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

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

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

\[\begin{align} 10^a \times 10^{\frac{b}{2}} \times 10^{2c} &=10^{a+\frac{b}{2}+2c}\\ &= 2 \times 2 \times 25\\ &=100\\ &=10^2, \end{align}\]

which implies

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