A graph can tell you how a familiar family was moved, reflected or stretched. In an absolute-value graph, the vertex identifies the horizontal and vertical shifts. A second point determines the signed vertical scale, and a third point checks the proposal. This is inference from supplied evidence, not simply plotting a formula you already know.
Predict
Predict whether two points determine a unique function. They determine a line if a linear model is assumed and their inputs differ. Without a family assumption, infinitely many functions can pass through them. Also predict whether a downward V could be caused by a horizontal shift alone. Its orientation requires a negative outside scale.
Experiment
Graphs F and G. Navy F is the parent absolute-value graph. Red G is an exact downward V with vertex (1,4) and zeros at −1 and 3.
Supplied Graph F is the parent y=|x|. Supplied Graph G is known to be a transformed absolute-value graph. Its exact landmarks are listed without providing its formula.
x
F(x)
G(x)
−1
1
0
0
0
2
1
1
4
2
2
2
3
3
0
Read the vertex of G from the plot, infer a rule of the form a|x−h|+k, and reserve one nonvertex point as a check. The lines extend beyond the displayed window; there are no finite domain endpoints.
Understand
For a parent graph y=f(x), the rule g(x)=a f(x−h)+k sends a parent point (u,v) to (u+h,av+k). The input must increase by h so that x−h equals the old input u. The output is then scaled by a and shifted by k. A negative a reverses vertical direction. This point map is often safer than memorizing a verbal left/right rule without checking it.
For absolute value, the parent vertex is (0,0), so the transformed vertex is (h,k). A nonvertex point (x,y) satisfies y−k=a|x−h|; divide by the nonzero distance |x−h| to find a. Use a third point to detect a reading or arithmetic error. The sign of a determines whether the vertex is a minimum or maximum.
For a quadratic known to have form a(x−h)²+k, its vertex similarly gives h,k, while another point determines a. This does not mean a graph uniquely distinguishes every imaginable sequence of transformations: for example f(2x)=4x² when f(x)=x², so a horizontal compression can produce the same quadratic as a vertical stretch. The inferred rule is precise, while the verbal transformation history may not be unique.
Domains transform too. If a parent is available only for u in a set D, then g is defined where x−h belongs to D. Outside scaling and translation do not create new input values. If a proposed graph extends past those transformed endpoints, it represents a different domain from the given function.
Finite observations without a family assumption are underdetermined. A line and a parabola can share two points; a more complicated function can share many sampled points. Report what is assumed about the family and what additional landmark was used to test it.
Apply
Construct a transformed graph from an unfamiliar parent point table, then reverse the process from a supplied graph. In both directions write the coordinate map, not just the transformation names. A written description should make clear which landmark locates a shift, which determines a scale and which independently checks the result.
Continue
Graph inference combines visual reading with a model assumption. The next lesson reverses input and output roles without first finding a formula, using unseen tables and graphs to read inverse values and decide whether an inverse function is actually available.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Supplied Graph G is known to have form a|x−h|+k. What is h?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Using Graph G’s vertex and point (2,2), what is a in a|x−h|+k?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Recover the full rule for Graph G from its supplied graph, and check a point not used to find the scale.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A quadratic is specified by vertex (−2,1) and passes through (0,9). Assuming y=a(x−h)²+k, recover the rule and describe five plotting landmarks.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Only the two points (0,1) and (2,5) are supplied, with no family assumption. Can a unique function be recovered? Give two different rules through them.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A parent graph f is defined only on [−2,3]. For g(x)=−3f(x−4)+2, give the new domain and the image of parent point (1,−2).
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
