Q12. - Algebra 2

Question

Write an exponential function whose graph passes through the points $\left(0,-18\right)\text{ and }\left(-2,-2\right)$.

Step-by-Step Solution

Answer

The exponential function passing through the given points is $y=-18{\left(3\right)}^x$.

Step 1. Write down the given information.

The given points are $\left(0,-18\right)\text{ and }\left(-2,-2\right)$.

Step 2. Concept used.

The exponential function is of the form $y=a{b}^x$ where $a\ne 0,b\ne 1\text{ and }b>0$.

Step 3. Satisfying the given points in exponential function.

On satisfying the points $\left(0,-18\right)\text{ and }\left(-2,-2\right)$ in exponential function $y=a{b}^x$.

$\begin{array}{c}y=a{b}^x\mathrm{....}\left(1\right)\\ \left(-18\right)=a{b}^{\left(0\right)}\mathrm{....}\left(\text{For the point }\left(0,-18\right)\right)\\ a=-\mathrm{18....}\left(2\right)\end{array}$

Similarly,

$\begin{array}{c}y=a{b}^x\\ \left(-2\right)=a{b}^{\left(-2\right)}\mathrm{....}\left(\text{For the point }\left(-2,-2\right)\right)\\ -2=\frac{a}{b^2}\\ -2=\frac{\left(-18\right)}{b^2}\mathrm{....}\left(\begin{array}{l}\text{From }\left(2\right)\\ a=-18\end{array}\right)\\ b^2=9\\ b=\mathrm{3....}\left(\because b>0\right)\end{array}$

Plugging $a=-18,b=3$ in (1) gives $y=-18{\left(3\right)}^x$ which is the required exponential function passing through the given points $\left(0,-18\right)\text{ and }\left(-2,-2\right)$.

Step 4. Conclusion.

The exponential function passing through the given points is $y=-18{\left(3\right)}^x$.