Graphs of Exponential Functions

The graph of an exponential function is a strictly increasing or decreasing curve that has a horizontal asymptote. Let's find out what the graph of the basic exponential function $y=a^x$ looks like:

(i) When $a>1,$ the graph strictly increases as $x.$ We know that $a^0=1$ regardless of $a,$ and thus the graph passes through $(0,1).$ Also observe that $\displaystyle{\lim_{x\to-\infty}a^x}=0,$ which implies that the graph has the $x$-axis as its asymptote. Using this information, we can draw the graph as shown below:

Imgur

Observe that the entire graph lies above the $x$-axis, because the range of $y=a^x$ is all positive reals.

(ii) Likewise, for $0<a<1,$ the graph strictly decreases as $x.$ We also have $a^0=1$ and $\displaystyle{\lim_{x\to \infty}a^x}=0,$ which gives the graph shown below:

Imgur

Again, the entire graph lies above the $x$-axis, since the range of $y=a^x$ is all positive reals.

In summary, the properties of the graph of an exponential function $y=a^x$ are as follows:

• The graph passes through $(0,1).$
• When $a>1,$ the graph strictly increases as $x,$ and is concave up.
• When $0<a<1,$ the graph strictly decreases as $x,$ and is concave up.
• The graph lies above the $x$-axis.
• The graph has the $x$-axis as its horizontal asymptote.

Which of the following is the graph of $\displaystyle y=\left(\frac{1}{5}\right)^x?$

exponential

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An exponential function $f(x)=a^x~(a>0, a\ne 1)$ has the following properties:

• Its domain is all real numbers and its codomain is all positive real numbers.
• The graph of $y=f(x)$ passes through the points $(0, 1)$ and $(1, a).$
• The asymptote of $y=f(x)$ is the $x$-axis.
• $f(x)$ is an increasing function for $a>1$ and a decreasing function for $0<a<1.$

Therefore, the graph of $\displaystyle y=\left(\frac{1}{5}\right)^x$ is $(B). \ _\square$

Which of the five graphs in the above problem represents $\displaystyle y=-a^x~(a>1)?$

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By applying the information given in the problem above, we have the following properties of the function $f(x)=-a^x~(a>1):$

• Its domain is all real numbers and its codomain is all negative real numbers.
• The graph of $y=f(x)$ passes through the points $(0, -1)$ and $(1, -a).$
• The asymptote of $y=f(x)$ is the $x$-axis.
• $f(x)$ is a decreasing function.

Therefore, the graph of $\displaystyle y=-a^x~(a>1)$ is $(E). \ _\square$

If the graph below represents $y=3^x,$ what is $b-a?$

y3x

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The graph shows $y=7$ for $x=a,$ which implies $3^a=7. \qquad (1)$
It also shows $y=63$ for $x=b,$ which implies $3^b=63. \qquad (2)$
Taking $(2)\div (1)$ gives

$$\begin{aligned} \frac{3^b}{3^a} &=3^{b-a}\\ &=\frac{63}{7}\\ &=9\\ &=3^2 \\ \Rightarrow b-a&=2. \ _\square \end{aligned}$$

If the graph below represents $y=10^x,$ what is $\displaystyle a+\frac{b}{2}+2c?$

ylog10

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The graph shows $y=2$ for $x=a,$ which implies $10^a=2. \qquad (1)$
Similarly, $y=4$ for $x=b,$ implying $10^b=4 \Rightarrow 10^{\frac{b}{2}}=\left(10^b\right)^{\frac{1}{2}}=4^{\frac{1}{2}}=2. \qquad (2)$
Finally, $y=5$ for $x=c,$ implying $10^c=5 \Rightarrow 10^{2c}=\left(10^c\right)^2=5^2=25. \qquad (3)$

Taking $(1) \times (2) \times (3)$ gives

$$\begin{aligned} 10^a \times 10^{\frac{b}{2}} \times 10^{2c} &=10^{a+\frac{b}{2}+2c}\\ &= 2 \times 2 \times 25\\ &=100\\ &=10^2, \end{aligned}$$

which implies

$$a+\frac{b}{2}+2c=2. \ _\square$$