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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > Unit 6

Lesson 4: Passport to Advanced Math: lessons by skill- Solving quadratic equations | Lesson
- Interpreting nonlinear expressions | Lesson
- Quadratic and exponential word problems | Lesson
- Manipulating quadratic and exponential expressions | Lesson
- Radicals and rational exponents | Lesson
- Radical and rational equations | Lesson
- Operations with rational expressions | Lesson
- Operations with polynomials | Lesson
- Polynomial factors and graphs | Lesson
- Graphing quadratic functions | Lesson
- Graphing exponential functions | Lesson
- Linear and quadratic systems | Lesson
- Structure in expressions | Lesson
- Isolating quantities | Lesson
- Function Notation | Lesson

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# Graphing exponential functions | Lesson

## What are exponential functions, and how frequently do they appear on the test?

**Note:**On your official SAT, you might not see a question about graphing exponential functions at all! At most, you'll see

**1 question**.

If you haven’t already mastered more frequently tested SAT skills, you may want to save this topic for later.

In an $f(x)={2}^{x}+1$ is an exponential function, because $x$ is an exponent of the base $2$ .

**exponential function**, the output of the function is based on an expression in which the input is in the exponent. For example,The graphs of exponential functions are

**nonlinear**—because their slopes are always changing, they look like**curves**, not straight lines:**You can learn anything. Let's do this!**

## How do I graph exponential functions, and what are their features?

### Graphing exponential growth & decay

### Using points to sketch an exponential graph

The best way to graph exponential functions is to find a few points on the graph and to sketch the graph based on these points.

To find a point on the graph, select an input value and calculate the output value. For example, for the function $f(x)={2}^{x}+1$ , if we want to find the $y$ -value when $x=1$ , we can evaluate $f(1)$ :

Since $f(1)=3$ , the point $(1,3)$ is a point on the graph.

We need to use the points to help us identify three important features of the graph:

- What is the
-intercept?$y$ - Is the slope of the graph positive or negative?
- What happens to the value of
as the value of$y$ becomes very large?$x$

#### The $y$ -intercept

Not only is the $y$ -intercept the easiest feature to identify, it also helps you figure out the rest of the features.

To find the $y$ -intercept, evaluate the function at $x=0$ .

For example, the $y$ -intercept of the graph of $f(x)={2}^{x}+1$ is:

#### The slope

An exponential function is either always increasing or always decreasing. If you have already evaluated $f(0)$ , try evaluating $f(1)$ .

- If
, then the slope of the graph is positive.$f(1)>f(0)$ - If
, then the slope of the graph is negative.$f(1)<f(0)$

For $f(x)={2}^{x}+1$ , since $f(0)=2$ and $f(1)=3$ , we can conclude that the slope of the graph is positive because $3>2$ .

#### The end behavior

End behavior is just another term for what happens to the value of $y$ as $x$ becomes very large in both the positive and negative directions. For the graph of an exponential function, the value of $y$ will always grow to positive or negative infinity on one end and approach, but not reach, a horizontal line on the other. The horizontal line that the graph approaches but never reaches is called the

**horizontal asymptote**.For $f(x)={2}^{x}+1$ :

- As
increases,$x$ becomes very large. The value of$f(x)$ on the right end of the graph approaches infinity.$y$ - As
decreases,$x$ becomes closer and closer to$f(x)$ , but it's always slightly larger than$1$ . The value of$1$ on the left end of the graph approaches, but never reaches,$y$ .$1$

#### Putting it all together

With the help of a few more points, $(-2,1.25)$ , $(-1,1.5)$ , and $(2,5)$ , we can sketch the graph of $f(x)={2}^{x}+1$ .

**Note:**if you're graphing by hand, it's more important to recognize that the value of

*exactly right*! You can use the points you identified to establish a trend and sketch out the curve.

To graph an exponential function:

- Evaluate the function at various values of
—start with$x$ ,$-1$ , and$0$ . Find additional points on the graph if necessary.$1$ - Use the points from Step 1 to sketch a curve, establishing the
-intercept and the direction of the slope.$y$ - Extend the curve on both ends. One end will approach a horizontal asymptote, and the other will approach positive or negative infinity along the
-axis.$y$

**Example:**Graph

### Try it!

## How do I identify features of exponential graphs from exponential functions?

### Graphs of exponential growth

### Identifying features of graphs from functions

#### The basic exponential function

Let's start with the basics!

The most basic exponential function has a base and an exponent:

Let's consider the case where $b$ is a positive real number:

- If
, then the slope of the graph is positive, and the graph shows$b>1$ **exponential growth**. As increases, the value of$x$ approaches infinity. As$y$ decreases, the value of$x$ approaches$y$ .$0$ - If
, then the slope of the graph is negative, and the graph shows$0<b<1$ **exponential decay**. In this case, as increases, the value of$x$ approaches$y$ . As$0$ decreases, the value of$x$ approaches infinity.$y$ - For all values of
, the$b$ -intercept is$y$ .$1$

The graphs of ${f(x)={1.5}^{x}}$ and ${f(x)={\left({\displaystyle \frac{2}{3}}\right)}^{x}}$ are shown below.

