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Q1 · Recognize quadratic tables by constant second differences and graph the parabola

Learn to recognize quadratic tables by constant second differences and graph the parabola through clear examples and targeted practice.

Ontario Grade 10 Mathematics

Quadratic Relations

Spotting Constant Second Differences in a Table and Drawing the Curve

You already know that a straight-line graph goes up or down by the same amount every step — that is a linear relationship. But what happens when the amount of change is itself changing? This lesson shows you how to catch that pattern in a table of values, name it as a quadratic relationship, and draw the smooth U-shaped curve it produces. No formulas beyond Grade 10 are needed. Everything builds on ideas you already met in Grade 9: plotting points, reading a graph, and calculating differences between numbers.

What you will learn

Identify whether a table of values shows a linear, quadratic, or neither relationship by calculating first and second differences.
Explain in plain words why constant second differences signal a quadratic relationship.
Plot a quadratic table of values on a coordinate grid and sketch the resulting parabola.
Read key features — vertex, axis of symmetry, and direction of opening — directly from a graph.

Grade 9 Bridge: First Differences and Linear Tables

Before looking at something new, recall a tool from Grade 9: the first difference. A first difference is simply the change in the output (

y

) value from one row to the next in a table, assuming the input (

x

) values are equally spaced by the same step every time.

For example, if

y

takes the values CAD 3, 7, 11, 15

as

x increases by

1

each time, the first differences are

7-3=4

11-7=4

, and

15-11=4

. Because every first difference equals

4

, the relationship is linear — its graph is a straight line.

The key rule to carry forward: equal first differences → linear relationship. You will need this contrast to understand what makes a quadratic table different.

A first difference is the change in $y$ between consecutive rows when $x$ increases by a fixed step.
Constant (equal) first differences identify a linear relationship.
Always check that the $x$ values are equally spaced before calculating any differences.

What Is a Second Difference and Why Does It Matter?

A second difference is the change in the first differences — in other words, you subtract consecutive first differences from each other. If the first differences are not constant but the second differences are all equal, the relationship is quadratic.

A quadratic relationship is one where the output is related to the square of the input. Its graph is a smooth, symmetrical U-shaped curve called a parabola. The word 'quadratic' comes from the Latin word for square.

Here is the logic in plain language: a linear relationship grows by the same amount each step (constant first differences). A quadratic relationship grows by a changing amount, but that change itself is constant. It is like a ball rolling faster and faster down a ramp — the speed increase per second stays the same even though the speed itself keeps growing.

The rule: if

x

values are equally spaced and all second differences equal the same non-zero number, the table represents a quadratic relationship. If even the second differences are unequal, the relationship is neither linear nor quadratic (at least not a simple one).

\Delta^2 y = constant ≠ 0

Second differences are found by subtracting consecutive first differences.
Constant, non-zero second differences → quadratic relationship.
The graph of a quadratic relationship is a parabola.
Always verify equal spacing in $x$ before trusting your difference calculations.

Reading a Quadratic Table Step by Step

Follow these four steps every time you analyse a table. Step 1: confirm that the

x

values go up by the same amount each row. Step 2: calculate the first differences (each

y

value minus the one above it). Step 3: calculate the second differences (each first difference minus the one above it). Step 4: decide — constant first differences means linear; constant second differences (and non-constant first differences) means quadratic.

A common trap is skipping Step 1. If the

x

values jump by different amounts, the difference method gives misleading results. Always check the

x

column first.

The size of the constant second difference does not change your conclusion — whether it is

2

-6

, as long as it is the same throughout the table, the relationship is quadratic. A negative constant second difference means the parabola opens downward.

Four-step process: check $x$ spacing → first differences → second differences → conclude.
A positive constant second difference produces an upward-opening parabola.
A negative constant second difference produces a downward-opening parabola.
Non-constant second differences mean the relationship is not quadratic.

Graphing the Parabola from a Table

Once you know a table is quadratic, you can plot its points and connect them with a smooth curve. Unlike a linear graph, you must never connect the points with straight line segments — the curve between them is rounded.

As you plot the points, look for symmetry. A parabola is perfectly symmetrical about a vertical line called the axis of symmetry. The turning point of the parabola — the highest or lowest point — is called the vertex. If the parabola opens upward, the vertex is the minimum (lowest) point. If it opens downward, the vertex is the maximum (highest) point.

To sketch neatly: plot all points from your table first, then draw a smooth continuous curve through them. Use the symmetry of the points to guide the curve — if one side of the table mirrors the other, your curve should too. Label the vertex and, if possible, draw a dashed vertical line through it to show the axis of symmetry.

You do not need an equation to read the graph. From the plotted parabola you can directly identify the vertex coordinates, state whether the parabola opens up or down, and estimate where the curve crosses the

x

-axis (the

x

-intercepts, also called the zeros of the relationship).

Connect plotted quadratic points with a smooth curve, never straight segments.
The vertex is the maximum or minimum point of the parabola.
The axis of symmetry is the vertical line passing through the vertex.
Key features — vertex, direction of opening, and zeros — can all be read directly from the graph.

