Construct cube-root and absolute-value graphs

Try

A cube-root graph and an absolute-value graph both pass through the origin, but they answer different questions. The cube root asks which signed number cubes to the input. Absolute value measures distance from zero and is never negative. Build each graph from exact landmarks before applying shifts or reflections.

Predict

Predict the output for a negative input in each family. The cube root of −8 is −2, while the absolute value of −8 is 8. Predict whether connecting cube-root landmarks with sharp straight segments gives the exact curve. It does not: those segments are only an approximation, whereas the two straight rays of an absolute-value graph are exact.

Experiment

Construct a function graph

Choose a family and predict a landmark, one output and any missing inputs. Then reveal the graph. Changing a setting hides the result so you can make a new prediction. This is exploration; record your construction and justification in the lesson practice.

Function familyAbsolute valueCube rootCeiling stepsExponentialLogarithm Vertical factor a-2-112 Horizontal shift h-2-1012 Vertical shift k-2-1012 Probe input x-6-4-2-101246

y = 1 × |x|

Domain: all real x; the display shows only −6 ≤ x ≤ 6. The factor a multiplies outputs; h shifts the input location and k shifts outputs.

Reveal graph and valuesReset construction

Same rule, numerical view

At x=1: predict y before revealing.

Selected inputs and outputs

x	y
-4	Hidden
-2	Hidden
0	Hidden
2	Hidden
4	Hidden

The vertex is (0, 0); compare the two arms after reflecting or stretching.

Axes and unit lengths stay fixed. Parts of the graph outside the square are clipped, not absent from the function. Continuous curves use 481 fixed input locations, including every integer landmark in the window. Logarithms also use inverse samples across the visible output range, including exact intersections with its borders, so the curve continues to the edge near its excluded boundary. The formula determines the domain and exact values. Ceiling segments and endpoint inclusion are drawn from their exact intervals. Rounded decimals are displays, not proof. Settings are temporary and do not award mastery.

The live playground above has independent settings; the supplied graphs below are separate fixed examples.

Use supplied parent Graph C and Graph V, together with the exact coordinate table. The live family widget is a separate bounded playground: first predict a transformed landmark, then reveal the graph and compare. Its visible window does not define the whole domain.

Input x	cube_root(x)	Absolute value of x
−8	−2	8
−1	−1	1
0	0	0
1	1	1
8	2	8

The two supplied graphs use different displayed vertical ranges to show their shapes clearly; compare numerical labels rather than visual steepness across the panels.

Understand

The real cube-root function y=cube_root(x) is defined for every real x because cubing is strictly increasing and attains every real value. Exact points come from cubing convenient outputs: choose y=−2,−1,0,1,2 and obtain x=−8,−1,0,1,8. Its domain and range are all real numbers. It increases through the origin, with negative output on negative inputs and positive output on positive inputs.

For y=a cube_root(x−h)+k, a parent point (u,v) moves to (u+h,av+k). The horizontal change is read from the inside expression, and the vertical changes from outside. For example y=2 cube_root(x−1)+3 sends the parent origin to (1,3) and the parent point (8,2) to (9,7). A negative a reflects vertically as well as scaling distances from the shifted midline.

The function y=|x| equals −x for x<0 and x for x≥0. Its exact graph therefore consists of two rays meeting at a corner. For y=a|x−h|+k, the vertex is (h,k). If a>0, the range is y≥k; if a<0, it is y≤k. The branch slopes are −a on the left and a on the right. The domain is all real inputs unless a context restricts it.

A graph construction should state axes and scale, landmarks, how points are connected, endpoint or ray behavior, domain and range. A coordinate description can communicate these precisely without a drawing tool. Plotting five points alone does not specify a unique function unless the family and connecting shape are also stated.

A square-root branch is not a whole sideways parabola

For the principal real square root, y=√x requires x≥0 and y≥0. Choose nonnegative outputs 0, 1, 2 and 3 to obtain exact parent landmarks (0,0), (1,1), (4,2), (9,3). The graph has an included endpoint and curves to the right; unlike a cube root it has no negative-input branch. Its inverse relation x=y² must retain y≥0.

For example, construct y=1−√(x−2). The point map sends (u,v) to (u+2,1−v), producing (2,1), (3,0), (6,−1), (11,−2). The domain is x≥2, range y≤1, and the endpoint (2,1) is filled. The curve decreases continuously as x increases, with ever smaller downward changes per equal input increment: rationalizing gives √(x+h)−√x=h/(√(x+h)+√x) for x≥0 and h>0. Draw a smooth curve through the landmarks, not exact joined straight segments. There is no y-intercept because x=0 is outside the domain; the x-intercept is (3,0).

Squaring 1−y=√(x−2) yields (1−y)²=x−2. That equation alone also permits y=1+√(x−2). Keep the sign restriction y≤1 to recover exactly the original branch. Graphing software must receive the branch formula or this restriction; a plot of the unrestricted squared relation is a different object. The changed practice case asks you to construct another reflected and scaled square-root branch, including exact intercepts.

Apply

Construct a shifted cube-root curve and a reflected absolute-value graph on paper or as written coordinate descriptions. Check at least one point away from the center after applying the point map. For absolute value, verify the intercepts by solving the piecewise rule. For a root curve, use exact cubes to avoid relying on rounded calculator points near the steep center.

Continue

The family tells you how to connect landmarks; the domain tells you which inputs belong. Keep the cube-root curve continuous and the absolute-value corner sharp. Next, a ceiling function introduces genuine jumps, where open and closed endpoint marks carry information that a smooth curve would erase.

Continue to Graph ceiling steps and discrete decisions.