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Analyse exponential and logarithmic graphs

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Features of exponential and logarithmic graphs

Start with the basic graph family and then map its features under each transformation. The asymptote, domain and range distinguish the two families and locate the translated graph

\[y=a b^{n\left(x-h\right)}+k\quad\text{and}\quad y=a\log_{b}\left(n\left(x-h\right)\right)+k,\quad a\ne0,\;n\ne0,\;b>0,\;b\ne1\]
  1. 1Read the exponential asymptote In a transformed exponential, the outside translation gives the horizontal asymptote \(\displaystyle y=a b^{n\left(x-h\right)}+k\quad\Longrightarrow\quad y=k\)
  2. 2Read the logarithmic asymptote In a transformed logarithm, the inside translation gives the vertical asymptote and the argument must be positive \(\displaystyle y=a\log_{b}\left(n\left(x-h\right)\right)+k\quad\Longrightarrow\quad x=h\)
  3. 3State domain and range An exponential has domain all real x and a range on one side of its horizontal asymptote. A logarithm has range all real y and a domain set by its positive argument \(\displaystyle \text{Exponential: }x\in\mathbb{R},\quad y>k\text{ when }a>0,\quad y<k\text{ when }a<0;\quad\text{logarithm: }n\left(x-h\right)>0,\quad y\in\mathbb{R}\)
  4. 4Map the transformations The absolute outside scale gives the vertical dilation and the absolute input scale gives the reciprocal horizontal dilation; negative scales reflect the graph \(\displaystyle y=2^{2\left(x-1\right)}+3\quad\Longrightarrow\quad \text{horizontal factor }\frac{1}{2},\text{ right 1, up 3}\)
Year 12, VCE Units 3 & 4, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Features of exponential and logarithmic graphs

Start with the basic graph family and then map its features under each transformation. The asymptote, domain and range distinguish the two families and locate the translated graph

\[y=a b^{n\left(x-h\right)}+k\quad\text{and}\quad y=a\log_{b}\left(n\left(x-h\right)\right)+k,\quad a\ne0,\;n\ne0,\;b>0,\;b\ne1\]
  1. Read the exponential asymptote In a transformed exponential, the outside translation gives the horizontal asymptote \(\displaystyle y=a b^{n\left(x-h\right)}+k\quad\Longrightarrow\quad y=k\)
  2. Read the logarithmic asymptote In a transformed logarithm, the inside translation gives the vertical asymptote and the argument must be positive \(\displaystyle y=a\log_{b}\left(n\left(x-h\right)\right)+k\quad\Longrightarrow\quad x=h\)
  3. State domain and range An exponential has domain all real x and a range on one side of its horizontal asymptote. A logarithm has range all real y and a domain set by its positive argument \(\displaystyle \text{Exponential: }x\in\mathbb{R},\quad y>k\text{ when }a>0,\quad y<k\text{ when }a<0;\quad\text{logarithm: }n\left(x-h\right)>0,\quad y\in\mathbb{R}\)
  4. Map the transformations The absolute outside scale gives the vertical dilation and the absolute input scale gives the reciprocal horizontal dilation; negative scales reflect the graph \(\displaystyle y=2^{2\left(x-1\right)}+3\quad\Longrightarrow\quad \text{horizontal factor }\frac{1}{2},\text{ right 1, up 3}\)

Worked example

State the horizontal asymptote and calculate the value at the given input

\[f(x)=3^{x+2}-4,\quad x=-1\]
  1. \(\displaystyle f(x)=3^{x+2}-4\quad\Longrightarrow\quad k=-4\)The outside translation gives the horizontal asymptote
  2. \(\displaystyle y=-4\)This is the horizontal asymptote
  3. \(\displaystyle x=-1\quad\Longrightarrow\quad x+2=1\)Substitute the given input to find the exponent
  4. \(\displaystyle f(-1)=3^{1}-4=3-4=-1\)Evaluate the power and then apply the outside translation

Common mistakes

Leaving a translated asymptote at zero
✕\(\displaystyle f(x)=2^{x-1}+3\quad\text{has horizontal asymptote }y=0\)✓\(\displaystyle f(x)=2^{x-1}+3\quad\text{has horizontal asymptote }y=3\)
The outside translation moves the horizontal asymptote from y = 0 to y = 3
Using the basic y-intercept
✕\(\displaystyle f(x)=2^{x-1}+3\quad\Rightarrow\quad f(0)=1\)✓\(\displaystyle f(x)=2^{x-1}+3\quad\Rightarrow\quad f(0)=2^{-1}+3=\frac{7}{2}\)
A y-intercept is found by substituting x = 0 into the complete transformed rule
Deciding growth from the coefficient sign alone
✕\(\displaystyle f(x)=3\left(\frac{1}{2}\right)^{x}+2\quad\Rightarrow\quad a>0\text{ so the graph is increasing}\)✓\(\displaystyle f(x)=3\left(\frac{1}{2}\right)^{x}+2\quad\Rightarrow\quad 0<\frac{1}{2}<1\text{ so the graph is decreasing}\)
The base and the input multiplier determine the exponential direction, while a negative outside scale can reverse it

Difficulty levels

introA basic exponential or logarithm is analysed with an unshifted asymptote and landmark points
standardOne translation or reflection is used to complete a feature table
challengeCombined transformations and reverse feature-to-rule reasoning determine an exact parameter
examA generated graph or rule is selected and two exact features are recorded

Questions about this topic

Why is an exponential asymptote horizontal?+

The exponential term approaches zero in one direction, leaving the outside translation as the horizontal asymptote

How does an inside translation affect a graph?+

A positive inside translation moves the graph right; for a logarithm it moves the vertical asymptote by the same amount

How can growth or decay be determined?+

Compare the base with 1, account for the sign of the input multiplier, and then check whether the outside scale reverses the direction

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 4: Logarithmic functions

In this chapter

Connect exponential and logarithmic formsApply logarithm laws and inverse identitiesEvaluate logarithms and antilogarithmsExpress one logarithmic quantity in terms of anotherUse a formula-bearing logarithmic scaleSolve exponential equations and inequalities exactlySolve exponential and logarithmic equationsSolve literal logarithmic and indicial equationsAnalyse exponential and logarithmic graphsConstruct exponential and logarithmic inverse pairsInterpret and use an exponential modelInterpret and use a logarithmic modelDetermine and use a logarithmic model