Find the graph of the following exponential function:
$y=-2 \cdot 2^x$
Answer (2)
The function y = − 2 × 2 x is an exponential function with a base of 2.
The negative coefficient reflects the graph across the x-axis, making it decrease as x increases.
The graph has a y-intercept at (0, -2).
The graph has a horizontal asymptote at y = 0 .
The graph is an exponential decay reflected across the x-axis, with a y-intercept at (0, -2) and a horizontal asymptote at y = 0 .
Explanation
Analyzing the Function We are asked to describe the graph of the exponential function y = − 2 × 2 x . Let's break down the components of this function to understand its behavior.
Understanding the Base The base of the exponential function is 2, which is greater than 1. This means that the function 2 x is an increasing exponential function. As x increases, 2 x also increases.
Considering the Coefficient The coefficient in front of the exponential term is -2. This has two effects on the graph:
The negative sign reflects the graph across the x-axis. So, instead of increasing above the x-axis, the graph will decrease below the x-axis.
The 2 stretches the graph vertically by a factor of 2. This means that the y-values are twice as far from the x-axis as they would be for the function − 2 x .
Finding the Y-Intercept Let's find the y-intercept of the graph. The y-intercept is the point where the graph intersects the y-axis, which occurs when x = 0 . Plugging in x = 0 into the equation, we get:
y = − 2 × 2 0 = − 2 × 1 = − 2
So, the y-intercept is at the point (0, -2).
Identifying the Asymptote Exponential functions have a horizontal asymptote. In this case, as x approaches negative infinity, 2 x approaches 0. Therefore, − 2 × 2 x also approaches 0. The horizontal asymptote is the line y = 0 (the x-axis). The graph approaches this line as x goes to negative infinity.
Summary of the Graph's Properties In summary, the graph of y = − 2 × 2 x is an exponential function that:
Decreases as x increases.
Is reflected across the x-axis.
Has a y-intercept at (0, -2).
Has a horizontal asymptote at y = 0 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have will grow exponentially over time. The function y = − 2 × 2 x is similar, but with a negative coefficient, which could represent exponential decay or a decrease in value over time. Understanding exponential functions helps us analyze and predict these types of trends.
The graph of the function y = − 2 ⋅ 2 x reflects an exponential decay, starting from a y-intercept of (0, -2) and approaching the horizontal asymptote at y = 0 . It decreases as x increases due to the negative coefficient. Overall, it exhibits characteristics of exponential functions but mirrored in the opposite direction due to the negative sign.
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