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Chapter 10: Q33. (page 528)

Write an exponential function whose graph passes through the points $(0, - 2) and (- 2, - 32)$ .

Short Answer

Expert verified

The exponential function passing through the given points is $y = - 2 {(\frac{1}{4})}^{x}$ .

Step by step solution

01

Step 1. Write down the given information.

The given points are $(0, - 2) and (- 2, - 32)$ .

02

Step 2. Concept used.

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

03

Step 3. Satisfying the given points in exponential function.

On satisfying the points $(0, - 2) and (- 2, - 32)$ in exponential function $y = a b^{x}$ .

$\begin{array}{l} y = a b^{x} .... (1) \\ (鈭� 2) = a b^{(0)} .... (Forthepoint (0, 鈭� 2)) \\ a = 鈭� 2.... (2) \end{array}$

Similarly,

$\begin{array}{l} y = a b^{x} \\ (鈭� 32) = a b^{(鈭� 2)} .... (Forthepoint (鈭� 2, 鈭� 32)) \\ 鈭� 32 = \frac{a}{b^{2}} \\ 鈭� 32 = \frac{(鈭� 2)}{b^{2}} .... (\begin{array}{l} From (2) \\ a = 鈭� 2 \end{array}) \\ b = \frac{1}{4} .... (鈭� b > 0) \end{array}$

Plugging $a = - 2, b = \frac{1}{4}$ in (1) gives $y = - 2 {(\frac{1}{4})}^{x}$ which is the required exponential function passing through the given points.

04

Step 4. Conclusion.

The exponential function passing through the given points is $y = - 2 {(\frac{1}{4})}^{x}$ .

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