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The graph of y=8(0.5)x3y=8(0.5)^{\frac{x}{3}} is shown in the xyxy-plane. Which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?

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Full solution

Q. The graph of y=8(0.5)x3y=8(0.5)^{\frac{x}{3}} is shown in the xyxy-plane. Which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?
  1. Identify Constants and Coefficients: Identify the constants and coefficients in the equation y=8(0.5)x/3y=8(0.5)x/3. The constants and coefficients in the equation are 88 and 0.50.5. The number 33 is also a constant that affects the exponent of 0.50.5.
  2. Determine Representations in Graph: Determine what each constant or coefficient represents in the graph.\newlineThe constant 88 is the vertical stretch factor, which affects the steepness of the graph. The coefficient 0.50.5 in the exponent represents the base of the exponential function, which determines the rate of decay or growth. The number 33 in the denominator of the exponent affects the horizontal stretch of the graph.
  3. Match Characteristics to Equation: Match the constants and coefficients to the characteristics of the graph.\newlineThe vertical stretch factor 88 is a constant that is directly visible in the equation as a multiplier of the exponential function. The base of the exponential function 0.50.5 is a coefficient that is also directly visible in the equation and indicates a decay since it is less than 11. The number 33 in the denominator of the exponent is a constant that modifies the horizontal stretch of the graph.

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