Trigonometric and Exponential Graphs with Worksheet

Trigonometric and Exponential Graphs

In mathematics, graphs are a visual representation of a function or relationship between two variables.
Trigonometric and exponential functions are two important types of functions in mathematics that are commonly plotted on graphs.

In this article, we will discuss:

What are Trigonometric Ratios?
What is an Exponent?
Steps for plotting Trigonometric Graphs (Sine, Cosine, Tan) and Exponential Graphs

Here is one more link to practice a few extra questions: Maths Genie Trigonometric and Exponential Graphs Questions

What are Trigonometric Ratios?

Trigonometric ratios are ratios that relate the sides of a right triangle to its angles.
The three main trigonometric ratios are sine, cosine, and tangent, commonly abbreviated as sin, cos, and tan, respectively.
These ratios are used in many applications, such as finding the height of a building, the distance between two points, and the angle of a slope.

Steps for plotting Trigonometric Graphs (Sine, Cosine, Tan)

To plot a trigonometric graph, we need to first understand the shape of the function.
The sine and cosine functions are periodic functions that oscillate between 1 and -1.
The tangent function has asymptotes at regular intervals.

Here are the steps to plot a trigonometric graph:

Step #1: Determine the period of the function by dividing 2π by the coefficient of x.
Step #2: Determine the amplitude of the function, which is the maximum value of the function.
Step #3: Determine the phase shift of the function, which is the horizontal displacement of the function from its original position.
Step #4: Plot the graph by plotting key points using the period, amplitude, and phase shift.

Solved Example:

Question 1: y = 3sin(2x)

Solution:

Step #1: The period of the function is 2π/2 = π.

Step #2: The amplitude of the function is 3.

Step #3: There is no phase shift in this function.

Step #4: To plot the graph, we can start by finding the key points for one period of the function.
- The maximum value of the function is 3 and occurs at π/2 and 3π/2.
- The minimum value of the function is -3 and occurs at π and 2π.
- We can then use these key points to sketch the graph

Question 2: y = 2cos (3x – π/4)

Solution:

Step #1: The period of the function is 2π/3.

Step #2: The amplitude of the function is 2.

Step #3: The phase shift of the function is π/12 units to the right.

Step #4: To plot the graph, we can find the key points for one period of the function.
- The maximum value of the function is 2 and occurs at π/3 and 5π/3.
- The minimum value of the function is -2 and occurs at 2π/3 and 4π/3.
- We can then shift the graph π/12 units to the right and sketch the graph

What is an Exponent?

An exponent is a mathematical notation that indicates how many times a number, called the base, should be multiplied by itself.
For example, 2³ means 2 multiplied by itself 3 times, or 2 x 2 x 2 = 8.
Exponents are used in many mathematical formulas, such as the compound interest formula and the formula for exponential growth.

Steps for plotting Exponential Graphs

To plot an exponential graph, we need to first understand the shape of the function.
An exponential function has a curve that starts at a certain point and either increases or decreases rapidly.

Here are the steps to plot an exponential graph:

Step #1: Determine the base of the function, which is the number that is being raised to power.
Step #2: Determine the y-intercept of the function, which is the value of the function when x = 0.
Step #3: Determine whether the function is increasing or decreasing by checking the sign of the base.
Step #4: Plot the graph by plotting key points using the base and y-intercept.

Solved Example

Question 1: y = 2^x

Solution:

- Base = 2
- Y-intercept = 1
- Increasing function

To plot this graph, we can choose some key points:

- When x = 0, y = 1
- When x = 1, y = 2
- When x = -1, y = 1/2

By plotting these points on a graph and connecting them, we obtain a rising exponential curve that asymptotically approaches the x-axis without intersecting it.

Question 2: y = 0.5^x

Solution:

- Base = 0.5
- Y-intercept = 1
- Decreasing function

To plot this graph, we can choose some key points:

- When x = 0, y = 1
- When x = 1, y = 0.5
- When x = -1, y = 2

By plotting these points on a graph and connecting them, we obtain a rising exponential curve that asymptotically approaches the x-axis without intersecting it.

Question 3: y = 3^x – 2

Solution:

- Base = 3
- Y-intercept = -1
- Increasing function

To plot this graph, we can choose some key points:

- When x = 0, y = 2
- When x = 1, y = 3
- When x = -1, y = 1/3

Plotting these points on a graph and connecting them, we get an increasing exponential curve that intersects the y-axis at -1.

Conclusion

Trigonometric and exponential graphs provide valuable insights into mathematical functions and relationships, making them indispensable tools for comprehension.
By grasping the shape and characteristics of these graphs, we gain the ability to solve a multitude of real-world problems in diverse fields including engineering, physics, and economics.
These graphs offer a visual representation of complex mathematical concepts, enabling us to analyze and interpret data with greater clarity and precision.
The understanding of trigonometric and exponential graphs equips us with practical problem-solving skills, empowering us to apply mathematical principles to real-life scenarios.
Incorporating trigonometric and exponential graphs into our toolkit expands our capacity to innovate, optimize processes, and make informed decisions in various professional domains.