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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
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