Q4 Compare quadratic and exponential graphs and interpret zero and negative exponents

Ontario Grade 10 Mathematics

Quadratic Relations

MPM2D · Study Topic Q4 · Mixed Difficulty

You already know from Grade 9 that a linear function grows by adding the same amount each time, and that its graph is a straight line. In this lesson you will compare two non-linear families — quadratic and exponential — that both curve upward but do so in very different ways. You will also learn two important exponent rules: what it means to raise a number to the power of zero, and what a negative exponent really means. These rules are not just shortcuts; they are the key to reading exponential graphs all the way back to and past the y-axis.

What you will learn

Identify and describe the key differences between the graphs of quadratic and exponential functions.
Evaluate expressions that contain zero or negative integer exponents.
Connect the rules for zero and negative exponents to the behaviour of exponential graphs.
Determine whether a table of values represents a quadratic or exponential relationship.

Grade 9 Bridge: What You Already Know About Exponents

In Grade 9 you practised writing repeated multiplication using exponent notation. For example,

2 \times 2 \times 2 = 2^3 = 8

. The base is

2

and the exponent (or power) is

3

, telling you how many times the base is used as a factor.

You also used the product rule: when you multiply powers with the same base, you add the exponents. In symbols,

a^m \times a^n = a^{m+n}

. This Grade 9 rule is the foundation for the two new rules in this lesson, so keep it in mind as you read on.

a^m × a^n = a^{m+n}

The exponent tells you how many times the base appears as a factor.
Product rule: $a^m \times a^n = a^{m+n}$ (same base, add exponents).
These rules hold for any non-zero base $a$ .

The Zero Exponent and the Negative Exponent Rules

Start with a pattern. Divide each term by the base as you move one step to the left:

2^3 = 8

,

2^2 = 4

,

2^1 = 2

. Each step divides by

2

. Continuing:

2^0 = 1

,

2^{-1} = \frac{1}{2}

,

2^{-2} = \frac{1}{4}

. The pattern never breaks — you just keep dividing by the base.

The zero exponent rule says that any non-zero base raised to the power zero equals

1

. In symbols,

a^0 = 1

for

a \neq 0

. You can confirm this with the product rule:

a^3 \times a^{-3} = a^0

, and at the same time

a^3 \times \frac{1}{a^3} = 1

, so

a^0

must equal

1

.

The negative exponent rule says that a negative exponent means the reciprocal of the positive version. In symbols,

a^{-n} = \frac{1}{a^n}

for

a \neq 0

and any positive integer

n

. So

3^{-2} = \frac{1}{3^2} = \frac{1}{9}

. A negative exponent does NOT make the result negative — it makes it a fraction less than

1

(when the base is greater than

1

).

a^{-n} = \frac{1}{a^{n}}

Zero exponent rule: $a^0 = 1$ for any $a \neq 0$ .
Negative exponent rule: $a^{-n} = \frac{1}{a^n}$ .
A negative exponent produces a fraction, never a negative number (when the base is positive).
Both rules follow directly from the pattern of dividing by the base one step at a time.

Quadratic Functions and Their Graphs

A quadratic function has the form

y = ax^2 + bx + c

, where the highest power on

x

is

2

. The simplest example is

y = x^2

. Its graph is a U-shaped curve called a parabola. The vertex is the turning point — the lowest point on the parabola when

a > 0

.

Key features to notice: the parabola is symmetric about a vertical line through its vertex; for large positive or negative

x

values,

y

grows larger and larger; and the function can have zero, one, or two

x

-intercepts depending on the specific equation.

How fast does

y = x^2

grow? Each time

x

increases by

1

, the amount added to

y

gets bigger and bigger, but it is not multiplied by a fixed amount. This is called polynomial growth. The second differences in a table of values for a quadratic are constant — a useful test you will apply shortly.

y = ax^2 + bx + c

Standard form: $y = ax^2 + bx + c$ ; highest power is $2$ .
Graph is a parabola — U-shaped, symmetric, with a vertex.
Second differences in a table of values are constant for a quadratic.
As x \to ±\infty, $y$ grows without bound, but not by a fixed multiplier each step.

