Stretches, Compressions, and Reflections of Exponential Curves


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Modifying the Exponential Function

Recall

The basic form of an exponential function is given by \(f(x) = c^x\) where \(c \gt 0\) and \(c \neq 1\).

 

Explore This 1

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Explore This 1 Summary

In this activity, you may have observed that the curve \(y = a \cdot 2^{bx}\) is a transformed version of \(y = 2^x\).

 

Explore This 1 Summary Continued

Did You Know?

The absolute value of a number \(a\) is denoted \(\vert a \vert\) and is equal to the number without the sign.

 

Explore This 1 Summary Continued

In this activity, you may have observed that the curve \(y = a \cdot 2^{bx}\) is a transformed version of \(y = 2^x\).

  • The curve is reflected in the \(x\)-axis if ​​​​​​\(a\) is negative.
  • The curve is reflected in the \(y\)-axis if \(b\) is negative.

 

Example 1 — Part A

Let \(f(x) = 3^x\).

 

Example 1 — Part B

Let \(f(x) = 3^x\). Find an equation for each of the following functions and then graph it alongside \(y = f(x)\).

  1. \(g(x) = 2 f(x) \)
  2. \(h(x) = f(2x)\)
  3. \(j(x) =f(-x)\)
  4. \(k(x) = -f(x)\)

Solution — Part B

 

Example 1 — Part C

Let \(f(x) = 3^x\). Find an equation for each of the following functions and then graph it alongside \(y = f(x)\).

  1. \(g(x) = 2 f(x) \)
  2. \(h(x) = f(2x)\)
  3. \(j(x) =f(-x)\)
  4. \(k(x) = -f(x)\)

Solution — Part C

The function \(j(x)\) is given by

\[j(x) = f(-x) \class{timed in1}{= 3^{-x}}\]

 

Example 1 — Part D

Let \(f(x) = 3^x\). Find an equation for each of the following functions and then graph it alongside \(y = f(x)\).

  1. \(g(x) = 2 f(x) \)
  2. \(h(x) = f(2x)\)
  3. \(j(x) =f(-x)\)
  4. \(k(x) = -f(x)\)

Solution — Part D

The function \(k(x)\) is given by

\[k(x) = -f(x) \class{timed in1}{= -3^{x}}\]

 

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Check Your Understanding 1


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Example 2

Let \(y=g(x)\) be the curve obtained by applying a horizontal stretch of magnitude \(2\) and a reflection in the \(x\)-axis to the curve \(y = f(x) = 4^x\). Determine an equation for \(g(x)\).

Solution

A horizontal stretch by a factor of \(2\) in the graph of a function is accomplished by multiplying the input by a factor of \(\dfrac{1}{2}\).

\[ x \rightarrow \frac{1}{2}x\]

A reflection in the \(x\)-axis is accomplished by negating the output.

\[f(x) \rightarrow -f(x)\]

These transformations can be combined as follows:

\[ f(x) \rightarrow -f\left(\frac{1}{2} x\right)\]

Therefore, the function \(g(x)\) is given by

\[ g(x) = -f\left(\frac{1}{2} x\right) = - 4^{\frac{x}{2}}\]

Check Your Understanding 2


Let \(f(x) = (((b)*(a))*(s))*2.718281828459045^x\) and let ‌\(y=g(x)\) be the curve obtained by applying the following transformations to \(y = f(x)\):

((((((((((f)*(b))*(S))*(t))*(a))*(r))*(t))*(L))*(i))*(s))*(t)

$qReflectionX(details...)

$qReflectionY(details...)

((((((((((q)*(V))*(s))*(t))*(r))*(e))*(t))*(c))*(h))*(fact(o)))*(r)

((((((((((q)*(H))*(s))*(t))*(r))*(e))*(t))*(c))*(h))*(fact(o)))*(r)

(((((((((f)*(b))*(S))*(t))*(o))*(p))*(L))*(i))*(s))*(t)

Determine an equation for ‌\(g(x)\).

Enter \(\frac{1}{3} \cdot 2 ^{-\frac{x}{2}}\) as "1/3(2)^(-x/2)".

\(g(x)\) =  There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

null

 

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