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Wize High School Grade 10 Math Textbook > Quadratic Relations

Quadratic x2x^2 vs Exponential 2x2^xGraphs

Properties of Quadratic GraphsPrevious Section
Factored Form of a Quadratic EquationNext Section
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Review: Exponent Rules (Laws of Exponents)

  1. Multiplying powers with the same base: an×am=an+m\large a^n\times a^m=a^{n+m}
  2. Dividing powers with the same base: an÷am=an−m\large a^n\div a^m=a^{n-m}
  3. Power of powers: (an)m=an×m\large (a^n)^m=a^{n\times m}
  4. Powers of products: (ab)n=(anbn)\large \left(ab\right)^n=\left(a^nb^n\right)
  5. Powers of quotients: (ab)n=(anbn)\large \left(\dfrac{a}{b}\right)^n=\left(\dfrac{a^n}{b^n}\right)
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Quadratic VS Exponential Graphs

Watch Out!
Many students mix up y=x2y=x^2 and y=2xy=2^x.
  • y=x2y=x^2 is a quadratic equation
  • y=2xy=2^x is an exponential equation

Let's take a look at the table of values for both equations:

But what about 20\bco{2^0}? 2−1\bco{2^{-1}}? 2−2\bco{2^{-2}}? or 2−exponent\bco{2^{-\text{exponent}}}?

Let's take a look at the pattern in the table of values for y=2xy=2^x.
  • What's happening to the y-value as x increases from 1 to 2, from 2 to 3, etc.?
    It is doubling!
  • What's happening to the y-value as x decreases from 3 to 2, from 2 to 1, etc.?
    It is halfing!

According to this pattern:
  • The value of 20=2^0=
    1
  • The value of 2−1=2^{-1}=
    1/2
  • The value of 2−2=2^{-2}=
    1/4
  • The value of 2−3=2^{-3}=
    1/8
  • The value of 2−4=2^{-4}=
    1/16

Wize Tip
In general, what is the rule for b0b^0 and b−nb^{-n}?
  • b0=1\large\boxed{b^0=1}
  • b−n=1bn\large\boxed{b^{-n}=\dfrac{1}{b^n}}

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Another way to Understand b0\bco{b^0} and b−n\bco{b^{-n}}

  • Observe that b0=bn−n=bnbn=1\large b^0=b^{n-n}=\dfrac{b^n}{b^n}=1, so we can confirm that b0=1\large b^0=1
  • Observe that b−n=b0−n=b0bn=1bn\large b^{-n}=b^{0-n}=\dfrac{b^0}{b^n}=\dfrac{1}{b^n}, so we can confirm that b−n=1bnb^{-n}=\dfrac{1}{b^n}

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Comparing Quadratic VS Exponential Graphs

Here are the table of values for the quadratic and exponential equations:

Here are the graphs for the two equations:

  • The x2x^2 graph is symmetric along the axis of symmetry; the 2x2^x graph is not symmetric
  • The x2x^2 graph decreases and then increases; the 2x2^x graph is always increasing
  • The x2x^2 graph has an x-intercept at x=0x=0; the 2x2^x graph does not have any x-intercepts
  • The x2x^2 graph has a y-intercept at x=0x=0; the 2x2^x graph has a y-intercept at y=1y=1

Practice: Zero Exponents

Evaluate the following powers.

a) 505^0

b) (−5)0\left(-5\right)^0

c) (15)0\left(\dfrac{1}{5}\right)^0

Practice: Negative Exponents

Rewrite the following powers without negative exponents, your final answer should be in rational form (your answer should be a fraction)

a) 3−43^{-4}

b) 2−32^{-3}

c) (23)−4\left(\dfrac{2}{3}\right)^{-4}

d) (−2)−3\left(-2\right)^{-3}

Practice: Negative Exponents

What value(s) of nn will make each of the following equations true?

a) 3n=273^n=27

b) 2n=1162^n=\dfrac{1}{16}

c) (−4)n=−164(-4)^n=-\dfrac{1}{64}