What Does the Function f(x) = |x| + 6 Graph Transformation Tell Us About g(x) = f(2/3 x)?

The graph of g is the graph of f stretched horizontally by a factor of 3/2, so every point (x, y) on f maps to (3/2 x, y) on g. Because f(x) = |x| + 6, the result is g(x) = |2/3 x| + 6, a wider V-shape with the same vertex at (0, 6) and a shallower slope of 2/3 on each arm instead of 1.

The mechanism behind the horizontal stretch

A function transformation of the form f(kx) acts on the input before the function rule runs. For g(x) = f(2/3 x), the input x is first multiplied by 2/3, then fed into f. Because 2/3 is less than 1, the input reaching f is smaller in magnitude than x itself, so f produces each output only after x travels farther from zero.

The reversal is the core idea. A multiplier inside the function that is less than 1 stretches the graph horizontally; a multiplier greater than 1 compresses it. The horizontal scale factor equals the reciprocal of the input multiplier, so 1 ÷ (2/3) = 3/2 = 1.5.

Work through one point to confirm. On f, the point (6, 12) satisfies f(6) = |6| + 6 = 12. On g, the output 12 occurs where 2/3 x = 6, which gives x = 9. So (6, 12) on f becomes (9, 12) on g. The y-value held still while the x-value grew by 1.5.

The vertex never moves under a horizontal transformation. Plug x = 0 into g and the input to f is 0, so g(0) = f(0) = 6. The minimum point stays at (0, 6), and the axis of symmetry stays on the y-axis.

Quantified context: slopes, points, and scale factors

The slope of each arm changes in a precise, calculable way. On f, the right arm rises 1 unit for every 1 unit right, giving slope 1. On g, the right arm rises 1 unit for every 1.5 units right, giving slope 2/3 ≈ 0.667. The left arm mirrors this at −2/3. Horizontal stretch flattens slopes by the same 2/3 factor.

The table below tracks matching points across both graphs. Each row shows a y-value and where it lands on each function.

The pattern is linear: the farther a point sits from the vertex, the more its x-coordinate stretches. A point 2 units from the vertex moves 1 unit outward; a point 24 units out moves 12 units outward. Nothing vertical changes, so y-intercepts, minimums, and range remain identical.

The domain and range stay put as well. Both functions accept all real numbers and output values of 6 or greater, written as [6, ∞). Only the horizontal geometry changes.

Edge cases and the counterintuitive detail

The counterintuitive point is that multiplying by 2/3, a number less than 1, makes the graph wider rather than narrower. Students reasonably expect a "smaller" multiplier to shrink a figure. But the multiplier compresses the input, so the function needs a larger x to reach any given output. Wider input spacing means a wider graph.

Order of operations matters when transformations stack. In g(x) = f(2/3 x), the multiplication happens inside f's parentheses, so the stretch is horizontal. Writing 2/3 · f(x) instead would scale the output and stretch the graph vertically by 2/3, a completely different shape. Position, not just value, determines the effect.

Nonlinear pieces behave consistently. Gluing two segments with different slopes, adding a constant, or shifting the vertex all preserve the 3/2 horizontal ratio. The stretch applies to the input coordinate regardless of what f does with it.

A common arithmetic slip is computing the scale factor backward. If a student divides 2/3 by 1 instead of taking 1 ÷ (2/3), they report a compression by 2/3 and describe the wrong graph. Verify by testing one point, as above, rather than trusting memory.

Misconceptions worth correcting

The most persistent error is confusing horizontal and vertical transformations. A coefficient inside the function argument changes x-behavior and uses the reciprocal scale; a coefficient outside changes y-behavior and uses the coefficient directly. The two never swap.

Another misconception treats the constant +6 as part of the transformation. It is not. The +6 belongs to f itself and survives unchanged into g, keeping the vertex at (0, 6). Horizontal transformations do not move the vertex sideways when the axis of symmetry is already the y-axis.

Some students assume the graph's area or "size" grows by 1.5 in every direction. Only horizontal distances scale. Vertical heights are untouched, so the shape is stretched, not enlarged uniformly.

Watch for these red flags when checking work: Reporting a compression by 2/3 instead of a stretch by 3/2 Moving the vertex to (0, 4) or (6, 6) by mistake Changing the range or minimum value Applying the 2/3 to y-values instead of x-values

Practical actions for graphing and checking

Follow a short routine to graph g accurately and catch errors early.

Rewrite g in explicit form: g(x) = |2/3 x| + 6, factoring the 2/3 out of the absolute value when useful. Plot the vertex first at (0, 6), since horizontal stretches leave it fixed. Compute two or three matching points using the 3/2 factor, such as (3, 8) and (9, 12). Draw straight arms with slope 2/3 to the right and −2/3 to the left. Check symmetry by confirming g(−x) = g(x) at one or two x-values. Compare the finished graph against f at one shared y-value to confirm the 1.5 spacing.

The threshold for calling the transformation a stretch versus a compression is simple: an input multiplier strictly between 0 and 1 stretches, exactly 1 leaves the graph unchanged, and greater than 1 compresses. For this problem, 2/3 falls in the stretch band, so the answer is a horizontal stretch by 1.5.

Limits of what a single transformation shows

This analysis covers one transformation applied to one absolute-value function. It does not address combined transformations, such as f(2/3 x + 4), where an inside shift interacts with the stretch and changes the order of operations. Those require factoring the argument first.

It also says nothing about vertical scaling, reflections, or non-polynomial functions where periodic behavior can make horizontal stretches look like phase shifts. The reciprocal rule holds generally, but visual interpretation can mislead near asymptotes or periodic features.

The reasoning here is algebraic and graphical, not data-derived, so no empirical claims or external sources are needed. Verify any specific graph against its equation rather than relying on appearance alone.