Determining the Rule for a Parabola Revision Notes for VCE SSCE Mathematical Methods

Determining the Rule for a Parabola

Introduction

Finding the equation of a parabola from information about its graph is a fundamental skill in working with quadratic functions. The key is identifying what information you have available and choosing the most appropriate form to use.

infoNote

The strategy you choose depends entirely on what information is given. Before starting any problem, always identify what you know: Are there x-intercepts? Do you know the turning point? Or just three random points?

The three useful forms

There are three main forms you can use to find the rule of a quadratic function, depending on what information you have:

Form 1: Factored form

y = a(x - e)(x - f)

Use this form when you know the two x-axis intercepts and the coordinates of one other point on the parabola.

In this form:

$e$ and $f$ are the x-intercepts (where the parabola crosses the x-axis)
$a$ is a coefficient that determines the shape and direction of the parabola

Form 2: Vertex form

y = a(x - h)^2 + k

Use this form when you know the coordinates of the turning point (vertex) and one other point on the parabola.

In this form:

$(h, k)$ are the coordinates of the turning point
$a$ is a coefficient that determines the shape and direction of the parabola

Form 3: General polynomial form

y = ax^2 + bx + c

Use this form when you know the coordinates of three points on the parabola.

In this form:

$a$ , $b$ , and $c$ are coefficients you need to find
$c$ is the y-intercept (where the parabola crosses the y-axis)

chatImportant

Choosing the right form is crucial for efficiency:

Two x-intercepts known → Use Factored form
Turning point known → Use Vertex form
Three random points known → Use General form

Choosing the wrong form can make the problem much harder than it needs to be!

Worked examples

Worked example: Finding the rule using x-intercepts

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Worked Example: Using X-Intercepts

A parabola has x-axis intercepts at $-3$ and $4$ , and it goes through the point $(1, 24)$ . Find the rule for this parabola.

Solution:

Since we know two x-axis intercepts, we use the factored form $y = a(x - e)(x - f)$ .

y = a(x + 3)(x - 4)

Now substitute the point $(1, 24)$ to find $a$ :

When $x = 1$ , $y = 24$ . Thus:

24 = a(1 + 3)(1 - 4)

24 = a(4)(-3)

24 = -12a

\therefore a = -2

The rule is $y = -2(x + 3)(x - 4)$ .

Explanation: We use the factored form because two x-axis intercepts are given. The x-intercept at $-3$ gives us the factor $(x + 3)$ , and the x-intercept at $4$ gives us the factor $(x - 4)$ . By substituting the additional point, we can solve for the coefficient $a$ .

infoNote

Notice that the x-intercept at $-3$ becomes the factor $(x + 3)$ , not $(x - 3)$ . The general pattern is: if the x-intercept is at $e$ , the factor is $(x - e)$ .

Worked example: Finding the rule using the turning point

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Worked Example: Using the Turning Point

The coordinates of the turning point of a parabola are $(2, 6)$ , and the parabola goes through the point $(3, 3)$ . Find the rule for this parabola.

Solution:

Since we know the turning point and one other point, we use the vertex form $y = a(x - h)^2 + k$ .

y = a(x - 2)^2 + 6

Now substitute the point $(3, 3)$ to find $a$ :

When $x = 3$ , $y = 3$ . Thus:

3 = a(3 - 2)^2 + 6

3 = a(1)^2 + 6

3 = a + 6

\therefore a = -3

The rule is $y = -3(x - 2)^2 + 6$ .

Explanation: We use the vertex form because the coordinates of the turning point are given. The turning point $(2, 6)$ tells us that $h = 2$ and $k = 6$ . By substituting the additional point into the equation, we can solve for $a$ . The negative value of $a$ indicates that this parabola opens downward.

Worked example: Finding the rule using three points

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Worked Example: Using Three Points

A parabola passes through the points $(1, 4)$ , $(0, 5)$ , and $(-1, 10)$ . Find the rule for this parabola.

Solution:

Since we know three points on the parabola, we use the general polynomial form $y = ax^2 + bx + c$ .

We substitute each point to create three equations:

When $x = 1$ , $y = 4$ :

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

When $x = 0$ , $y = 5$ :

5 = c \text{ ... (2)}

When $x = -1$ , $y = 10$ :

10 = a - b + c \text{ ... (3)}

Now we solve these simultaneous equations. From equation (2), we know that $c = 5$ .

