Which Graph Represents an Exponential Function

Which Graph Represents an Exponential Function

An exponential function in Mathematics is a function that has a constant value raised to the power of an argument (unremarkably the abiding “e,” which is approximately equivalent to ii.71828.). There are several real-life applications of the exponent role. Read on to notice them and more about exponential role properties, how to graph exponential functions, formulas, graphs, examples, do problems, and more.

Here is what nosotros volition cover in the article:

  • What is an exponential function? Exponential function formula
  • Exponential part derivative
  • Backdrop of exponential functions
  • Graph exponential functions
  • How to detect an exponential function from a graph?

What is An Exponential Part? Exponential Office Formula

An exponential function has the following form

f (10) = ax.

Here,

“x” stands for a variable and is a real number.

“a” represents a constant and is chosen the office’s base. The base should be greater than 0, i.eastward., a>0, which is not equal to one. The transcendental number “e” is approximately equal to ii.71828 and is mostly used as the base of the exponential function.

Exponential Functions Examples

Some examples of exponential functions are:

  • f(x) = 3x+2
  • f(x) = 5x
  • f(x) = 2e2x
  • f(x) = (1/ 4)x
    = 4-x
  • f(x) = 0.sevenx

Exponential Function Derivative

The derivative of exponential functions can exist given as follows:

d(ex)/dx = eastten

Besides, the exponential function f(10) =due eastten  has a special property. Then, the derivative of the exponential part is the function itself as given beneath:

f ‘(10) = ex = f(x)

Properties of Exponential Functions

Yous must empathize the backdrop of the exponential functions to perform calculations. The key properties are as follows:

  • Rule of product

ax
ay
= aten+y

e.thousand., 52
10 v3
= five2+3
⇒ v5
= 3125

  • Rule of quotient

ax/ay
= aten-y

e.g., 54
x vtwo
= 54-2

⇒ 5two
= 25

  • Power dominion

(ax)y
= axy

e.g. (v2)3
= 52×3

⇒ 56
= fifteen,625

  • Power of a production

axb10=(ab)ten

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⇒ 22
32
 = (2 ten 3)ii

⇒ 62
 = 36

  • Power of fractions

(a/b)10= ax/bten

⇒ (6/2)2
= half-dozentwo
/22

⇒ 36/4 = 9

  • Negative Exponent dominion

a-10= 1/ ax

2-2= 1/ 2ii

= 1/four

  • Zip exponent rule

a=1

eastward.grand. 5=1

Here a> 0 and b>0, ten and y are real numbers.

Example one:  Solve iv

x


= 4


5


Solution: Since the given bases are the same (i.e., 4), we volition equate the powers. Answer: The value of x is 5.
Example ii: Solve 5

1-x


= 5


5


Solution: Since the given bases are the aforementioned (i.east., 5), we volition equate the powers.
1-10 = v
-x = v-1
-10 = 4
x =-4
Answer: The value of x is -4.

Graph Exponential Functions

Which graph represents an exponential function? An exponential function graph is an upward curve, as shown in the post-obit image. This graph is always nonlinear every bit its slopes are always changing.


Hither, x > 1, the value of y = fn(ten) volition increase when we increase the values of (north). Also, the bend volition get steeper equally the exponent increases. The charge per unit of growth volition increase likewise.

How to Graph Exponential Functions?

Now, we will learn how to graph exponential functions. 1 of the all-time ways to graph exponential functions is past finding a few graph points and sketching the graph based on those points.

For finding a point on the graph, we will get-go select an input value. Now, calculate the output value from the input value. For case, for the office
f (x) = two


x

+1.

To find the value of y when x =1, we can use f(one)

f (one) = 21
+ 1

  = iii

So, we have our outset point for the graph now, that is (1, 3).