Linking the Table to the Shape of the Graph

The table and the graph tell the same story in different languages. When first differences are increasing (getting larger), the curve is rising more and more steeply — you are on the right side of an upward parabola climbing away from the vertex. When first differences are decreasing (getting smaller and eventually negative), the curve is slowing down, reaching a peak, then falling — typical of a downward-opening parabola.

This connection helps you check your graph: if the first differences in your table go from negative to positive (or from large negative to zero then positive), expect the parabola to dip to a minimum and then rise. If they go from positive down to negative, expect a maximum.

Practise reading graphs as well as drawing them. Given a parabola on a grid, you should be able to write the table of values for integer

x

inputs, compute the differences, and confirm the second difference is constant. This back-and-forth between table and graph deepens your understanding of what a quadratic relationship really is.

Rising first differences correspond to a curve that is getting steeper upward.
First differences changing from positive to negative signal a maximum vertex.
First differences changing from negative to positive signal a minimum vertex.
You can move in both directions: table → graph, or graph → table.

Comparing Linear and Quadratic Tables at a Glance

Feature	Linear Table	Quadratic Table
First differences	All equal (constant)	Change from row to row
Second differences	All zero	All equal (constant, non-zero)
Graph shape	Straight line	Parabola (U-shape)
Direction clue	Positive slope → rises; negative → falls	Positive 2nd diff → opens up; negative → opens down
Vertex exists?	No turning point	Yes — minimum or maximum point

Worked example

Example 1 — Identifying a Quadratic Table and Naming Key Numbers

A table of values is shown below. Determine whether the relationship is linear, quadratic, or neither. If it is quadratic, state the constant second difference.

x

: 0,\ 1,\ 2,\ 3,\ 4

y

: 1,\ 4,\ 11,\ 22,\ 37

Check that x values are equally spaced
Look at the $x$ column: CAD 0, 1, 2, 3, 4. Each value increases by $1$ , so the spacing is equal. The difference method is valid.
Calculate the first differences
Subtract each $y$ value from the one below it to get the first differences.
$4-1=3,\ 11-4=7,\ 22-11=11,\ 37-22=15$
Check whether first differences are constant
The first differences are CAD 3, 7, 11, 15. These are not all equal, so the relationship is not linear.
Calculate the second differences
Subtract each first difference from the one below it.
$7-3=4,\ 11-7=4,\ 15-11=4$
Draw a conclusion
All three second differences equal $4$ . Because the second differences are constant and non-zero, the relationship is quadratic. The constant second difference is $4$ . Since $4$ is positive, the parabola opens upward.

Answer: The relationship is quadratic. The constant second difference is

4

, and the parabola opens upward.

Check: Double-check one second difference: first differences were

3

and

7

, so

7 - 3 = 4

. Then

11 - 7 = 4

and

15 - 11 = 4

. All equal — confirmed.

Worked example

Example 2 — Building and Graphing a Quadratic Table

A ball is dropped from a height. The table below records its distance fallen (in metres) at each second.

t

(s): 0,\ 1,\ 2,\ 3,\ 4

d

(m): 0,\ 5,\ 20,\ 45,\ 80

(a) Confirm the relationship is quadratic by checking second differences.
(b) Plot the points on a coordinate grid and describe the shape of the graph.

Check equal spacing in t
The $t$ values are CAD 0, 1, 2, 3, 4 — each step is $1$ second. Equal spacing confirmed.
Find the first differences for d
Subtract consecutive $d$ values to find how much distance is added each second.
$5-0=5,\ 20-5=15,\ 45-20=25,\ 80-45=35$
Find the second differences
Subtract consecutive first differences.
$15-5=10,\ 25-15=10,\ 35-25=10$
State the conclusion for part (a)
Every second difference equals $10$ , which is constant and non-zero. The relationship is quadratic. The constant second difference is $10$ .
Plot the points and describe the graph for part (b)
Plot the five ordered pairs $(0, 0)$ , $(1, 5)$ , $(2, 20)$ , $(3, 45)$ , $(4, 80)$ with $t$ on the horizontal axis and $d$ on the vertical axis. Connect them with a smooth curve (not straight segments). The curve starts at the origin, rises slowly at first, and then rises more and more steeply. This is one arm of an upward-opening parabola. The vertex is at $(0, 0)$ — the starting point — and there is no downward portion visible in this context because distance fallen cannot be negative.

Answer: (a) The constant second difference is

10

, confirming a quadratic relationship. (b) The graph is the right arm of an upward-opening parabola starting at the origin, curving steeply upward as time increases.

Check: Verify: first differences CAD 5, 15, 25, 35 increase by

10

each time — correct. Also confirm

d

t=3

: the pattern gives

0 + 5 + 15 + 25 = 45

✓, and at

t=4

45 + 35 = 80

✓.

Common mistakes and how to avoid them

Calculating differences when the x values are not equally spaced, then drawing a false conclusion.

Correction: Always check that the x column increases by the same amount every row before calculating any differences. If the spacing is unequal, the method does not apply.

