Analyse exponential and logarithmic graphs

What should be identified first when analysing a transformed exponential or logarithmic graph?

Function

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

Change h through all five settings and compare key points

Which description works for every displayed shift?

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

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 \(\displaystyle x\in\mathbb{R},\quad y>k\text{ when }a>0\quad\text{or}\quad n\left(x-h\right)>0\)
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 \(\displaystyle y=2^{2\left(x-1\right)}+3\quad\Longrightarrow\quad \text{horizontal factor }\frac{1}{2},\text{ right 1, up 3}\)

Transformed rule

\(\displaystyle y=2^{x-1}+3,\quad x=2\)

For the transformed rule shown, calculate the exponent when x = 2.

Transformed rule

\(\displaystyle y=2^{x-1}+3,\quad x=2\)

Calculate the y-coordinate when x = 2.

Which statement gives the horizontal asymptote of the transformed function?

Construct the curve, then apply its domain

Prepare exact key points, intercepts, domain/range and asymptotes before joining the curve. Use transformations and logarithm laws, not differentiation. Keep an unrestricted tail distinct from a specified finite endpoint

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics without calculus Substitute labelled points and use tables or simple inequalities to check the shape and end behaviour. A plotted model is feedback, not your submitted drawing
4. 4Transform the coordinates For y=A f(bx)+c with A and b nonzero, a parent point (u,v) maps to (u/b,Av+c). A negative input multiplier reverses the input direction. Exponential/logarithmic bases must be positive and not equal to one; every logarithm argument must be positive \(\displaystyle (u,v)\mapsto(u/b,Av+c)\)
5. 5Check your drawing Full drawings and explanations are ungraded. Compare your work with the feedback; short entries check individual values only

Exponential key points and the omitted bound

Map the parent points (−1,1/2),(0,1),(1,2): they become (0,−3/2),(1,−1),(2,0). The curve increases, stays above y=−2 and approaches that horizontal asymptote to the left. Domain R; range (−2,∞). The asymptote is not an attained minimum

Worked rule

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

Evaluate the function at x = 1.

Your complete exponential or logarithmic drawing

Sketch the function by hand from parent key points. Label domain, range, intercepts and asymptote, and show the direction of growth/decay. Explain why it has no x-intercept

Review the whole graph and its domain

Points (−1,4),(0,2),(1,4/3). Domain R, range y>1, horizontal asymptote y=1. The curve decreases from unbounded positive values towards 1 as x increases. There is no x-intercept because 3^(−x)+1>1.

Logarithms preserve their input restriction

Require x+1>0, so the vertical asymptote is x=−1 and the domain is (−1,∞). Map parent points (1/e,−1),(1,0),(e,1) to (1/e−1,−2),(0,−1),(e−1,0). The curve increases, with range R. It falls without bound near x=−1 and grows without bound to the right

Worked rule

\(\displaystyle f(x)=\ln(x+1)-1\)

Find f(0) exactly

Your complete exponential or logarithmic drawing

Sketch the function. Use exact powers of ten to choose points, state domain/range, identify the vertical asymptote and find any intercepts. Explain why a y-intercept is unavailable

Review the whole graph and its domain

Domain x>2, range R, vertical asymptote x=2. Points (21/10,0),(3,1),(12,2); x-intercept 21/10. There is no y-intercept because x=0 is outside the domain. The increasing curve approaches −∞ near 2 and is unbounded above as x increases

A specified domain is not a viewing window

The specified domain is [1/2,4), not the full positive half-line. Draw the increasing logarithmic segment from the filled point (1/2,−1) through (1,0) and (2,1) to the open point (4,2). Its range is [−1,2). The x-intercept is (1,0); there is no y-intercept. The unrestricted parent has vertical asymptote x=0, but this restricted graph never approaches zero and has no infinite branch there. Do not extend the segment beyond either domain endpoint merely because the axes continue

Worked rule

\(\displaystyle p(x)=\log_2 x,\quad \frac12\le x<4\)

What is the included minimum output of the restricted model p?

Your complete exponential or logarithmic drawing

On separate axes, construct g(x)=2^(−2x)+1 for −1≤x≤1, and h(x)=log₁/₂(x)+1 for 1/2≤x<4. For each, use the transformation of exact parent points, label every intercept and included/excluded endpoint, state the exact domain and range, and show whether it increases or decreases. Also sketch each on its maximal real domain and label its asymptote. Explain what is lost when the graph is restricted, and why a finite plotting window alone would not impose a new mathematical domain

Review the whole graph and its domain

For restricted g, the points are (−1,5),(−1/2,3),(0,2),(1/2,3/2),(1,5/4), with both endpoints filled. It decreases, has domain [−1,1], range [5/4,5], y-intercept (0,2) and no x-intercept. On R it instead has range (1,∞) and horizontal asymptote y=1. For h, log₁/₂x=−log₂x: points (1/2,2) filled, (1,1), (2,0) and (4,−1) open. It decreases with domain [1/2,4), range (−1,2], x-intercept (2,0) and no y-intercept. On x>0 it has range R and vertical asymptote x=0; it rises without bound near zero and falls without bound to the right. Neither restricted segment approaches its parent asymptote. Parent points map by (u,v)→(−u/2,v+1) for g and (u,v)→(u,−v+1) from log₂ for h.