Investment Graphs Revision Notes for HSC SSCE Mathematics Standard

Investment Graphs

Simple interest graphs

When you invest money with simple interest, the interest earned grows at a constant rate. This means that each year (or time period), you earn exactly the same amount of interest. When we graph this relationship, it creates a straight line.

To create a simple interest graph, you place the time period ( $n$ ) on the horizontal axis and the interest earned ( $I$ ) on the vertical axis. Because the interest increases by the same amount each period, the graph will be linear (a straight line).

infoNote

Simple interest graphs always produce a straight line because the interest earned remains constant for each time period. This makes them easy to predict and calculate!

Steps to construct a simple interest graph

Follow these three steps to draw a simple interest graph:

Construct a table of values for $I$ and $n$ using the simple interest formula
Draw a number plane with $n$ on the horizontal axis and $I$ on the vertical axis, then plot the points
Join the points to make a straight line

lightbulbExample

Worked Example: Constructing a Simple Interest Graph

Question: Draw a graph showing the amount of simple interest earned over a period of 10 years if $1000 is invested at 8% p.a. Use the graph to estimate the interest earned after 7.5 years.

Solution:

Step 1: Write the simple interest formula

$I = Prn$

Step 2: Substitute $P = 1000$ , $r = 0.08$ and $n$ into the formula

$I = 1000 \times 0.08 \times n$

$I = 80n$

Step 3: Draw a table of values for $I$ and $n$

Step 4: Let $n = 0, 2, 4, 6, 8, 10$ and find the interest ( $I$ ) using $I = 80n$

$n$	0	2	4	6	8	10
$I$	0	160	320	480	640	800

Step 5: Draw a number plane with $n$ as the horizontal axis and $I$ as the vertical axis

Step 6: Plot the points $(0, 0)$ , $(2, 160)$ , $(4, 320)$ , $(6, 480)$ , $(8, 640)$ and $(10, 800)$

Step 7: Draw a straight line between the points (simple interest graphs are linear)

Step 8: Read the graph to estimate $I$ when $n = 7.5$

From the graph, $I = 600$ when $n = 7.5$

Step 9: Write the answer in words

Interest after 7.5 years is approximately $600.

Compound interest graphs

When you invest money with compound interest, the interest earned increases at an accelerating rate. This is because you earn interest not only on your original investment but also on the interest that has already been added. Each time period, the amount of interest earned gets larger.

To create a compound interest graph, you place the compounding time periods ( $n$ ) on the horizontal axis and the interest earned ( $I$ ) on the vertical axis. Because the interest increases by different amounts each period, the graph will be an exponential curve, not a straight line.

infoNote

Unlike simple interest, compound interest graphs form a J-shaped curve that gets steeper over time. This reflects the accelerating growth as you earn "interest on interest"!

Steps to construct a compound interest graph

Follow these three steps to draw a compound interest graph:

Construct a table of values for $I$ and $n$ using the compound interest formula
Draw a number plane with $n$ on the horizontal axis and $I$ on the vertical axis, then plot the points
Join the points to make an exponential curve

lightbulbExample

Worked Example: Constructing a Compound Interest Graph

Question: Draw a graph showing the interest earned over a period of 10 years if $1000 is invested at a compound interest rate of 8% p.a. Use the graph to estimate the interest earned after 7.5 years.

Solution:

Step 1: Write the future value and interest earned formulas

$FV = PV(1 + r)^n$

$I = FV - PV$

Step 2: Substitute $PV = 1000$ , $r = 0.08$ and $n$ into the formulas

$FV = 1000 \times (1.08)^n$

$I = FV - PV$

$I = PV(1 + r)^n - PV$

$I = 1000(1.08)^n - 1000$

Step 3: Draw a table of values for $n$ , $FV$ and $I$

Step 4: Let $n = 0, 2, 4, 6, 8, 10$ and find the future value and interest earned

$n$	0	2	4	6	8	10
$FV$	1000	1166	1360	1587	1851	2159
$I$	0	166	360	587	851	1159

Step 5: Draw a number plane with $n$ as the horizontal axis and $I$ as the vertical axis

Step 6: Plot the points $(0, 0)$ , $(2, 166)$ , $(4, 360)$ , $(6, 587)$ , $(8, 851)$ and $(10, 1159)$

Step 7: Draw an exponential curve (not a straight line) between the points

Step 8: Read the graph to estimate $I$ when $n = 7.5$ years

From the graph, $I = 780$ when $n = 7.5$

Step 9: Write the answer in words

Interest after 7.5 years is about $780.

Key differences between the graphs

Comparing the two worked examples above, you can see that:

Both investments started with $1000 at 8% p.a.
After 7.5 years, simple interest earned approximately $600
After 7.5 years, compound interest earned approximately $780
The simple interest graph is a straight line (linear)
The compound interest graph is a curve that gets steeper over time (exponential)
Compound interest earns more because interest is calculated on the growing total, not just the original principal

chatImportant

Common Mistake to Avoid:

Students often confuse which type of graph to draw! Remember:

Simple = Straight (constant increase each period)
Compound = Curve (accelerating increase each period)

Always check the question carefully to identify which type of interest is being used.

Exam tip

chatImportant

When constructing investment graphs, always remember to label your axes clearly. The horizontal axis should show the time period ( $n$ ) and the vertical axis should show the interest earned ( $I$ ). Make sure you know which type of graph to draw: simple interest produces a straight line, while compound interest produces an exponential curve.

Remember!

bookmarkSummary

Key Points to Remember:

Simple interest graphs are linear: They form a straight line because interest increases by the same amount each period
Compound interest graphs are exponential: They form a curve that gets steeper over time because interest is earned on both principal and accumulated interest
Axis labels are important: Always put time period ( $n$ ) on the horizontal axis and interest earned ( $I$ ) on the vertical axis
Use the correct formula: Simple interest uses $I = Prn$ , while compound interest uses $I = PV(1+r)^n - PV$
Compound interest grows faster: Over the same time period at the same interest rate, compound interest will always earn more than simple interest

Investment Graphs (HSC SSCE Mathematics Standard): Revision Notes