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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

y = x² Versus y = 2ˣ: Comparing Graphs, and Zero and Negative Exponents

1.4About 75 minutes
Learning goals
  • I can compare the graphs and tables of y = x² and y = 2ˣ using technology.
  • I can describe how the two graphs differ in symmetry, intercepts, turning points and growth.
  • I can explain why a⁰ = 1 and a⁻ⁿ = 1/aⁿ using patterns and the exponent laws.
  • I can evaluate powers with zero and negative exponents.

Words to know

base
The number that is multiplied in a power: in 2⁵, the base is 2.
exponent
The number of times the base is used as a factor: in 2⁵, the exponent is 5.
zero exponent
For any a ≠ 0, a⁰ = 1.
negative exponent
For any a ≠ 0, a⁻ⁿ = 1/aⁿ, the reciprocal of aⁿ.
exponential relation
A relation such as y = 2ˣ in which the variable is in the exponent; the y-values have a constant ratio.
horizontal asymptote
A horizontal line that a graph gets closer and closer to without reaching it, such as the x-axis for y = 2ˣ.

Two curves that start out looking alike

The graphs of y = x² and y = 2ˣ both curve upward, and they even pass through some of the same points. But they are very different relations. In y = x² the variable is the base and the exponent is fixed at 2. In y = 2ˣ the base is fixed at 2 and the variable is the exponent. Today you will compare their graphs and tables using technology, and use the table of y = 2ˣ to explain what zero and negative exponents mean.

Remember from Grade 9

In Grade 9 you used patterns to see that a⁰ = 1 and a⁻ⁿ = 1/aⁿ, and you used the exponent laws aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Today you will see these rules again inside the table and graph of y = 2ˣ.

Building the table of y = 2ˣ

Start with exponents you know: 2³ = 8, 2² = 4, 2¹ = 2. Each time the exponent goes down by 1, the value is divided by 2. Keep the pattern going:

  • 2⁰ = 2 ÷ 2 = 1
  • 2⁻¹ = 1 ÷ 2 = 1/2
  • 2⁻² = 1/2 ÷ 2 = 1/4
  • 2⁻³ = 1/8

So a zero exponent gives 1, and a negative exponent gives a reciprocal: 2⁻ⁿ = 1/2ⁿ. A negative exponent never makes the value negative.

2³= 8÷22²= 4÷22¹= 2÷22⁰= 1÷22⁻¹= 1/2÷22⁻²= 1/4÷22⁻³= 1/8Each step left to right: subtract 1 from the exponent, divide the value by 2.
The halving pattern of the powers of 2. Following it past 2¹ forces 2⁰ = 1 and 2⁻ⁿ = 1/2ⁿ.
Worked example

Example 1: The exponent laws agree

Use the exponent laws to explain why 5⁰ = 1 and why 2⁻³ = 1/8.

  1. Zero exponent: 5³ ÷ 5³ = 5³⁻³ = 5⁰ by the quotient law. But any non-zero number divided by itself is 1: 125 ÷ 125 = 1. So 5⁰ = 1.
  2. Negative exponent: 2² ÷ 2⁵ = 2²⁻⁵ = 2⁻³ by the quotient law. As a fraction: 4/32 = 1/8. So 2⁻³ = 1/8 = 1/2³.
  3. The same reasoning works for any non-zero base a: a⁰ = 1 and a⁻ⁿ = 1/aⁿ. (The base must not be 0, because you cannot divide by 0.)

a⁰ = 1 and a⁻ⁿ = 1/aⁿ for any a ≠ 0; the pattern and the exponent laws give the same answer.

Worked example

Example 2: Evaluating powers

Evaluate: (a) 4⁻² (b) (−7)⁰ (c) 10⁻³ (d) (1/2)⁻³ (e) 3⁰ + 3⁻¹

  1. (a) 4⁻² = 1/4² = 1/16.
  2. (b) Any non-zero base to the exponent 0 is 1, including negative bases: (−7)⁰ = 1.
  3. (c) 10⁻³ = 1/10³ = 1/1 000 = 0.001.
  4. (d) (1/2)⁻³ = 1 ÷ (1/2)³ = 1 ÷ 1/8 = 8. A negative exponent flips the fraction: (1/2)⁻³ = 2³.
  5. (e) 3⁰ + 3⁻¹ = 1 + 1/3 = 4/3.

(a) 1/16 (b) 1 (c) 0.001 (d) 8 (e) 4/3

Watch the brackets

(−7)⁰ = 1 because the whole bracket is the base. But −7⁰ means −(7⁰) = −1, because the exponent only applies to the 7. In the same way, (−2)² = 4 but −2² = −4.

Comparing y = x² and y = 2ˣ

Graph both in Desmos or on a graphing calculator, and compare their tables.

x−3−2−10123456
y = x²9410149162536
y = 2ˣ1/81/41/21248163264
Tables of values for y = x² and y = 2ˣ.
xy−3−2−101234504812162024(2, 4)(4, 16)about (−0.77, 0.59)solid: y = x²dashed: y = 2ˣ
y = x² (solid) and y = 2ˣ (dashed). They cross three times: at (2, 4), at (4, 16) and at about (−0.77, 0.59). After x = 4, 2ˣ pulls away and stays above x².
Worked example

Example 3: Comparing the features

Compare the key features of y = x² and y = 2ˣ.

