chevron icon chevron icon chevron icon chevron icon

Graphs Of Functions And Graphical Solution

In this particular article, we will be learning about graphs of functions and graphical solutions. We will learn about:

  • Graphs of exponential functions.
  • Solving an equation by graphical method.

This article is specifically written to serve the requirements for Secondary 3 Mathematics.

Graphs of Exponential Functions y=kax

  • Exponential means a number to the power \(x\).
    Example: \(2^x \)
     
  • A curve is called the exponential curve, when there is a sudden sharp increase in the curve.
     
  • The curve should not touch the x-axis. It can be above or below the x-axis.

 

There are different types of graphs present. Those are explained below.

  1. \(y=ka^x\), where \(a > 1\), \(k >0\), which means the value of \(k\) will be a positive number.

Graphs Functions And Graphical Solution Image 1

If \(k\) is positive, the curve will be above the x-axis.

Let us consider the equation, \(y = 2^x\)

For example, if the value of \(x\) is as shown below,   

\(x\) \(-1\) \(0\) \(1\) \(2\) \(3\)
\(y\) \(?\) \(?\) \(?\) \(?\) \(?\)

To find \(y\), substitute the value of \(x\) in the equation \(y = 2^x\).

\(\begin{align*} x &=-1 \\[2ex] y &= 2^{-1} \\[2ex] y &= \frac {1}{2} \end{align*}\)

Calculate the rest by using the above method.

Therefore, we get the following table,

\(\displaystyle{x}\) \(-1\) \(0\) \(1\) \(2\) \(3\)
\(\displaystyle{y}\) \(\displaystyle{\frac {1}{2}}\) \(1\) \(2\) \(4\) \(8\)

The value of \(y\) increases sharply. When plotted as the graph, the curve also will increase sharply.

 

  1. \(y=ka^x\), where \(a > 1\), \(k <0\), which means the value of \(k\) will be a negative number.

Graphs Functions And Graphical Solution Image 1

If \(k\) is negative, the curve will be below the x-axis.

 

Question 1:

A number of bacteria are introduced to a culture. The number of bacteria \(y\), in the culture \(t\) hours after they are first introduced is given by the formula \(y = 30 \times 2^t\).

The table shows some corresponding values of \(t\) and \(y\), correct to 3 significant figures.

\(\displaystyle{t}\) \(\displaystyle{0.5 }\) \(\displaystyle{1}\) \(\displaystyle{1.5}\) \(\displaystyle{2}\) \(\displaystyle{2.5}\) \(\displaystyle{3}\) \(\displaystyle{3.5}\) \(\displaystyle{4}\)
\(\displaystyle{y}\) \(\displaystyle{42.4}\) \(\displaystyle{60}\) \(p\) \(\displaystyle{120}\) \(\displaystyle{170}\) \(\displaystyle{240}\) \(\displaystyle{339}\) \(\displaystyle{480}\)

 

  1. How many bacteria are introduced to the culture at the start?

Solution:

Since we have to calculate the bacteria introduced at the start, the time \(t\) is \(0\).

Substitute \(t=0\), in the equation \(y = 30 \times 2^t\).

\(y = 30 \times 2^0\)

The value of any number with power \(0\) is \(1\).

\(\begin{align*} y &= 30 \times 1 \\ \\ y &= 30 \end{align*}\)

The number of bacteria introduced at the start is \(30\).

 

  1. Calculate the value of \(p\).

Solution:

The value of \(t\) corresponding to the \(p\) is \(1.5\) from the table.

Therefore, when \(t=1.5\), \(y=p\).

Substituting the above value in the equation \(y = 30 \times 2^t\).

\(p = 30 \times 2^{1.5}\)

Solving the above equation will give the following answer

\(p =84.9\)

 

  1. Using a scale of \(2\,cm\), to represent \(\mathrm{1\,unit}\), draw a horizontal t-axis for \(0 ≤ t ≤ 4\).
    Using a scale of \(1\,cm\), to represent \(\mathrm{50\,unit}\), draw a horizontal y-axis for \(0 ≤ y ≤ 500\).
    On your axes, draw a graph to show the number of bacteria in the culture for \(0 ≤  t ≤ 4\).

Solution:

The specifications about the graph scale is given in the question. Following those we are drawing a graph. And plotting the values given in the table. Join the plotted value to get the graph curve.

Graphs Functions And Graphical Solution Image 3

 

  1. Use your graph to find how many hours it takes for the number of bacteria to reach \(300\).

Solution:

  • Mark \(300\) in the y- axis.
  • Draw a horizontal line from this point till it meets the curve.
  • Then draw a line towards the t-axis.
  • Note the value of the point it meets at the t-axis.

Graphs Functions And Graphical Solution Image 4

Therefore the hours it takes for the number of bacteria to reach \(300\) is \(\mathrm{3.3 \,hours}\).

Conclusion

In this article, we learned about graphs of exponential functions and how to solve an equation using a graphical method as per the Secondary 3 Mathematics syllabus. 

 

Continue Learning
Further Trigonometry Quadratic Equations And Functions
Linear Inequalities Laws of Indices
Coordinate Geometry Graphs Of Functions And Graphical Solution
Applications Of Trigonometry
Resources - Academic Topics
icon expand icon collapse Primary
icon expand icon collapse Secondary
icon expand icon collapse
Book a free product demo
Suitable for primary & secondary
select dropdown icon
Our Education Consultants will get in touch with you to offer your child a complimentary Strength Analysis.
Book a free product demo
Suitable for primary & secondary
icon close
Default Wrong Input
Get instant access to
our educational content
Start practising and learning.
No Error
arrow down arrow down
No Error
*By submitting your phone number, we have
your permission to contact you regarding
Geniebook. See our Privacy Policy.
Success
Let’s get learning!
Download our educational
resources now.
icon close
Error
Error
Oops! Something went wrong.
Let’s refresh the page!
Claim your free demo today!
Geniebook CTA Illustration Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Geniebook CTA Illustration Geniebook CTA Illustration
close icon
close icon
Turn your child's weaknesses into strengths
Trusted by over 220,000 students.
 
Arrow Down Arrow Down
 
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.