Substitute $c = 5$ into equations (1) and (3):

4 = a + b + 5

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

10 = a - b + 5

5 = a - b \text{ ... (3')}

Now we can add equations (1') and (3'):

-1 + 5 = (a + b) + (a - b)

4 = 2a

\therefore a = 2

Substitute $a = 2$ into equation (1'):

-1 = 2 + b

\therefore b = -3

The rule is $y = 2x^2 - 3x + 5$ .

Explanation: When we have three points but no clear pattern like x-intercepts or a turning point, we use the general polynomial form. This gives us three equations with three unknowns. By using substitution and elimination methods, we can solve for all three coefficients $a$ , $b$ , and $c$ .

infoNote

The point $(0, 5)$ immediately gives us the y-intercept, which is the value of $c$ . This simplifies our work significantly! Always look for points where $x = 0$ as they make calculations easier.

Worked example: Finding equations from graphs

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Worked Example: Finding Equations from Graphs

Find the equation of each of the following parabolas:

Solution:

Part a:

This parabola has its vertex at the origin, so it is of the form $y = ax^2$ .

We can see that the point $(2, 5)$ is on the parabola. Substituting:

5 = a(2)^2

5 = 4a

\therefore a = \frac{5}{4}

The rule is $y = \frac{5}{4}x^2$ .

Part b:

This parabola is symmetric about the y-axis, so it is of the form $y = ax^2 + c$ .

From the graph, the y-intercept is at $(0, 3)$ , so $c = 3$ .

The point $(-3, 1)$ is on the parabola:

1 = a(-3)^2 + 3

1 = 9a + 3

-2 = 9a

\therefore a = -\frac{2}{9}

The rule is $y = -\frac{2}{9}x^2 + 3$ .

Part c:

This parabola has x-intercepts at $0$ and $3$ , so it is of the form $y = ax(x - 3)$ .

The point $(-1, 8)$ is on the parabola:

8 = a(-1)(-1 - 3)

8 = a(-1)(-4)

8 = 4a

\therefore a = 2

The rule is $y = 2x(x - 3)$ .

Part d:

This parabola is of the general form $y = ax^2 + bx + c$ .

From the graph, the y-intercept is $2$ , so $c = 2$ .

The parabola appears to have x-intercepts at $-1$ and $1$ . We can verify using these points:

At $(-1, 0)$ :

0 = a(-1)^2 + b(-1) + 2

0 = a - b + 2 \text{ ... (1)}

At $(1, 2)$ :

Wait, the maximum appears to be at approximately $x = 1$ with $y = 2$ . Let me use visible points instead.

From the graph, we can see the parabola goes through $(-1, 0)$ and the maximum is at approximately $(1, 2)$ , with y-intercept at $2$ .

At $(-1, 0)$ :

0 = a - b + 2 \text{ ... (1)}

At $(1, 2)$ :

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

Adding equations (1) and (2):

0 + 2 = (a - b + 2) + (a + b + 2)

2 = 2a + 4

2a = -2

\therefore a = -1

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

0 = -1 - b + 2

0 = 1 - b

\therefore b = 1

The rule is $y = -x^2 + x + 2$ .

chatImportant

Key strategies for finding rules from graphs:

Look for x-intercepts first - if you can identify two of them, use the factored form
Check if the vertex is at a special position (like the origin) - this can simplify your equation
Look for symmetry about the y-axis - this means $b = 0$ in the general form
Identify the y-intercept - this gives you the value of $c$ in the general form
The sign of $a$ determines whether the parabola opens upward (positive) or downward (negative)

Remember!

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Key Points to Remember:

Choose the right form based on available information: Factored form for x-intercepts, vertex form for the turning point, and general form for three random points
The coefficient $a$ determines the parabola's shape: Positive $a$ means the parabola opens upward, negative $a$ means it opens downward, and the magnitude of $a$ affects how wide or narrow the parabola is
Systematic substitution is key: Always substitute known point coordinates carefully into your chosen form and solve for the unknown coefficient(s)
When using the general form with three points: Set up three equations and use substitution or elimination to solve the simultaneous equations systematically
Read graphs carefully: Identify all key features including intercepts, turning points, and any points with clear coordinates before choosing your approach

Determining the Rule for a Parabola (VCE SSCE Mathematical Methods): Revision Notes