  1. Shape and symmetry. y = x² is a parabola, symmetric about the y-axis (x = 0). y = 2ˣ has no axis of symmetry: the left side is almost flat, the right side shoots up.
  2. Vertex and turning. y = x² has a vertex (0, 0) where it turns from decreasing to increasing. y = 2ˣ never turns: it increases for every x.
  3. Intercepts. y = x² has a zero at x = 0 and y-intercept 0. y = 2ˣ has no zeros (2ˣ is always positive) and y-intercept 1 (since 2⁰ = 1).
  4. Left side. As x becomes very negative, x² grows large again, but 2ˣ gets closer and closer to 0 without ever reaching it (1/8, 1/16, 1/32, …). The x-axis is a line it approaches but never touches, called a horizontal asymptote.
  5. Differences. For y = x² the second differences are constant (2). For y = 2ˣ the first differences are 1, 2, 4, 8, 16, … (the same as the values), the second differences are the same again, and no level of differences is ever constant. Instead, each y is 2 times the previous one: a constant ratio.
  6. Growth. Between x = 2 and x = 4, x² is larger (at x = 3: 9 > 8). For x > 4, 2ˣ is larger, and the gap grows fast: at x = 10, x² = 100 but 2ˣ = 1 024.

y = x² is symmetric with a vertex and constant second differences; y = 2ˣ is always increasing, always positive, has y-intercept 1, a constant ratio, and eventually grows much faster.

Just for fun

A famous legend: a king offered a reward and the inventor of chess asked for 1 grain of rice on the first square of the board, 2 on the next, 4 on the next, doubling each time. On the 64th square alone that would be 2⁶³ grains (about 9 quintillion), hundreds of times more rice than the whole world grows in a year. With x² instead, the 64th square would need only 64² = 4 096 grains.

Hands-on activity

Folding paper: powers of 2

You need: a large sheet of paper (newspaper works well) and a pencil.

  1. Count the layers of paper: 1 layer means 2⁰. Fold it in half: 2 layers (2¹). Fold again: 4 layers (2²). Keep folding and record the number of layers after each fold until you cannot fold any more.
  2. Unfold. After n folds, what fraction of the whole sheet is one small rectangle? (After 1 fold it is 1/2 = 2⁻¹.) Record the fractions for each number of folds.
  3. Make a table with columns: folds n, layers 2ⁿ, and fraction of the sheet 2⁻ⁿ. Find the first differences of the layers column.
  4. Compare with the squares table (1, 4, 9, 16, 25, …). Which grows faster at first? Which wins in the end?

What to look for: layers 1, 2, 4, 8, 16, 32, 64, … and fractions 1, 1/2, 1/4, 1/8, … Most people can fold an ordinary sheet only about 7 or 8 times: 2⁷ = 128 layers is too thick to bend. The first differences of the layers (1, 2, 4, 8, …) repeat the same doubling pattern, so they are never constant. Squares grow faster for a little while (at n = 3: 9 > 8), but 2ⁿ passes them after n = 4.

Your turn

1

Evaluate 6⁰ and 6⁻¹.

2

Evaluate 2⁻⁴ as a fraction and as a decimal.

3

Use the quotient law to simplify 3⁴ ÷ 3⁶, then evaluate.

4

Evaluate (−3)⁰ and −3⁰. Explain why they differ.

5

Give the y-intercept of y = x² and of y = 2ˣ.

6

For which whole-number values of x is 2ˣ = x²?

More practice: warm-up

1

Evaluate 9⁰.

2

Evaluate 5⁻².

3

Evaluate 10⁻².

4

Does y = 2ˣ have an axis of symmetry?

5

Does y = 2ˣ ever equal 0?

More practice: on your own

1

Evaluate (2/3)⁻².

2

Evaluate 2⁻¹ + 4⁻¹ + 8⁰.

3

Simplify x⁵ × x⁻⁵ (x ≠ 0) and explain your answer.

4

Find the first differences of y = 2ˣ for x = 0 to 5. What do you notice?

5

For which x-values between 2 and 4 is x² greater than 2ˣ? Test x = 2.5, 3 and 3.5.

Challenge

1

Explain why 0⁻¹ cannot be given a value.

2

At x = 10, how many times larger is 2ˣ than x²? At x = 20?

3

Compare y = x² and y = 2ˣ on the left side. Describe what happens to each as x goes from 0 to −10.

Check your understanding

Quiz

1.

What is 7⁰?

2.

What is 3⁻²?

3.

Which feature does y = x² have but y = 2ˣ does not?

4.

The y-intercept of y = 2ˣ is…

5.

Which statement about the tables is true?

6.

Which is the value of −5⁰?

For parents and teachers

Big idea: Expectation 1.4 asks students to compare the graphs of y = x² and y = 2ˣ using technology, and to determine the meaning of zero and negative exponents (from the table of y = 2ˣ and from the exponent laws). Students met these exponent rules in Grade 9; here they connect them to a graph and see why exponential growth eventually beats quadratic growth.

How to help: Graph both curves together in Desmos and zoom out. Ask "Which one wins?" at x = 3, 5 and 10. The paper-folding activity is a quick, memorable demonstration.

Common difficulties:

  • Thinking a negative exponent makes a negative number (2⁻³ = −8). It means a reciprocal: 1/8.
  • Thinking 2⁰ = 0. Use the halving pattern: 2¹ ÷ 2 = 1.
  • Confusing −5⁰ with (−5)⁰.
  • Calling y = 2ˣ a parabola because it curves. Ask: does it have a vertex or an axis of symmetry?