Question

Which function is graphed below?

Answer 1

The general basic exponential function is of the form,

$y=na^{kx} ---(1)$

The function has a $y$ intercept of $(0,3)$

This point must satisfy equation (1). So let us do the substitution to obtain,

$3=na^{k(0)}$

$\Rightarrow 3=n \times a^{0}$

$\Rightarrow 3=n \times 1$

$\Rightarrow n=3$

Now our equation 1 becomes

$y=3a^{kx} ---(2)$

Also note that the graph passes through, $(1,1)$, hence must also satisfy its equation. Substituting this point into equation (2), gives us

$1=3a^{k(1)}$

$\Rightarrow \frac{1}{3}=a^{k}$

Equation (2) can be rewritten as;

$y=3(a^{k})^x ---(3)$

Now let us substitute the value of ${a}^{k}=\frac{1}{3}$ into equation (3) to obtain,

$y=3(\frac{1}{3})^x$

Hence the correct answer is option B

Answer 2

The graph illustrates an exponential function.

The function of the graph is: $\mathbf{y = 3(\frac 13)^x}$

An exponential function is represented as:

$\mathbf{y = ab^x}$

From the graph, we have the following points

$\mathbf{(x,y) = (1,1)}$

$\mathbf{(x,y) = (0,3)}$

Substitute $\mathbf{(x,y) = (0,3)}$ in $\mathbf{y = ab^x}$

$\mathbf{3 = ab^0}$

$\mathbf{3 = a\times 1}$

$\mathbf{3 = a}$

Rewrite as:

$\mathbf{a = 3}$

Substitute $\mathbf{(x,y) = (1,1)}$ in $\mathbf{y = ab^x}$

$\mathbf{1 = ab^1}$

$\mathbf{1 = ab}$

Substitute $\mathbf{a = 3}$

$\mathbf{1 = 3b}$

Divide both sides by 3

$\mathbf{b = \frac 13}$

Substitute $\mathbf{a = 3}$ and $\mathbf{b = \frac 13}$ in $\mathbf{y = ab^x}$

$\mathbf{y = 3(\frac 13)^x}$

Hence, the exponential function is: $\mathbf{y = 3(\frac 13)^x}$

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