Graph transformed exponentials and logarithms

Try

An exponential graph can approach a horizontal line without touching it. A logarithmic graph can approach a vertical line while its outputs grow without bound. Shifts move those asymptotes along with the curve. To construct either family, combine exact landmarks with domain and end behavior; a few plotted points alone can hide the most important feature.

Predict

Predict the effect of replacing base 2 by base 1/2. Since powers of 1/2 decrease as the exponent increases, the exponential becomes decreasing. Its inverse logarithm also decreases. A base between zero and one does not make the logarithm undefined; positive input is still required, and the base must be positive and different from one.

Experiment

Supplied Graph E shows E(x)=2^(x−1)−2, and Graph L shows L(x)=log2(x+2)−1. The dashed lines are asymptotes, not additional function branches.

Graph	Exact landmarks	Asymptote
E	(0,−3/2),(1,−1),(2,0),(3,2)	y=−2
L	(−3/2,−2),(−1,−1),(0,0),(2,1),(6,2)	x=−2

Read one intercept from each supplied curve and check it algebraically. These curves are different transformed families, not claimed inverses of one another. The plotting window displays only part of their domains.

Understand

For y=b^(x−h)+k with b>0 and b≠1, the exponential part is always positive. Therefore the domain is all real numbers, the horizontal asymptote is y=k and the range is y>k. If an outside negative factor is included, the range lies below k instead. For b>1 the basic curve increases; for 0<b<1 it decreases. Locate an x-intercept by solving b^(x−h)=−k, which requires a positive right side.

For y=log_b(x−h)+k, the logarithm’s input must be positive, so x>h. The vertical asymptote is x=h and the range is all real numbers. Use the inverse relationship b^u=v exactly when log_b(v)=u. Convenient input offsets are powers of b: x=h+b^u gives output u+k. This generates exact landmarks without rounding a logarithm on a calculator.

For the decreasing base 1/2, increasing x makes log_(1/2)(x) smaller. As its positive input approaches zero, its output approaches positive infinity; as the input grows, the output approaches negative infinity. These end directions reverse the familiar base-two logarithm, but its domain remains positive.

Shifts act on different parts of the rule. Adding inside a logarithm changes its domain boundary and moves the graph horizontally. Adding outside changes output heights and leaves the domain boundary unchanged. A graph that looks nearly flat or vertical near a window edge does not prove an output is attained or that the domain ends there. Use the algebraic inequality to decide endpoint inclusion.

Apply

Produce one shifted decreasing exponential and one shifted decreasing logarithm. Your graph description should list axes, at least three exact landmarks, asymptote, domain, range, intercepts and both end directions. If an intercept does not exist, explain why. Use a calculator only for an intercept that lacks a convenient exact landmark, and label its approximation.

Continue

The same point-mapping ideas used for roots and absolute value apply here, but domains and asymptotes must move consistently. Next, work backward: use supplied graphs to infer a transformation, then test it against an additional landmark rather than assuming one visible feature determines the whole rule.

Continue to Recover transformations from supplied graphs.