This lecture will cover exponential parent functions, transformations of
exponential functions, rules of exponents, and solving exponential
equations.
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Two additional functions will be added to the list of parent functions, one of
these being the exponential function.
Exponential parent functions are functions of the form f(x) = b^x, where b is
known as the base of the function. The base must fall into one of two ranges
of values: either b is greater than 1 or b is between 0 and 1. 1 is excluded
from the set of values for b, since b=1 would correspond to a constant
function. The value of the base of an exponential parental function
determines the direction of the graph.
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If the base is greater than 1, then the graph of the function increases from
left to right. Notice the general shape of the graph. The graph begins
increase more and more rapidly as x increases.
It is easy to see that the domain is defined for all real numbers. However, the
range of this function is composed of all positive numbers.
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The graph of an exponential function decreases if the base falls between 0
and 1.
The graph of this function looks almost like a mirror image of the graph of
the previous function.
The domain and range are the same for this function.
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The graph of an exponential parent function can be used to identify the base
of the function. Remember that at x = 1, b^1 = b. So the base of this function
can be found by locating the point corresponding to x =1. This point occurs
at (1,4). So b^1 = 4. Thus the base of this exponential function is 4.
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Additional, more complex exponential functions can be graphed by applying
transformations previously seen to the exponential parent functions.
Horizontal shifts behave in the same way as before. Replacing x in the
exponent with x+h shifts the graph of f(x) = b^x to the left h units. Replacing
x with x-h shifts the graph of the parent function to the right h units.
Vertical shifts are easily identified. Adding a value k to f(x) = b^x shifts the
graph upward k units, and subtracting a value k from the parent function
shifts the graph downward k units.
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Exponential parent functions may also be reflected about the y-axis by
replacing x with –x. If b^x is multiplied by -1, then the graph of the parent
function is reflected about the x-axis.
An exponential parent function may also be stretched or compressed
vertically by a factor of c.
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Example:
Following the previously graphed transformations of functions, start by
identifying the parent function. The parent function in this case is f(x) =3^x.
Two transformations have then been applied: first the graph of f(x) = 3^x has
been shifted left 1 unit and then stretched vertically by a factor of 2.
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Example:
The parent function here is another familiar exponential function: f(x) =
(1/2)^x. Once again, we have two transformations: reflection about the x-axis
and shifted down 2 units.
Notice that the range has changed after the transformations have been
applied, but the domain remains the same.
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An important constant that appears in the discussion of exponential
functions is e. e is approximately 2.718281828. The value of this constant,
like pi, is stored in most scientific calculators.
This constant will appear again in the discussion of exponential functions
and logarithmic functions.
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At this point, it is useful to discuss the rules of exponents in preparation for
solving exponential equations.
First, the Product Rule tells us that when multiplying exponential expressions
with the same base, we add the exponents. Similarly, when taking the
quotient of two exponential expressions with the same base, we take the
difference of the exponents of the terms as the exponent of the resulting
expression. The negative exponent rule shows how to evaluate expressions
with negative exponents. This rule can be extended to the case where a
term with a negative exponent appears in the denominator: 1/b^-m =
1/(/1/b^m) =b^m.
The zero exponent rule indicates that for any base other than 0, b^0 =1.
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The next three rules are applicable when raising an exponential expression
to another power. If an exponential term is raised to another power, the
Power Rule states to take the product of the exponents.
The final two rules, the Power of a Product and the Power of a Quotient
rules indicate that an exponent is applied to each term in a product or a
quotient. This same principle does NOT apply to sums and differences,
however.
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Rational exponents represent a special case in the discussion of exponential
expressions. Rational exponent notation is actually an alternate way of
writing expressions involving radicals. The denominator of the rational
exponent represents the index of the root that is to be taken, and the
numerator represents the actual power that this root is to be raised to.
For example, 16^3/4 is equivalent to the fourth root of 16, raised to the third
power. The fourth root of 16 is 2. Completing the simplification, we end up
with 8.
