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(Video) Exponential Graph

by HEGARTYMATHS

Here's a video by HEGARTYMATHS going over the graph of an exponential function.

$\large{\text{Summary}}$

The graph of an exponential function never goes to zero but gets very very close. On the other side the numbers begin to grow incredibly fast. This is called "growing exponentially".

Exponential Function Graph

Let's graph $$f(x)=2^x$$ So let's first make a t-table of values. $$\begin{array}{c|l} x&y\\ \hline -2&2^{-2}=\frac1{2^2}=\frac14\\ -1&2^{-1}=\frac12\\ 0&2^0=1\\ 1&2\\ 2&4\\ 3&8 \end{array}$$ Let's plot these points now:

Now we connect them to give is our graph. Notice that the graph keeps approaching $0$ but never reaches it. This means we have a horizontal asymptote at $y=0$.

We can also see $$\begin{align} \text{Domain: }&(-\infty,\infty)\\ \text{Range: }&(0,\infty) \end{align}$$

So we see with exponential functions, that they increase dramatically. As you see each $x$ step you take you actually double the $y$ value.

An example of exponential growth is population size over time. It is easy to think about population size in medieval times and compare it to now and picture this graph!

(Video) Exponential Fraction Graphs

by mathman1024

Here's a video by mathman1024 where he graphs an exponential function where the base is between zero and one.

$\large{\text{Summary}}$

The function he looks at is: $f(x)=\bigg(\dfrac12\bigg)^x$. In this problem our base is $\frac12$ which is between zero and one, so our graph will look at little different. He starts by making a t-table. $$\begin{array}{c|c} x&y\\ \hline -1&2\\ 0&1\\ 1&\frac12 \end{array}$$ Let's plot these points and see what happens.

So it seems as we go to the left now, the points begin growing by $\times 2$. In fact, this graph is the reflection of $f(x)=2^x$ over the $y$-axis.

Just like with $f(x)=2^x$ there's still a horizontal asymptote at $y=0$. Also the domain and range are still the same: $$\begin{align} &\text{Domain: }(-\infty,\infty)\\ &\text{Range: }(0,\infty) \end{align}$$

(Videos) More Complicated Exponential Function Graphs

by mathman1024

In this video by mathman1024 he graphs a more complicated exponential function by applying the different transformation.

$\large{\text{Summary}}$

The function he graphs is: $f(x)=2^{x-2}-5$. The first step is to figure out the base function. For this problem the base function is clearly $2^x$.

Next we notice there's a $-2$ inside the base function: $f(x)=2^{x\color{#8925AE}{-2}}-5$. This is the horizontal transformation. If it's positive it means we move the graph to the left and if it's negative we move the graph to the right. Since this is negative, we move our graph two units to the right.

The final transformation is the $-5$ outside the function: $f(x)=2^{x-2}\color{#8925AE}{-5}$. This is the vertical transformation. If it's positive we move the graph up and if it's negative we move the graph down. Since this is negative, we move the graph five units down.

Graphing Exponential Equations

An exponential equation is a function where the variable is the exponent:

$$y=a^x$$

In this equation, $a$ is a positive real number, and it's called the base. The graph of the equation creates a smooth curve that increases or decreases quickly.

To know if the graph is increasing or decreasing:

• If $a>1$, the graph shows exponential growth, meaning it curves upward as $x$ increases.
• If $0<a<1$, the graph shows exponential decay, meaning it curves downward as $x$ increases.

Every exponential graph passes through the point $(0,1)$ because any number raised to the power of $0$ is $1$. The graph will also have a horizontal asymptote, usually at $y=0$. That means that the curve gets close to the $x$-axis, but never touches it.

The best way to graph an exponential equation is to make a small table of values for $x$, plug them into the equation, and find the matching $y$ values. Then, plot the points on a coordinate plane and connect them with a smooth curve.