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The graph of is shown in the -plane. Which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?
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Q. The graph of is shown in the -plane. Which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?
- Identify Characteristics: The equation given is . We need to identify the characteristics of the graph that are represented by the constants or coefficients in this equation.
- Vertical Stretch or Compression: The number in the equation is the coefficient of the exponential expression . This coefficient represents the vertical stretch or compression of the graph. In this case, it is a vertical stretch since it is greater than .
- Rate of Decay: The base of the exponent, , is a constant that affects the rate of decay of the graph since it is an exponential decay function (because 0 < 0.25 < 1). This constant determines how quickly the graph decreases as increases.
- Horizontal Asymptote: The graph of an exponential function of the form , where is a positive constant and 0 < b < 1, will always have a horizontal asymptote at . This is not explicitly shown as a coefficient or constant in the equation, but it is an inherent characteristic of all exponential decay functions.
- Explicit Characteristics: The characteristics displayed as constants or coefficients in the equation are the vertical stretch factor and the rate of decay . These are the only characteristics that are explicitly shown in the equation.
