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 \(\begin{align*} \boldsymbol {y=ka^x} \end{align*}\)

• 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.

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,   

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

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

Calculate the rest by using the above method.

Therefore, we get the following table,

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

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

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.

(a) 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 bacterias introduced at the start is \(30\).

(b) 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

\(p =84.9\)

(c) Using a scale of \(2\,cm\), to represent \(1\,unit\), draw a horizontal t-axis for \(0 ≤ t ≤ 4\).

     Using a scale of \(1\,cm\), to represent \(50\,unit\), 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.

(d) 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.

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

Conclusion

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