Determining the Rule for a Function From Its Graph Revision Notes for VCE SSCE Mathematical Methods

Determining the Rule for a Function From Its Graph

Introduction

When we have enough information about a curve, we can work out its equation. This is called determining the rule of the function. The information we need typically comes from knowing specific points that lie on the curve, along with understanding what type of function it is (such as a hyperbola or square root function).

This is an important skill because it allows us to move from a visual representation of a function to its algebraic form. In exams, you might be given a graph with some key features marked, and you'll need to find the equation.

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This skill bridges the gap between graphs and equations. Once you can determine a rule from a graph, you can make predictions, find exact values, and work with the function algebraically.

Hyperbolas of the form $y = \frac{a}{x} + b$

A hyperbola is a curve with two branches and two asymptotes. When a hyperbola has the form $y = \frac{a}{x} + b$ , we need to find the values of the parameters $a$ and $b$ .

To find these two unknown values, we need two pieces of information. Usually, this means we need the coordinates of two points that lie on the curve. By substituting each point into the equation, we create two equations that we can solve simultaneously.

Here's what a typical hyperbola looks like:

This hyperbola has a vertical asymptote at $x = 1$ and a horizontal asymptote at $y = 2$ . The asymptotes give us clues about the transformed equation. The curve passes through the point $(0, 1)$ .

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Asymptotes provide valuable information

The horizontal asymptote tells you the value of $b$ in the equation $y = \frac{a}{x} + b$ . If the horizontal asymptote is at $y = 2$ , then $b = 2$ .

Sometimes the hyperbola equation might be written in a slightly different form, such as $y = \frac{A}{x + b} + B$ . The process for finding the parameters is the same - substitute known points and solve the resulting equations.

Square root functions

Square root functions can also be transformed and we may need to determine their rules. The simplest form is $y = a\sqrt{x}$ , which is a dilation of the basic square root function.

More complex forms include horizontal and vertical translations, such as:

y = a\sqrt{x - h} + b

where:

$a$ affects the vertical stretch or compression
$h$ represents a horizontal shift
$b$ represents a vertical shift

For the basic form $y = a\sqrt{x}$ , we only need one point (other than the origin) to find the value of $a$ . However, for transformed versions with more parameters, we need more points. Generally, we need as many equations as we have unknown parameters.

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Match equations to unknowns

You need as many equations as you have unknown parameters. Two unknowns? You need two equations. Three unknowns? You need three equations. Always check you have enough information before starting!

Method: Using simultaneous equations

The standard method for determining the rule of a function follows these steps:

Identify the general form of the equation based on the type of curve
Substitute the coordinates of each known point into the general equation
Create a system of simultaneous equations (one equation for each point)
Solve the system using elimination or substitution
Write the final equation with the values you've found

The key principle is: you need as many equations as you have unknown parameters. Two unknowns? You need two equations. Three unknowns? You need three equations.

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Substitute first, solve second

Always substitute all the coordinates to create your equations first. Only then should you solve the simultaneous equations. This systematic approach reduces errors.

Worked example: Hyperbola

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Worked Example: Finding the equation of a hyperbola

The points $(1, 5)$ and $(4, 2)$ lie on a curve with equation $y = \frac{a}{x} + b$ . Find the values of $a$ and $b$ .

Step 1: Substitute the first point

When $x = 1$ , $y = 5$ , so substituting into the equation:

5 = a + b \text{ ... (1)}

Step 2: Substitute the second point

When $x = 4$ , $y = 2$ , so substituting into the equation:

2 = \frac{a}{4} + b \text{ ... (2)}

Step 3: Solve by elimination

Now we subtract equation (2) from equation (1):

5 - 2 = a + b - \frac{a}{4} - b

3 = a - \frac{a}{4}

3 = \frac{4a - a}{4}

3 = \frac{3a}{4}

Therefore, $a = 4$

Step 4: Find the second unknown

Substituting $a = 4$ into equation (1):

5 = 4 + b

Therefore, $b = 1$

Answer: The equation of the curve is y = 4/x + 1.

Worked example: Square root function

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Worked Example: Finding the equation of a square root function

The points $(2, 1)$ and $(10, 6)$ lie on a curve with equation $y = a\sqrt{x - 1} + b$ . Find the values of $a$ and $b$ .

Step 1: Substitute the first point

When $x = 2$ , $y = 1$ , so substituting into the equation:

1 = a\sqrt{2 - 1} + b

1 = a\sqrt{1} + b

1 = a + b \text{ ... (1)}

Step 2: Substitute the second point

When $x = 10$ , $y = 6$ , so substituting into the equation:

6 = a\sqrt{10 - 1} + b

6 = a\sqrt{9} + b

6 = 3a + b \text{ ... (2)}

Step 3: Solve by elimination

Subtracting equation (1) from equation (2):

6 - 1 = 3a + b - a - b

5 = 2a

Therefore, $a = \frac{5}{2}$

Step 4: Find the second unknown

Substituting into equation (1) to find $b$ :

1 = \frac{5}{2} + b

b = 1 - \frac{5}{2}

b = -\frac{3}{2}

Answer: The equation of the curve is y = (5/2)√(x - 1) - 3/2.

Remember!

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Key Points to Remember:

You need as many equations as you have unknown parameters. Two points give you two equations.
Always substitute the coordinates carefully into the general form of the equation.
Use simultaneous equations techniques (elimination or substitution) to solve for the unknowns.
For hyperbolas in the form $y = \frac{a}{x} + b$ , the horizontal asymptote is at $y = b$ .
Check your answer makes sense by substituting one of the original points back into your final equation.

Determining the Rule for a Function From Its Graph (VCE SSCE Mathematical Methods): Revision Notes