Exponential Functions and Their Graphs

An exponential function has the form

y = a \cdot b^x

, where the variable

x

is in the exponent,

a

is the initial value (the

y

-intercept when

x = 0

), and

b > 0

,

b \neq 1

is called the base or growth/decay factor. The simplest example is

y = 2^x

.

When

b > 1

, the function grows — each step multiplies

y

by the same factor

b

. This is called exponential growth. When

0 < b < 1

, each step multiplies by a factor less than

1

, so

y

shrinks — this is exponential decay. In either case, the ratio of consecutive

y

-values in a table is constant. That constant ratio is the key test for an exponential relationship.

Crucially, the exponential graph never crosses the

x

-axis and never goes below it (for positive

a

). As

x

moves in the negative direction, the negative exponent rule tells you what happens:

y

approaches zero but stays positive. This horizontal line that the curve gets infinitely close to but never reaches is called an asymptote. For

y = 2^x

, the asymptote is the line

y = 0

(the

x

-axis).

At

x = 0

,

y = a \cdot b^0 = a \cdot 1 = a

. This confirms that the zero exponent rule sets the

y

-intercept of any exponential function equal to

a

.

y = a · b^{x}

Standard form: $y = a \cdot b^x$ ; variable is in the exponent.
Consecutive $y$ -values in a table have a constant ratio equal to $b$ .
Graph has a horizontal asymptote (usually $y = 0$ ); never crosses it.
The $y$ -intercept is always $a$ because $b^0 = 1$ .
Negative $x$ -values produce fractions using the negative exponent rule, explaining why the graph flattens near zero.

Comparing the Two Graphs Side by Side

Both

y = x^2

and

y = 2^x

are increasing for positive

x

values, and both curve upward. However, exponential growth eventually far outpaces quadratic growth. For small

x

the parabola can sit above the exponential curve, but for large

x

the exponential overtakes and stays ahead forever.

The shapes also differ near

x = 0

. The parabola

y = x^2

has a flat bottom (vertex) at the origin, and it is symmetric: the left side mirrors the right side. The exponential

y = 2^x

has no flat bottom and is not symmetric. On the left side, instead of going back up, it hugs the

x

-axis and approaches zero.

To tell the two types apart from a table: compute first differences (

\Delta y

). If the first differences are not constant but the second differences of those first differences are constant, you have a quadratic. If neither set of differences is constant but the ratio

\frac{y_{n+1}}{y_n}

is constant, you have an exponential.

Both curves rise for large positive $x$ , but exponential growth eventually dominates.
Quadratic: symmetric parabola with a vertex; exponential: one-sided curve with an asymptote.
Table test — quadratic: constant second differences; exponential: constant ratio of consecutive $y$ -values.
The negative- $x$ side behaves completely differently: parabola rises again; exponential decays toward zero.

Quadratic vs. Exponential: Key Comparison

Feature	Quadratic y = x²	Exponential y = 2ˣ
General form	$y = ax^2 + bx + c$	$y = a \cdot b^x$
Where is x?	Base (x is squared)	Exponent
Graph shape	Parabola, symmetric, vertex	One-sided curve, no vertex
Left-side behaviour	Rises back up (symmetric)	Approaches y = 0 (asymptote)
Table test	Constant second differences	Constant ratio of y-values
Value at x = 0	$0^2 = 0$ (for y = x²)	$a \cdot b^0 = a$ (for y = a·bˣ)

Worked example

Evaluating Zero and Negative Exponents

Evaluate each expression without a calculator. (a)

5^0

(b)

4^{-3}

(c)

3 \cdot 2^{-4}

Apply the zero exponent rule to part (a)
Any non-zero base raised to the power $0$ equals $1$ . The base here is $5$ , which is not zero, so the rule applies directly.
$5^0 = 1$
Apply the negative exponent rule to part (b)
A negative exponent means take the reciprocal of the positive-exponent version. Rewrite $4^{-3}$ as $\frac{1}{4^3}$ , then compute $4^3 = 4 \times 4 \times 4 = 64$ .
$4^{-3} = \frac{1}{4^3} = \frac{1}{64}$
Handle the coefficient in part (c)
Only the base $2$ is raised to the $-4$ power; the coefficient $3$ in front is not affected by the exponent. First rewrite $2^{-4}$ as $\frac{1}{2^4}$ , then compute $2^4 = 16$ , and finally multiply by $3$ .
$3 · 2^{-4} = 3 · \frac{1}{16} = \frac{3}{16}$

Answer: (a)

1

(b)

\frac{1}{64}

(c)

\frac{3}{16}

Check: Check (b) using the product rule:

4^{-3} \times 4^3 = 4^0 = 1

. Indeed

\frac{1}{64} \times 64 = 1

. ✓

Worked example

Identifying a Function Type from a Table and Connecting to Graph Behaviour

The table below shows values for two mystery functions, P and Q. Determine whether each is quadratic or exponential, write its equation, and describe one key difference between their graphs.

