Search topics

Search

Analyse exponential and logarithmic graphs

Start a set of 8
Step 5 of 5

Which statement gives the correct horizontal asymptote?

Year 12, VCE Units 3 & 4, Chapter 2, 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

\[y=a b^{n\left(x-h\right)}+k\quad\text{and}\quad y=a\log_{b}\left(n\left(x-h\right)\right)+k\]
  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 \(\displaystyle x\in\mathbb{R},\quad y>k\text{ when }a>0\quad\text{or}\quad n\left(x-h\right)>0\)
  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 y = −4.
  2. \(\displaystyle y=-4\)Therefore the horizontal asymptote is y = −4.
  3. \(\displaystyle x=-1\quad\Longrightarrow\quad x+2=1\)Substitute x = −1 to find the exponent at the requested input
  4. \(\displaystyle f(-1)=3^{1}-4=3-4=-1\)Evaluate the power and then apply the outside translation, giving f(−1) = −1.

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 k as a horizontal limiting value y=k.

How does an inside translation affect a graph?+

In x-h, a positive h moves the graph right by h; for a logarithm it also moves the vertical asymptote to x=h.

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 2: Functions and graphs

In this chapter

Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model