the y-intercept is at the point (0, 2).

Step 2: Identify the function's behavior

The base of our exponential function is \( \frac{1}{4} \), which is less than one. This means the function represents an exponential decay: as x increases, y will decrease. The function is also multiplied by a factor of 2, which will stretch it vertically.

Step 3: Plot more points

We already have the y-intercept but it's beneficial to plot additional key points for improved accuracy of the sketch. Let's plot the points for x=1 and x=-1. For x=1, \(y=2 \cdot\left(\frac{1}{4}\right)^{1} = \frac{1}{2}\), so we have point (1, \(\frac{1}{2}\)). For x=-1, \(y=2 \cdot\left(\frac{1}{4}\right)^{-1} = 2 \cdot 4 = 8\), so we have point (-1, 8).

Step 4: Sketch the graph

Using the points from step 1 and step 3 and understanding the behavior of the function from step 2, we can sketch the curve of the function. The curve passes through points (0, 2), (1, \(\frac{1}{2}\)) and (-1, 8) and decreases as x increases.

Key Concepts

Graphing Functions

Understanding how to graph functions is a crucial skill in mathematics. It allows us to visualize how the function behaves and changes over different values of the variable. Graphing an exponential function like \(y=2 \cdot\left(\frac{1}{4}\right)^{x}\) involves a few steps:

• Identifying the y-intercept: This is where the graph crosses the y-axis, found by setting \(x=0\).
• Plotting key points: In addition to the y-intercept, it helps to find a few other points by substituting different values for \(x\).
• Understanding the slope and shape: The base of the function, whether it's greater or less than one, determines if it's growth or decay.

In this exercise, you start by plotting the y-intercept at (0, 2). Additional points such as (1, \(\frac{1}{2}\)) and (-1, 8) help define the curve, making it easier to sketch the correct graph.

Exponential Decay

Exponential decay is a pattern of reduction that can be represented with an exponential function. For the given function, \(y=2 \cdot\left(\frac{1}{4}\right)^{x}\), the base of the exponential (\(\frac{1}{4}\)) is less than one, which is indicative of decay.

In exponential decay:

• The function decreases as \(x\) increases.
• The rate of decay is determined by the base. A smaller fraction results in a faster decay.
• The constant multiplier (2 in this case) affects the vertical stretch but not the decay rate.

Graphically, exponential decay functions slope downward from left to right. This inverse relationship between the base and the exponent results in the curve approaching but never actually reaching zero.

Y-Intercept

The y-intercept of a function is a critical point that indicates where the graph crosses the y-axis, helping with initial plotting. For exponential functions like \(y=2 \cdot\left(\frac{1}{4}\right)^{x}\), finding the y-intercept is straightforward:

• Set \(x\) to zero and simplify the expression. For this function: \(y=2 \cdot\left(\frac{1}{4}\right)^{0}=2\cdot1=2\).
• The y-intercept is thus at the point (0, 2).

The y-intercept provides a starting reference to sketch the graph. It's crucial in defining the initial position and helps in understanding how the function initially behaves relative to the y-axis. Make sure to always begin your graphing from the y-intercept to ensure accuracy.