Function P:

x = 0, 1, 2, 3, 4

gives

y = 1, 3, 9, 27, 81

.
Function Q:

x = 0, 1, 2, 3, 4

gives

y = 0, 1, 4, 9, 16

.

Test Function P for a constant ratio
Divide each $y$ -value by the one before it: $\frac{3}{1} = 3$ , $\frac{9}{3} = 3$ , $\frac{27}{9} = 3$ , $\frac{81}{27} = 3$ . The ratio is constantly $3$ , so Function P is exponential with base $3$ . At $x = 0$ , $y = 1$ , so the initial value $a = 1$ .
$y = 1 · 3^{x} = 3^{x}$
Test Function Q — check first differences
Subtract consecutive $y$ -values of Q: $1 - 0 = 1$ , $4 - 1 = 3$ , $9 - 4 = 5$ , $16 - 9 = 7$ . The first differences (CAD 1, 3, 5, 7) are not constant, so Q is not linear.
Check second differences of Function Q
Now subtract consecutive first differences: $3 - 1 = 2$ , $5 - 3 = 2$ , $7 - 5 = 2$ . The second differences are all $2$ , which is constant. This confirms Function Q is quadratic. At $x = 0$ , $y = 0$ , and the values match $x^2$ , so the equation is $y = x^2$ .
$y = x^{2}$
Describe one key graph difference
For Function P ( $y = 3^x$ ), substitute $x = -2$ : $3^{-2} = \frac{1}{9}$ . The graph approaches the $x$ -axis from above as $x$ decreases — it has a horizontal asymptote at $y = 0$ . For Function Q ( $y = x^2$ ), at $x = -2$ : $(-2)^2 = 4$ . The parabola rises symmetrically on the left side instead of flattening out. This is the clearest visual difference between the two graphs on the left half of the coordinate plane.
$3^{-2} = \frac{1}{9}, (-2)^2 = 4$

Answer: Function P is exponential:

y = 3^x

. Function Q is quadratic:

y = x^2

. Key difference: for negative

x

, the exponential graph flattens and approaches

y = 0

(asymptote), while the parabola rises back up symmetrically.

Check: Verify P at

x = 4

:

3^4 = 81

✓. Verify Q at

x = 4

:

4^2 = 16

✓. Constant ratio for P:

81 \div 27 = 3

✓. Second difference for Q:

7 - 5 = 2

✓.

Common mistakes and how to avoid them

Writing

a^0 = 0

because "zero means nothing."

Correction: Any non-zero base to the power zero equals

1

, not

0

. Remember:

a^0 = 1

. The exponent zero does not make the result zero.

Thinking

2^{-3} = -8

because the exponent is negative.

Correction: A negative exponent means reciprocal, not a negative result.

2^{-3} = \frac{1}{2^3} = \frac{1}{8}

, which is a positive fraction.

Applying the exponent to the coefficient, e.g. writing

3 \cdot 2^{-1} = 6^{-1} = \frac{1}{6}

.

Correction: Only the base

2

is affected by the exponent. The coefficient

3

stays separate:

3 \cdot 2^{-1} = 3 \cdot \frac{1}{2} = \frac{3}{2}

.

Confusing the table test: checking first differences for both quadratic and exponential.

Correction: For quadratic, check that second differences are constant. For exponential, check that the ratio of consecutive y-values is constant — not the differences.

Assuming the exponential graph eventually crosses the x-axis for negative x.

Correction: The exponential graph approaches the x-axis but never touches it. The line

y = 0

is a horizontal asymptote. Negative exponents produce fractions close to zero, never zero itself.