Using the points on a graph, we can identify the following important features of the graph:

  • y-intercept
  • Whether the slope of the graph positive or negative?
  • How does the value of y alter with an increase in the value of x?
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The y-intercept

The y-intercept of an exponential graph is important as it helps us identify a number of other features. We have to evaluate the function at x = 0, to find the value of the y-intercept.

f(ten) = 210
+ 1

f(0) = two+1

= 1+ 1

= two

Slope

To determine the slope of the graph, we use f (0) and f (1). The slope is either increasing or decreasing. The following two statements will help you determine the slope of the exponential function graph.

  • When f (1) > f (0), and then the graph has a positive slope.
  • When f (one) < f (0), then the graph has a negative slope.

In the above-mentioned graph example —  f (x) = iiten+1, f(1) = three and f(0) = two. Since f (1) > f(0), the slope of the graph is positive.

Finish behavior

The term terminate-behavior refers to the relation between 10 and y. We study what happens to the value of y when x becomes very large in positive or negative directions.

When we graph exponential functions, the value of y grows to positive or negative infinity towards one terminate. Information technology approaches but does not achieve the horizontal line. This horizontal line that the exponential part graph approaches simply fails to achieve is called the horizontal asymptote.

To graph exponential function,  f (x) = 2x+one, nosotros will calculate a few more points

f (-two) = two-2+one

= 1.25

f (-i) = 2-1+1

= 1.five

f (two) = twoii+1

= five

So, the points are (2,5), (-2, one.25), and (-1, 1.5). Now, we tin graph the exponential function.


Steps to Graph Exponential Function

The post-obit steps will help you graph exponential functions easily:

  • Step 1: We volition evaluate the exponential function for different values of x. We volition begin with 10= -i, 0, one, and observe boosted points if required.
  • Pace two: At present, nosotros volition use the points to sketch a graph bend, establishing the direction of the gradient and the y-intercept.
  • Step 3: We will extend the bend on both ends. While one terminate will reach a horizontal asymptote, the other volition approach negative or positive infinity forth the y-axis.
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How to Find Exponential Role from a Graph?

Nosotros tin can discover the exponential role equation from a given graph. It is a multi-step process. Every graph will provide different information depending on its type. We can decipher some information from the given graph itself, and we can then solve for other requirements for the exponential graph equation. Here is a list of the variables we must expect for in the given graph:

a – It volition be given, or we tin solve it by employing algebra.

b – It volition be given, or we tin solve for it using algebra.

c – If we assume x = 0 and ignore c, then the value of y will be equal to the y-intercept. Next, we will count the number of units the y-value is from the y-centrality. This number will give u.s. the value of “c.”

d – We solve it.

yard – It is equal to the horizontal asymptote value.

Let us consider an instance to completely understand the process of how to ding exponential function from a graph.

Instance: Discover exponential function ( y= ab

x

) from the given graph.


Solution:
For this graph solution, we will have to notice the values of a and b.

To solve for a, we will cull a point from the graph that eliminates b, as we don’t know b yet. Then, we will pick up the y-intercept, i.eastward., (0,3).

Since y = ab10
(given)

= 3 = ab

= iii = a1

So, a = 3

To solve for b, nosotros will selection upwardly another point from the graph, say (i,six).

Since y = abx
(given)

= vi = 3b1
( Nosotros will substitute the value of a that nosotros had calculated in the previous stride.)

So, b = 2

To write the concluding equation, we will substitute the values we have calculated in the given equation course.

As, y = ab10
and a = iii, b = 2

The equation will exist y = iii(two)x

Practice Problems

Question 1:
Solve the following equations:

a.62x
= half dozen25

b.24-x
= 25

c.41
= 4 + iv5
– 4

Question 2:
Given the role f(x)= 3x, evaluate each of the following.

a.f(3)

b.f(0)

c.f4/5

d.f(1)

Question iii: Given the role f (ten) = (iv/3)ten

a.Evaluate f (-3)

b.Evaluate f (iv)

Question 4: Sketch the graph for f(x) = e-x
+ two

Question 5: Sketch the graph for y = 5(2)x

Which Graph Represents an Exponential Function

Source: https://www.turito.com/learn/math/exponential-functions-and-their-graphs