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Example:
In part a, a product is being raised to a power. According to the Power of a
Product rule, the exponent outside of the parentheses needs to be applied to
each part of the product:5^-4(x^3)^-4 y^-4 (z^-2)^-4. Now use the Power
Rule to simplify the individual exponents: 5^-4x^-12y-4z^8. Rewriting this
expression so that we don’t have any negative exponents: z^8/(5^4x^12y^4)
= z^8/(625x^12y^4). The expression has now been simplified.
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Example:
In part b, the Power of a Quotient rule seems to be the most appropriate rule
to begin with. So using the Power of a Quotient rule, the exponent of the
quotient is applied to each one of the factors: 4^3(a^-2)^3b^3/7^3. Now
using the Power Rule to simplify the numerator and also evaluate the
numeric factors: 4^3 = 64 and 7^3 = 343: 64a^-6b^3/343. Since the
expression has a factor with a negative exponent, one more step must be
taken before this expression is fully simplified: =64b^3/343a^6.
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Example:
Part c shows the product of two exponential expressions. Before multiplying,
however, the exponent being applied to the second expression needs to be
taken first. Using the Power of a Product Rule: (xy^5/18z^-2)(15^2x^-6z^8).
Now multiply using the Product Rule to multiply the factors with x together:
=15^2x^-5y^5z^8 / 18z^-2.
Notice that the Quotient Rule can be used to simplify this expression and
that 15^2 = 225: =225x^-5y^5z^10/18.
225 and 18 have a common factor of 9. So dividing each by 9 and applying
the rule for negative exponents, the simplified form of the expression for part
c is: =25y^5z^10/2x^5
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To solve exponential equations for a variable, two more rules of exponents
related to equality needs to be added to our list.
First, if b^m = b^n, where the base is positive, then m = n. So, in other
words, if two exponential terms are equal and the bases of each term are
the same, then the exponents are equal, provided that the bases are
positive.
The second rule covers the case where the exponents are equal. Provided
that the bases are positive, a =b.
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Example:
In part a, (x+7)^5 = (4-6x)^5. Here there is one exponential expression on
each side of the equality. The exponents of each expression are equal. So,
by the second rule, the bases are equal, provided that they are both positive.
So x+7 = 4-6x. This rule results in having a linear equation, which can easily
be solved.
7x + 7 = 4
7x = -3
x = -3/7
Since the bases were assumed to be positive, a check needs to be made to
see that the bases are actually equal by plugging in the value for x into each
of the bases. Notice that each expression gives the same value. So x = -3/7
is the solution to this equation.
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Example:
In part b 8.1^3x-5 = 8.1^9. In this equation, the bases are positive and are
equal. So, by the first rule of equality, the exponents must be equal:
3x-5 = 9
3x = 14
x = 14/3
So the solution to this exponential equation is x = 14/3.
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Example:
In part c, the bases are the same, but the expression on the left side of the
equality symbol needs to be simplified. Using the Product Rule to simplify
this side: e^10+3x = e^x^2
Remember that e is a positive constant. So, by the first rule of equality,
10+3x = x^2, a quadratic equation!
So, 0 = x^2 – 3x – 10, with the expression on the right side being factored
into (x-5)(x+2).
Setting each of these factors equal to zero, results in x-5 = 0 and x+2 = 0.
This results in a pair of solutions to the equation: x=5 and x=-2.
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Example:
In part d neither the first rule nor the second rule of equality apply here. So
when faced with an equation like this, try to rewrite the bases on each side
of the equality so that they are equal. The bases here are 9 and 27. Notice
that both terms can be written as an exponential expression in terms of 3: 9
= 3^2 and 27 = 3^3. Now substitute these equivalent expressions into the
original equation:
(3^2)^x = (3^3)^2x-5
Using the Power Rule to simplify:
3^2x = 3^(6x-15)
Now that each side has been rewritten in terms of a common base, use the
first rule of equality. Since the bases are equal, the exponents must also be
equal:
2x = 6x -15.
-4x = -15
So x = 15/4.
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