Questions: Find a formula for the exponential function passing through the points (-1, 3/4) and (1,12).

Solution Steps

To find the formula for an exponential function passing through two points, we can use the general form of an exponential function \( y = ab^x \). We will substitute the given points into this equation to create a system of equations. Solving this system will give us the values of \( a \) and \( b \).

1. Substitute the first point \((-1, \frac{3}{4})\) into the equation to get the first equation.
2. Substitute the second point \((1, 12)\) into the equation to get the second equation.
3. Solve the system of equations to find the values of \( a \) and \( b \).

To find the formula for the exponential function passing through the given points, we will follow these steps:

An exponential function can be expressed in the form: \[ y = ab^x \] where \( a \) is a constant, \( b \) is the base of the exponential, and \( x \) is the variable.

We have two points: \((-1, \frac{3}{4})\) and \((1, 12)\). We will substitute these points into the exponential function to create a system of equations.

For the point \((-1, \frac{3}{4})\): \[ \frac{3}{4} = ab^{-1} \] \[ \frac{3}{4} = \frac{a}{b} \quad \text{(Equation 1)} \]

For the point \((1, 12)\): \[ 12 = ab^1 \] \[ 12 = ab \quad \text{(Equation 2)} \]

We have the following system of equations:

1. \(\frac{3}{4} = \frac{a}{b}\)
2. \(12 = ab\)

From Equation 1, solve for \(a\): \[ a = \frac{3}{4}b \]

Substitute \(a = \frac{3}{4}b\) into Equation 2: \[ 12 = \left(\frac{3}{4}b\right)b \] \[ 12 = \frac{3}{4}b^2 \]

Multiply both sides by \(\frac{4}{3}\) to solve for \(b^2\): \[ b^2 = 12 \times \frac{4}{3} \] \[ b^2 = 16 \]

Take the square root of both sides: \[ b = 4 \]

Substitute \(b = 4\) back into the expression for \(a\): \[ a = \frac{3}{4} \times 4 \] \[ a = 3 \]

Final Answer