How to find the asymptote of an exponential function

Exponents can be variables. Variable exponents obey all the properties of exponents listed in Properties of Exponents.

An exponential function is a function that contains a variable exponent. For example, f (x) = 2x and g(x) = 5ƒ3x are exponential functions. We can graph exponential functions. Here is the graph of f (x) = 2x:

We can translate this graph. For example, we can shift the graph down 3 units and left 5 units. Here is the graph of f (x) = 2x+5 - 3:

We can stretch and shrink the graph vertically by multiplying the output by a constant--see Stretches. For example, f (x) = 3ƒ2x is stretched vertically by a factor of 3:

We can also graph exponential functions with other bases, such as f (x) = 3x and f (x) = 4x. We can think of these graphs as differing from the graph of f (x) = 2x by a horizontal stretch or shrink: when we multiply the input of f (x) = 2x by 2, we get f (x) = 22x = (22)x = 4x. Thus, the graph of f (x) = 4x is shrunk horizontally by a factor of 2 from f (x) = 2x:

The graph of f (x) = ax does not always differ from f (x) = 2x by a rational factor. Thus, it is useful to think of each base individually, and to think of a different base as a horizontal stretch for comparison purposes only.

The graph of an exponential function can also be reflected over the x-axis or the y-axis, and rotated around the origin, as in Heading .

The general form of an exponential function is f (x) = cƒax-h + k, where a is a positive constant and a≠1. a is called the base. The graph has a horizontal asymptote of y = k and passes through the point (h, c + k).

1. Complex Numbers
1. Adding and Subtracting Complex Numbers
2. Conjugate of Complex Numbers
3. Multiplying and Dividing Complex Numbers
2. Quadratic Equations
1. Solving Quadratic Equations with Complex Solutions
2. Number and Nature of Solutions of Quadratic Equations
3. Functions
1. Evaluate a function
2. Domain and Range of a function
3. Operations on functions
4. One-to-one functions
5. Inverse of a function
6. Absolute Value function
7. Even and Odd functions
8. Transformations of functions
9. Piecewise functions
10. Rate of Change of a Function
4. Polynomials Functions
1. Dividing Polynomials
2. Factoring Polynomials
3. Factoring Polynomials Using Special Polynomials
4. Solving Polynomial Equations
5. Solving Polynomial Inequalities
6. Graphing Polynomials Functions
5. Rational Expressions, Equations, Inequalities and Functions
1. Simplifying Rational Expressions
2. Solving Rational Equations
3. Solving Rational Inequalities
4. Graphing Rational Functions
6. Trigonometry and Trigonometric Functions
1. Converting Degrees into Radians and Vice Versa
2. Evaluating Exact Values of Trigonometric Ratios
3. Using Trigonometric Identities to Simplify Trigometric Expressions
4. Solving Trigonometric Equations
5. Graphing Trigonometric Functions
7. Logarithmic and Exponential Functions
1. Simplifying Exponnential Expressions
2. Using Relationship Between Logarithmic and Exponential Expressions
3. Simplifying Logarithmic Expressions
4. Solving Logarithmic and Exponential Equations
5. Graphing Logarithmic and Exponential Functions

Algebra 2 problems with detailed solutions

1. Complex Numbers

Problem 1-1
Let z = 2 - 3 i where i is the imaginary unit. Evaluate z z* , where z* is the conjugate of z , and write the answer in standard form.

Detailed Solution.

Problem 1-2
Evaluate and write in standard form \( \dfrac{1-i}{2-i} \) , where i is the imaginary unit.

Detailed Solution.

2. Quadratic Equations

Problem 2-1
Find all solutions of the equation \( x(x + 3) = - 5 \).

Detailed Solution.

Problem 2-2
Find all values of the parameter m for which the equation \( -2 x^2 + m x = 2 m \) has complex solutions.

Detailed Solution.

3. Functions

Problem 3-1
Let \( f(x) = - x^2 + 3(x - 1) \). Evaluate and simplify \( f(a-1)\).

Detailed Solution.

Problem 3-2
Write, in interval notation, the domain of function \(f\) given by \(f(x) = \sqrt{x^2-16} \).

Detailed Solution.

Problem 3-3
Find and write, in interval notation, the range of function \(f\) given by \(f(x) = - x^2 - 2x + 6 \).