#### How do we shift the horizontal asymptote?

The $y$ -value of every exponential graph approaches positive or negative infinity on one end and a constant on the other. We can change the constant value $y$ approaches by introducing a constant term to the function:

- For
, the value of$f(x)={b}^{x}$ approaches infinity on one end and the constant$y$ on the other.$0$ - For
, the value of$f(x)={b}^{x}+d$ approaches infinity on one end and$y$ on the other.$d$

The graphs of ${f(x)={1.5}^{x}}$ and ${f(x)={1.5}^{x}+2}$ are shown below.

- For
, as${f(x)={1.5}^{x}}$ decreases, the value of$x$ approaches$y$ .${0}$ - For
, as${f(x)={1.5}^{x}+2}$ decreases, the value of$x$ approaches$y$ .${2}$

#### How do we shift the $y$ -intercept?

We can change the $y$ -intercept of the graph either by introducing a constant term (as above) or introducing a coefficient for the exponential term:

- For
, the$f(x)={b}^{x}+d$ -intercept is$y$ .$1+d$ - For
, the$f(x)=a\cdot {b}^{x}$ -intercept is$y$ . In this form,$a\cdot 1=a$ is also called the$a$ **initial value**. - For
, the$f(x)=a\cdot {b}^{x}+d$ -intercept is$y$ .$a+d$

The graphs of ${f(x)={1.5}^{x}+2}$ , ${f(x)=2\cdot {1.5}^{x}}$ , and ${f(x)=2\cdot {1.5}^{x}+2}$ are shown below.

- For
, the${f(x)={1.5}^{x}+2}$ -intercept is$y$ .${1+2=3}$ - For
, the${f(x)=2\cdot {1.5}^{x}}$ -intercept is$y$ .${2\cdot 1=2}$ - For
, the${f(x)=2\cdot {1.5}^{x}+2}$ -intercept is$y$ .${2+2=4}$

### Try it!

## You turn!

## Things to remember

For $f(x)={b}^{x}$ , where $b$ is a positive real number:

- If
, then the slope of the graph is positive, and the graph shows$b>1$ **exponential growth**. As increases, the value of$x$ approaches infinity. As$y$ decreases, the value of$x$ approaches$y$ .$0$ - If
, then the slope of the graph is negative, and the graph shows$0<b<1$ **exponential decay**. In this case, as increases, the value of$x$ approaches$y$ . As$0$ decreases, the value of$x$ approaches infinity.$y$ - For all values of
, the$b$ -intercept is$y$ .$1$

To shift the horizontal asymptote:

- For
, the value of$f(x)={b}^{x}$ approaches infinity on one end and the constant$y$ on the other.$0$ - For
, the value of$f(x)={b}^{x}+d$ approaches infinity on one end and$y$ on the other.$d$

To shift the $y$ -intercept:

- For
, the$f(x)={b}^{x}+d$ -intercept is$y$ .$1+d$ - For
, the$f(x)=a\cdot {b}^{x}$ -intercept is$y$ . In this form,$a\cdot 1=a$ is also called the$a$ **initial value**. - For
, the$f(x)=a\cdot {b}^{x}+d$ -intercept is$y$ .$a+d$

## Want to join the conversation?

- How do you answer the question with only the base and x??(2 votes)
- how do you get the equation of an exponential graph(2 votes)
- how would you graph a number if the x exponet is a diffrent number like negative 3 like for ex: f(X)= 2(3)^x-3 +2 ??(0 votes)
- Hi Angelina,

If I understand your equation correctly, it is 6 to the power of (x-3) plus 2. In other words;

f(x) = 6^(x-3) + 2.

To graph this you would do the same process as the other equations. Plug in a x value, and solve for y.

e.g. x = 4. y = 6^(4 - 3) + 2

y = 6^1 + 2

y = 6 + 2

y = 8

Lastly, if the x value is less than three, then you'll have a negative exponent. This may cause some confusion but don't be afraid as it's easier than it may seem. When a number is to the power of a negative number, it is simply 1 / x^n. Heres an example:

2^(-4)

= 1 / (2^4)

= 1 / 16

I believe there are other khan academy lessons which show this concept.

So using this, we can solve your equation when x is less than 3.

e.g. x = 1

y = 6^(1-3) + 2

y = 6^(-2) + 2

y = (1 / 6^2) + 2

y = (1 / 36) + 2

y = ((1 + 72) / 36)

y = 73 / 36

I hope this helps! :)(4 votes)