Concluding the relationship is quadratic just because the first differences are not constant, without actually checking the second differences.

Correction: Non-constant first differences only rule out a linear relationship. You must compute the second differences and confirm they are all equal before calling the relationship quadratic.

Connecting plotted points on a parabola with straight line segments instead of a smooth curve.

Correction: A parabola is a smooth, continuous curve. Always draw it freehand as a rounded curve passing through each point, not as a series of connected line segments.

Confusing the vertex with any point that looks central in the table, rather than identifying it from the graph as the highest or lowest point.

Correction: The vertex is defined as the turning point of the parabola on the graph — the one point where the curve changes direction. Locate it visually on the plotted graph, not just by looking at the middle row of the table.

Assuming a positive constant second difference always means the y values are increasing throughout the table.

Correction: A positive second difference means the parabola opens upward, but the y values may still decrease on the left side of the vertex before increasing on the right side. Look at the full picture of the graph.

Lesson summary

A first difference is the change in $y$ between consecutive rows; constant first differences identify a linear relationship.
A second difference is the change between consecutive first differences; a constant, non-zero second difference identifies a quadratic relationship.
Always confirm that $x$ values are equally spaced before applying the difference method.
The graph of a quadratic relationship is a parabola — a smooth, symmetrical U-shaped curve.
The vertex is the highest or lowest point of the parabola; the axis of symmetry is the vertical line through the vertex.
Key graph features — vertex, direction of opening, and approximate zeros — can be read directly from a plotted parabola without needing an equation.

Check your understanding

Question 1

A table has equally spaced

x

values. Its first differences are CAD 2, 5, 8, 11. What type of relationship does the table show?

Linear, because the first differences are all positive.
Quadratic, because the second differences are constant at $3$ .
Neither linear nor quadratic, because the first differences are not equal.
Quadratic, because the first differences are increasing.

Show answer and explanation

Quadratic, because the second differences are constant at

3

The second differences are

5-2=3

8-5=3

, and

11-8=3

— all equal to

3

. A constant, non-zero second difference confirms a quadratic relationship. Option D is a partial observation but not the correct reasoning; you must verify the second differences are constant, not just note that first differences increase.

Question 2

Which statement correctly describes the vertex of a parabola?

The point where the parabola crosses the $y$ -axis.
The point where the parabola crosses the $x$ -axis.
The highest or lowest point of the parabola, where the curve changes direction.
The midpoint of the two $x$ -intercepts, located on the $x$ -axis.

Show answer and explanation

The highest or lowest point of the parabola, where the curve changes direction.

The vertex is the turning point of the parabola — the single point where the curve stops going in one direction and reverses. It sits on the axis of symmetry and is either the minimum (for an upward-opening parabola) or the maximum (for a downward-opening parabola). It is not necessarily on either axis.

Question 3

A table of values has

x

values CAD 1, 2, 3, 4, 5 and

y

values

10, 7, 4, 1, -2

. What is the relationship?

Quadratic, because the $y$ values are decreasing.
Linear, because the first differences are all equal to $-3$ .
Quadratic, because the second differences are constant.
Neither, because the $y$ values become negative.

Show answer and explanation

Linear, because the first differences are all equal to

-3

The first differences are

7-10=-3

4-7=-3

1-4=-3

, and

-2-1=-3

. All first differences equal

-3

, which is constant, so the relationship is linear — its graph is a straight line with a negative slope. The fact that

y

values are negative or decreasing does not make a relationship quadratic.

Question 4

The constant second difference in a quadratic table is

-6

. What does this tell you about the parabola's graph?

The parabola opens downward because the second difference is negative.
The parabola opens upward because the second difference is non-zero.
The vertex is at $y = -6$ .
The parabola crosses the $x$ -axis at $x = -6$ .

Show answer and explanation

The parabola opens downward because the second difference is negative.

A negative constant second difference means the first differences are decreasing — the rate of change is getting smaller. This produces a parabola that opens downward, with a maximum vertex at the top. The value

-6

is the second difference, not a coordinate of the vertex or an intercept.

Key terms

First difference: The change in the output ( $y$ ) value between two consecutive rows of a table, when the input ( $x$ ) values are equally spaced.
Second difference: The change between consecutive first differences. A constant, non-zero second difference identifies a quadratic relationship.
Quadratic relationship: A relationship in which the output is connected to the square of the input. Its table has constant second differences and its graph is a parabola.
Parabola: The smooth, symmetrical U-shaped (or inverted U-shaped) curve that is the graph of a quadratic relationship.
Vertex: The turning point of a parabola — the lowest point if the parabola opens upward, or the highest point if it opens downward.
Axis of symmetry: The vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
Direction of opening: Whether a parabola curves upward (like a U) or downward (like an upside-down U), determined by the sign of the constant second difference.
Zeros (x-intercepts): The points where the parabola crosses the horizontal axis, meaning the output value equals zero. They can be read from a graph.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic Q1. It is a study resource, not an official curriculum publication.

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