Detailed Solution.

Problem 3-4
Let \(f(x) = \sqrt{x - 2} \) and \(g(x) = x^2 + 2 \); evaluate \( (f_o g)(a - 1) \) for \( a \lt 1 \).

Detailed Solution.

Problem 3-5
Which of the following is a one-to-one function?(There may be more than one answer).
a) \(f(x) = - 2 \)     b) \(g(x) = \ln(x^2 - 1) \)     c) \(h(x) = |x| + 2 \)     d) \(j(x) = 1/x + 2 \)     e) \(k(x) = \sin(x) + 2 \)     f) \(l(x) = ln(x - 1) + 1 \)

Detailed Solution.

Problem 3-6
What is the inverse of function f given by \(f(x) = \dfrac{-x+2}{x-1}\)?

Detailed Solution.

Problem 3-7
Classify the following functions as even, odd or neither.
a) \(f(x) = - x^3 \)     b) \(g(x) = |x|+ 2 \)     c) h(x) = \( \ln(x - 1) \)

Detailed Solution.

Problem 3-8
Function \(f \) has one zero only at \(x = -2\). What is the zero of the function \(2f(2x - 5) \)?

Detailed Solution.

Problem 3-9
Which of the following piecewise functions has the graph shown below?
a) \( f(x) = \begin{cases} x^2 & \text{if} \; x \ge 0 \\ 2 & \text{if} \; -2 \lt x \lt 0\\ - x + 1& \text{if} \; x \le -2 \end{cases} \)     b) \( g(x) = \begin{cases} x^2 & \text{if} \; x \gt 0 \\ 2 & \text{if} \; -2 \lt x \le 0\\ - x + 1& \text{if} \; x \le -2 \end{cases} \)     c) \( h(x) = \begin{cases} x^2 & \text{if} \; x \gt 0 \\ 2 & \text{if} \; -2 \lt x \lt 0\\ - x + 1 & \text{if} \; x \lt -2 \end{cases} \)





Detailed Solution.

Problem 3-10
Calculate the average rate of change of function \( f(x) = \dfrac{1}{x} \) as x changes from \( x = a\) to \( x = a + h \).

Detailed Solution.

4. Polynomials

Problem 4-1
Find the quotient and the remainder of the division \( \dfrac{-x^4+2x^3-x^2+5}{x^2-2} \).

Detailed Solution.

Problem 4-2
Find \( k \) so that the remainder of the division \( \dfrac{4 x^2+2x-3}{2 x + k} \) is equal to \( -1 \)?

Detailed Solution.

Problem 4-3
\( (x - 2) \) is one of the factors of \( p(x) = -2x^4-8x^3+2x^2+32x+24 \). Factor \(p\) completely.

Detailed Solution.

Problem 4-4
Factor \( 16 x^4 - 81 \) completely.

Detailed Solution.

Problem 4-5
Find all solutions to the equation \( (x - 3)(x^2 - 4) = (- x + 3)(x^2 + 2x) \)

Detailed Solution.

Problem 4-6
Solve the inequality \( (x + 2)(x^2-4x-5) \ge (-x - 2)(x+1)(x-3)\)

Detailed Solution.

Problem 4-7
The graph of a polynomial function is shown below. Which of the following functions can possibly have this graph?
a) \( y = -(x+2)^5(x-1)^2 \)     b) \( y = 0.5(x+2)^3(x-1)^2 \)     c) \( y = -0.5(x+2)^3 (x-1)^2 \)     d) \( y = -(x+2)^3(x-1)^2 \)





Detailed Solution.

Problem 4-8
Which of the following graphs could possibly be that of the function f given by \( f(x) = k (x - 1)(x^2 + 4) \) where k is a negative constant? Find k if possible.





Detailed Solution.

5. Rational Expressions, Equations, Inequalities and Functions

   Problem 5-1
   Write as a single rational expression: \( \dfrac{x^2+3x-5}{(x-1)(x+2)} - \dfrac{2}{x+2} - 1 \).

   Detailed Solution.

   Problem 5-2
   Solve the equation: \( \dfrac{- x^2+5}{x-1} = \dfrac{x-2}{x+2} - 4 \).

   Detailed Solution.