Lesson summary

A zero exponent always gives $1$ : $a^0 = 1$ for any non-zero base $a$ . This sets the y-intercept of every exponential function $y = a \cdot b^x$ equal to $a$ .
A negative exponent means the reciprocal of the positive version: $a^{-n} = \frac{1}{a^n}$ . The result is a positive fraction when the base is positive — never a negative number.
A quadratic function ( $y = ax^2 + bx + c$ ) produces a symmetric parabola. Its table of values has constant second differences.
An exponential function ( $y = a \cdot b^x$ ) produces a curve with a horizontal asymptote. Its table of values has a constant ratio between consecutive y-values.
The two graphs look similar for large positive x (both rise steeply), but differ completely for negative x: the parabola rises again, while the exponential curve flattens toward its asymptote.
Use the table tests — constant second differences for quadratic, constant ratio for exponential — to identify which family a set of data belongs to.

Check your understanding

Question 1

What is the value of

6^0

?

$0$
$6$
$1$
$-6$

Show answer and explanation

1

6^0 = 1

by the zero exponent rule. Any non-zero base raised to the power zero equals

1

.

Question 2

Which expression is equal to

5^{-2}

?

$-25$
$\frac{1}{25}$
$-\frac{1}{25}$
$25$

Show answer and explanation

\frac{1}{25}

5^{-2} = \frac{1}{5^2} = \frac{1}{25}

. A negative exponent means the reciprocal; it does not produce a negative value.

Question 3

A table of values has

y

-values CAD 2, 6, 18, 54, 162. What type of function does this represent?

Linear, because the values increase steadily.
Quadratic, because the second differences are constant.
Exponential, because each y-value is multiplied by the same factor.
Neither, because the values grow too quickly.

Show answer and explanation

Exponential, because each y-value is multiplied by the same factor.

Each y-value is multiplied by

3

:

6 \div 2 = 3

,

18 \div 6 = 3

,

54 \div 18 = 3

,

162 \div 54 = 3

. A constant ratio means the function is exponential.

Question 4

How does the graph of

y = 3^x

behave as

x

takes larger and larger negative values?

It rises steeply, mirroring the right side.
It crosses the x-axis and becomes negative.
It approaches the line $y = 0$ but never reaches it.
It approaches the line $y = 1$ but never reaches it.

Show answer and explanation

It approaches the line

y = 0

but never reaches it.

For negative

x

, the negative exponent rule gives values like

3^{-1} = \frac{1}{3}

,

3^{-2} = \frac{1}{9}

, getting closer and closer to

0

but always staying positive. The x-axis (

y = 0

) is the horizontal asymptote.

Key terms

Exponent: A number written above and to the right of a base that tells how many times the base is multiplied by itself. In $2^3$ , the exponent is $3$ .
Zero exponent rule: The rule that any non-zero base raised to the power of zero equals $1$ : $a^0 = 1$ .
Negative exponent rule: The rule that $a^{-n} = \frac{1}{a^n}$ . A negative exponent produces a reciprocal, not a negative number.
Quadratic function: A function of the form $y = ax^2 + bx + c$ whose graph is a parabola. The highest power of $x$ is $2$ .
Exponential function: A function of the form $y = a \cdot b^x$ where the variable $x$ is in the exponent and $b > 0$ , $b \neq 1$ .
Asymptote: A line that a graph gets closer and closer to but never touches or crosses. Exponential graphs have a horizontal asymptote.
Parabola: The U-shaped (or inverted U-shaped) curve that is the graph of a quadratic function. It has a vertex and is symmetric about a vertical line.
Second differences: The differences of the first differences in a table of values. If the second differences are constant and non-zero, the relationship is quadratic.

Continue through MPM2D

View the complete Ontario Grade 10 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic Q4. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

Q4 · Compare quadratic and exponential graphs and interpret zero and negative exponents

Ontario Grade 10 Mathematics

MPM2D · Study Topic Q4 · Mixed Difficulty

What you will learn

Grade 9 Bridge: What You Already Know About Exponents

The Zero Exponent and the Negative Exponent Rules

Quadratic Functions and Their Graphs

Exponential Functions and Their Graphs

Comparing the Two Graphs Side by Side

Quadratic vs. Exponential: Key Comparison

Evaluating Zero and Negative Exponents

Identifying a Function Type from a Table and Connecting to Graph Behaviour

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MPM2D

About this lesson