MPM2D Comparing y = x squared and y = 2 to the x: Notes and Practice

A summer job offers you a choice of two bonus plans. Plan A pays $x^{2}$ dollars on day $x$ . Plan B pays $2^{x}$ dollars on day $x$ . On day 3, Plan A pays 9 dollars and Plan B pays 8, so A looks slightly better. On day 10, Plan A pays 100 dollars, but Plan B pays $2^{10} = 1024$ dollars. On day 20, Plan A pays 400 dollars and Plan B pays over a million. The two rules look almost the same on paper (a 2 and an $x$ swap places), but they grow in completely different ways. In $y = x^{2}$ , the base $x$ changes and the exponent stays at 2: this is a quadratic relation. In $y = 2^{x}$ , the base stays at 2 and the exponent changes: this is an exponential relation. Comparing them also answers a question you may have wondered about: what could $2^{0}$ or $2^{- 3}$ possibly mean?

x	−3	−2	−1	0	1	2	3	4	5	6
y = x²	9	4	1	0	1	4	9	16	25	36
y = 2ˣ	1/8	1/4	1/2	1	2	4	8	16	32	64

Tables of values for y = x squared and y = 2 to the x

Look at how each row changes. For $y = x^{2}$ from $x = 0$ on, the first differences are $1, 3, 5, 7, 9, 11$ and the second differences are all $2$ , as for every quadratic with $a = 1$ . For $y = 2^{x}$ from $x = 0$ on, the first differences are $1, 2, 4, 8, 16, 32$ : they double, so they never settle to a constant, and neither do the second differences ( $1, 2, 4, 8, 16$ ). What is constant is the ratio: each $y$ -value is exactly $2$ times the one before it. Adding the same amount each step is linear; changing by a constant second difference is quadratic; multiplying by the same number each step is exponential.

Fig 1The graphs of y = x² and y = 2ˣ on one grid with x from −3 to 5 and y from 0 to 20, beside a table of values. They meet at (2, 4), at (4, 16) and at one point with x between −1 and 0. Between x = 2 and x = 4 the parabola is higher; after x = 4 the exponential curve stays above it.

Ay = x²

Parabola, symmetric about the axis $x = 0$
Vertex $(0, 0)$ , minimum value $0$
One zero, at $x = 0$ ; $y$ -intercept $0$
Decreases for $x < 0$ , increases for $x > 0$
Second differences constant ( $2$ )

By = 2ˣ

Not symmetric; always increasing
No vertex, no maximum or minimum value
No zeros: $2^{x} > 0$ for every $x$ ; $y$ -intercept $1$
To the left it gets closer and closer to $0$ without reaching it: the x-axis is a horizontal asymptote
Constant ratio ( $\times 2$ each step)

Which one is bigger depends on where you look. The curves cross three times: at $(2, 4)$ and $(4, 16)$ , where the tables match, and once for a negative $x$ between $- 1$ and $0$ (at about $x = - 0.77$ ). Between $x = 2$ and $x = 4$ the parabola is higher (at $x = 3$ : $9 > 8$ ). After $x = 4$ the exponential curve wins and keeps pulling away, because each step of 1 doubles $2^{x}$ , while $x^{2}$ grows by a smaller and smaller factor: from $x = 4$ to $x = 5$ , $x^{2}$ is multiplied by only $\frac{25}{16} \approx 1.56$ . In the long run, repeated doubling beats squaring.

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Choose the parent x squared, then switch to the parent 2 to the x, leaving every slider at its starting value. The sim shows one parent at a time on the same axes, with y from −10 to 10, so you can compare the shapes near the origin and the point (2, 4); the crossing at (4, 16) is above the window.

VideoA quadratic can lead at first, but an exponential relation that keeps doubling always overtakes it in the end.Watch on YouTube

What do 2⁰ and negative exponents mean?

An exponent counts repeated multiplication: $2^{3} = 2 \times 2 \times 2 = 8$ . That definition makes no sense for an exponent of $0$ or $- 2$ ; you cannot multiply "zero 2s" or "negative two 2s". But the table of $y = 2^{x}$ shows what those values must be for the pattern to continue. Read the table from right to left: each time the exponent goes down by 1, the value is divided by 2. Keep going past $2^{1}$ :

2^{3} = 8 \div 2 2^{2} = 4 \div 2 2^{1} = 2 \div 2 2^{0} = 1 \div 2 2^{- 1} = \frac{1}{2} \div 2 2^{- 2} = \frac{1}{4} \div 2 2^{- 3} = \frac{1}{8}

The exponent laws give the same answers. For powers with the same base, $a^{m} \times a^{n} = a^{m + n}$ and $a^{m} \div a^{n} = a^{m - n}$ . Divide a power by itself: $2^{3} \div 2^{3} = 2^{3 - 3} = 2^{0}$ , but also $2^{3} \div 2^{3} = 8 \div 8 = 1$ . So $2^{0} = 1$ . Now divide by a bigger power: $2^{2} \div 2^{5} = 2^{2 - 5} = 2^{- 3}$ , but also $\frac{4}{32} = \frac{1}{8} = \frac{1}{2 ^{3}}$ . So $2^{- 3} = \frac{1}{2 ^{3}}$ . The same reasoning works for any base except $0$ , because dividing by $0^{3} = 0$ is impossible.

a^{0} = 1 a^{- n} = \frac{1}{a ^{n}} (a \neq = 0)

Power	Meaning	Value
$5^{- 2}$	$\frac{1}{5 ^{2}}$	$\frac{1}{25} = 0.04$
$1 0^{- 3}$	$\frac{1}{1 0 ^{3}}$	$\frac{1}{1000} = 0.001$
$3^{- 1}$	$\frac{1}{3 ^{1}}$	$\frac{1}{3}$
$(- 4)^{0}$	any non-zero base to the exponent 0	$1$
$(\frac{1}{2})^{- 1}$	$\frac{1}{1/2}$	$2$

Zero and negative exponents in action

A negative exponent is not a negative number

2^{- 3}

\frac{1}{8}

, a small positive number, not

- 8

and not

- 6

. A negative exponent means "one over" the matching positive power. Also watch the brackets with exponent 0:

(- 3)^{0} = 1

because the base is

- 3

, but

- 3^{0} = - (3^{0}) = - 1

because the exponent applies only to the 3.

Comparing y = x squared and y = 2 to the x

Key things to remember

What do 2⁰ and negative exponents mean?

Worked examples

Where marks get lost

Flashcards 8

Practice