Question

Write an exponential function \(y=a b^x\) whose graph passes through the given points. \((1,2),(3,50)\)

Short answer

The exponential function that passes through the given points is \(y=\frac{2}{5} * 5^x\).

Step-by-step solution

Substitute the Coordinates

Take the given points \((1,2)\) and \((3,50)\) and substitute them into the formula \(y=ab^x\). For \((1,2)\), substitute \(x=1\) and \(y=2\), which will give \(2=a*b\). For \((3,50)\), replace \(x=3\) and \(y=50\) to get \(50=a*b^3\).

Solving for the Variable b

Divide the equation \(50=a*b^3\) by \(2=a*b\) equation, this will give \(25=b^2\). Then, we solve for \(b\) by taking the square root of both sides, which gives \(b=5\). Since \(b\) must be positive, we discard the negative value and keep the positive value \(5\).

Solving for the Variable a

Next is to solve for \(a\). Substitute \(b=5\) into the equation \(2=a*b\), this will give \(a=\frac{2}{5}\). The constants are then \(a=\frac{2}{5}\) and \(b=5\).

Key concepts

Coordinate Substitution in Exponential Functions

When working with exponential functions, such as \( y = ab^x \), coordinate substitution is a handy technique. It involves replacing variables with actual numbers to find unknowns.
To practice this, take given points from a graph or set - here it's \((1, 2)\) and \((3, 50)\) - and substitute them into the equation.
For the point \((1, 2)\):- Substitute \( x = 1 \) and \( y = 2 \) into \( y = ab^x \) to get the equation \( 2 = a \, b^1 \). For the point \((3, 50)\):- Substitute \( x = 3 \) and \( y = 50 \) into \( y = ab^x \), leading to \( 50 = a \, b^3 \). These resulting equations will enable us to identify the unknown constants through further calculations. Making these substitutions helps connect tangible numerical values with the abstract variables in your formula.

Solving Equations for Constants

After you've successfully made your substitutions, the next step is determining the constants \(a\) and \(b\). This is done by solving the equations you've built from your substitutions.
By looking at the system of equations: - \( 2 = ab \)- \( 50 = ab^3 \)You can solve for \( b \) by dividing the second equation by the first: - \( \frac{50}{2} = \frac{a b^3}{ab} \Rightarrow 25 = b^2 \)Take the square root of both sides to find \( b \): - \( b = 5 \)Since the exponential functions only make sense with a positive base, discard any negative roots, keeping \( b = 5 \).
Now substitute \( b = 5 \) into the first equation \(2 = ab\) to find \( a \): - \( a = \frac{2}{5} \)This process not only finds the constants but also validates them against your conditions, ensuring the results are appropriate for the function's context.

Graphing Exponential Functions

Once the constants \( a \) and \( b \) are known, they form the exponential equation: \( y = \frac{2}{5} \, 5^x \). Graphing this function shows its behavior over different values of \( x \).
An exponential graph typically starts small and grows rapidly if \( b > 1 \), as it does in this case since \( b = 5 \). The constant \( a \) affects the function's initial value or vertical starting point.- For \( x = 0 \), the output can be calculated as: - \( y = \frac{2}{5} \, 5^0 = \frac{2}{5} \)- With increasing \( x \), \( y \) grows quickly because of the \( b^x \) term.Plot key points like \((1,2)\) and \((3,50)\), confirming the function's validity. Knowing these constants and seeing them graphically can deepen your understanding of how exponential functions operate and why they are seen increasing rapidly in both mathematics and real-life applications.