To write an exponential equation from a graph, identify key points and use them to determine the equation's parameters. Here's a step-by-step guide:
Steps to Determine the Exponential Equation
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Identify Two Points: Locate two distinct points on the graph with clear, integer coordinates. Let's call them (x₁, y₁) and (x₂, y₂).
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Check for the Y-Intercept: If one of your points is the y-intercept, (0, a), then 'a' is the initial value. This simplifies the equation significantly. The exponential equation takes the form f(x) = a(b)x, where 'a' is the initial value and 'b' is the growth/decay factor.
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Substitute into General Form (If No Y-Intercept): If neither of your points is the y-intercept (0, a), substitute the coordinates of both points into two equations with the general form f(x) = a(b)x. This will give you two equations with two unknowns, 'a' and 'b'.
- Equation 1: y₁ = a(b)x₁
- Equation 2: y₂ = a(b)x₂
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Solve for 'a' and 'b':
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Solve for 'a' in one equation: Choose one of the equations (e.g., Equation 1) and solve for 'a': a = y₁ / (bx₁).
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Substitute into the other equation: Substitute the expression for 'a' you just found into the other equation (Equation 2). This will result in an equation with only 'b' as the unknown.
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Solve for 'b': Solve the resulting equation for 'b'. This often involves algebraic manipulation and potentially taking roots.
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Substitute 'b' back to find 'a': Once you've found 'b', substitute its value back into the equation you used to solve for 'a' (e.g., a = y₁ / (bx₁)) to find the value of 'a'.
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Write the Exponential Function: Now that you have the values of 'a' and 'b', substitute them into the general exponential function form: f(x) = a(b)x. This is your equation.
Example
Let's say our graph contains the points (1, 6) and (2, 12).
- Points: (1, 6) and (2, 12)
- No y-intercept given.
- Substitute:
- 6 = a(b)¹
- 12 = a(b)²
- Solve:
- a = 6/b
- 12 = (6/b) * b²
- 12 = 6b
- b = 2
- a = 6/2 = 3
- Equation: f(x) = 3(2)x
