Transformations of the exponential function

Exponential and logarithmic functions, grade 12 (4 units) Lesson 3 of 20Learning

Transformations of the exponential function

From

\displaystyle 3^x

\displaystyle -2\cdot3^{x-1}+4

The example below builds $\displaystyle {q(x)}={-2\cdot3^{x-1}+4}$ from $\displaystyle {f(x)}={3^x}$ in three steps, each a transformation of the graph before it:

$\displaystyle {g(x)}={3^{x-1}}={f(x-1)}$ : the graph of $\displaystyle f$ shifted 1 to the right. It passes through $\displaystyle (1,1)$ ; the asymptote stays $\displaystyle {y}={0}$ .
$\displaystyle {h(x)}={-2\cdot3^{x-1}}={-2g(x)}$ : stretched vertically by 2 and reflected in the $\displaystyle x$ -axis. Now it is negative and decreasing; the asymptote is still $\displaystyle {y}={0}$ .
$\displaystyle {q(x)}={-2\cdot3^{x-1}+4}={h(x)+4}$ : shifted 4 up. The asymptote moves with it, to $\displaystyle {y}={4}$ .

\displaystyle f(x)=3^{x}

\displaystyle g(x)=3^{x-1}

\displaystyle h(x)=-2\cdot3^{x-1}

\displaystyle q(x)=-2\cdot3^{x-1}+4

\displaystyle y=4

The properties of

\displaystyle q

The properties of $\displaystyle {q(x)}={-2\cdot3^{x-1}+4}$ follow from the steps:

The domain: every $\displaystyle x$ .
The horizontal asymptote: $\displaystyle {y}={4}$ , on the left.
The point on the $\displaystyle y$ -axis: $\displaystyle {q(0)}={-2\cdot3^{-1}+4}={-\frac{2}{3}+4}={3\frac{1}{3}}$ .
Decreasing for every $\displaystyle x$ , and $\displaystyle q(x)<4$ for every $\displaystyle x$ .
It meets the $\displaystyle x$ -axis where $\displaystyle {3^{x-1}}={2}$ – an equation solved with a logarithm, in a later lesson.

The transformations

Each change in the formula moves the graph of $\displaystyle f$ in its own way:

The formula	The graph of $\displaystyle f$	The asymptote $\displaystyle {y}={0}$ becomes
$\displaystyle f(x)+k$	shifted $\displaystyle k$ up ( $\displaystyle k<0$ : down)	$\displaystyle {y}={k}$
$\displaystyle f(x-h)$	shifted $\displaystyle h$ to the right ( $\displaystyle h<0$ : to the left)	$\displaystyle {y}={0}$
$\displaystyle -f(x)$	reflected in the $\displaystyle x$ -axis	$\displaystyle {y}={0}$
$\displaystyle f(-x)$	reflected in the $\displaystyle y$ -axis	$\displaystyle {y}={0}$
$\displaystyle k\cdot f(x)$ , $\displaystyle k>0$	stretched vertically ( $\displaystyle k<1$ : shrunk)	$\displaystyle {y}={0}$
$\displaystyle \lvert f(x)\rvert$	its part below the $\displaystyle x$ -axis reflected up	$\displaystyle {y}={0}$

Only a vertical shift moves a horizontal asymptote.
$\displaystyle f(-x)$ for $\displaystyle {f(x)}={3^x}$ is $\displaystyle {3^{-x}}={\left(\frac{1}{3}\right)^x}$ : increasing becomes decreasing.

The absolute value

\displaystyle \lvert2^x-4\rvert

The graph of $\displaystyle 2^x-4$ is below the $\displaystyle x$ -axis for $\displaystyle x<2$ ; the absolute value reflects that part up.

The point on the $\displaystyle x$ -axis: $\displaystyle {2^x}={4}$ , $\displaystyle {x}={2}$ . There the graph has a sharp corner, a minimum point $\displaystyle (2,0)$ .
On the left the graph comes close to $\displaystyle {y}={4}$ , the reflection of the asymptote $\displaystyle {y}={-4}$ .
Decreasing for $\displaystyle x<2$ , increasing for $\displaystyle x>2$ .

\displaystyle f(x)=2^{x}-4

\displaystyle g(x)=|2^{x}-4|

\displaystyle y=4

Exercise 1

The function: $\displaystyle {g(x)}={2^{x+1}-8}$ .

Mark the points where the graph of $\displaystyle g$ meets the axes.

\displaystyle g(x)=2^{x+1}-8

Solve the equation $\displaystyle {g(x)}={0}$ .

Write the equation of the asymptote of $\displaystyle g$ , and say whether $\displaystyle g$ is increasing or decreasing.

Exercise 2

The board shows the graph of $\displaystyle {f(x)}={2^x}$ and the graph of a transformation of it.
Write the formula of the second graph.

The second graph passes through $\displaystyle (2,2)$ and $\displaystyle (3,3)$ , its asymptote $\displaystyle {y}={1}$ .

\displaystyle f(x)=2^{x}

\displaystyle y=\,?

\displaystyle y=1

\displaystyle (2,2)

\displaystyle (3,3)

The second graph passes through $\displaystyle (0,2)$ and $\displaystyle (1,1)$ , its asymptote $\displaystyle {y}={3}$ .

\displaystyle f(x)=2^{x}

\displaystyle y=\,?

\displaystyle y=3

\displaystyle (0,2)

\displaystyle (1,1)

Exercise 3

The function: $\displaystyle {h(x)}={\lvert3^x-9\rvert}$ .

Mark the points where the graph of $\displaystyle h$ meets the axes.

\displaystyle h(x)=|3^{x}-9|

Write the asymptote of $\displaystyle h$ , its minimum point, and where $\displaystyle h$ is increasing and where decreasing.

Exercise 4

The function: $\displaystyle {k(x)}={5-\left(\frac{1}{2}\right)^x}$ .

Mark the point where the graph of $\displaystyle k$ meets the $\displaystyle y$ -axis.

\displaystyle k(x)=5-\left(\frac{1}{2}\right)^{x}

From which transformations of $\displaystyle \left(\frac{1}{2}\right)^x$ is $\displaystyle k$ built? Write its asymptote, and whether it is increasing or decreasing.

Formulas: exponential and logarithmic functions, grade 12 (4 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals and the area

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C$ $\displaystyle m\neq0$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$