   Problem 5-3
   Solve the inequality: \( \dfrac{1}{x-1}+\dfrac{1}{x+1} \ge \dfrac{3}{x^2-1} \).

   Detailed Solution.

   Problem 5-4
   Find the horizontal and vertical asymptotes of the function: \( y = \dfrac{3x^2}{5 x^2 - 2 x - 7} + 2 \).

   Detailed Solution.

   Problem 5-5
   Which of the following rational functions has an oblique asymptote? Find the point of intersection of the oblique asymptote with the function.
   a) \( y = -\dfrac{x-1}{x^2+2} \)     b) \( y = -\dfrac{x^4-1}{x^2+2} \)     c) \( y = -\dfrac{x^3 + 2x ^ 2 -1}{x^2- 2} \)     d) \( y = -\dfrac{x^2-1}{x^2+2} \)

   Detailed Solution.

   Problem 5-6
   Which of the following graphs could be that of function \( f(x) = \dfrac{2x-2}{x-1} \)?

   
   
   

   Detailed Solution.

6. Trigonometry and Trigonometric Functions

Problem 6-1
A rotating wheel completes 1000 rotations per minute. Determine the angular speed of the wheel in radians per second.

Detailed Solution.

Problem 6-2
Determine the exact value of \( sec(-11\pi/3) \).

Detailed Solution.

Problem 6-3
Convert 1200� in radians giving the exact value.

Detailed Solution.

Problem 6-4
Convert \( \dfrac{-7\pi}{9} \) in degrees giving the exact value.

Detailed Solution.

Problem 6-5
What is the range and the period of the the function \( f(x) = -2\sin(-0.5(x - \pi/5)) - 6 \)?

Detailed Solution.

Problem 6-6
Which of the following graphs could be that of function given by: \( y = - \cos(2x - \pi/4) + 2 \)?





Detailed Solution.

Problem 6-7
Find a possible equation of the form \( y = a \sin(b x + c) + d \) for the graph shown below.(there are many possible solutions)





Detailed Solution.

Problem 6-8
Find the smallest positive value of x, in radians, such that \( - 4 \cos (2x - \pi/4) + 1 = 3 \)

Detailed Solution.

Problem 6-9
Simplify the expression: \( \dfrac{\cot(x)\sin(x) + \cos(x) \sin^2(x)+\cos^3(x)}{\cos(x)} \)

Detailed Solution.

7. Logarithmic and Exponential Functions

Problem 7-1
Simplify the expression \( \dfrac{4x^2 y^8}{8 x^3 y^5} \) using positive exponents in the final answer.

Detailed Solution.

Problem 7-2
Evaluate the expression \( \dfrac{3^{1/3} 9^{1/3}}{4^{1/2}} \).

Detailed Solution.

Problem 7-3
Rewrite the expression \( \log_b(2x - 4) = c \) in exponential form.

Detailed Solution.

Problem 7-4
Simplify the expressiomn: \( \log_a(9) \cdot \log_3(a^2) \)

Detailed Solution.

Problem 7-5
Solve the equation \( \log(x + 1) - log(x - 1) = 2 \log(x + 1) \).

Detailed Solution.

Problem 7-6
Solve the equation \( e^{2x} + e^x = 6 \).

Detailed Solution.

Problem 7-7
What is the horizontal asymptote of the graph of \( f(x) = 2 ( - 2 - e^{x-1}) \)?

Detailed Solution.

Problem 7-8
What is the vertical asymptote of the graph of \( f(x) = log(2x - 6) + 3 \)?

Detailed Solution.

Problem 7-9
Match the given functions with the graph shown below?
A) \( y = 2 - 0.5^{2x-1} \)     B) \( y = 0.5^{2x-1} \)     C) \( y = 2 - 0.5^{-2x+1} \)     D) \( y = 0.5^{-2x+1} \)






Detailed Solution.

Problem 7-10
Match the given functions with the graph shown below?
A) \( y = 2+ln(x-2) \)     B) \( y=-log_2(x+1)-1 \)     C) \( y = -ln(-x) \)     D) \( y = y=-log_3(x+1)-1 \)






Detailed Solution.

More References and Links

How do you find the asymptote of an exponential equation?

How do you find the horizontal asymptote of an exponential function?

How do you find the vertical asymptote of an exponential function?