Concavity Under Vertical and Horizontal Transformations

Why Transformations Matter for Concavity

A graph can be shifted, reflected, or stretched without changing its basic shape in arbitrary ways. In particular, these transformations have predictable effects on where the graph is concave up or concave down, and on the coordinates of its inflection points. Knowing those effects lets you analyze a transformed function without starting its concavity analysis from scratch.

The key distinction is between changing the graph vertically and changing it horizontally. A vertical reflection reverses concavity: it turns concave up into concave down and vice versa. A horizontal reflection does not reverse the concavity type at corresponding points, although it swaps which side of the graph corresponds to the left or right side of the original.

A General Transformation Formula

Let \(f\) be a function, and consider the transformed function

where \(A\) and \(B\) are nonzero constants and \(H\) and \(K\) are constants. The transformed function is defined at each \(x\) for which \(B(x-H)\) belongs to the domain of \(f\).

In this formula, \(H\) gives a horizontal translation, and \(K\) gives a vertical translation. The factor \(B\) controls horizontal scaling and reflection: its magnitude changes horizontal scale, while a negative value also reflects the graph across the vertical axis relative to its horizontal center. The factor \(A\) controls vertical scaling and reflection: a negative value reflects the graph across the horizontal axis.

For a twice-differentiable function, the sign of \(f''\) identifies concavity on an interval: \(f''(x)>0\) means concave up there, and \(f''(x)<0\) means concave down there. A change in the sign of \(f''\) across a point is a useful way to confirm a change in concavity when the second derivative exists on both sides.

How the Second Derivative Changes

Differentiate \(g(x)=A f(B(x-H))+K\) using the chain rule. The constants \(A\) and \(K\) do not depend on \(x\), and the inner function \(B(x-H)\) has derivative \(B\). Differentiating a second time introduces another factor of \(B\).

This formula requires \(f\) to be twice differentiable at the input \(B(x-H)\). The factor \(B^2\) is positive because \(B\ne 0\). Therefore, horizontal scaling—including horizontal reflection—does not itself reverse the sign of the second derivative at corresponding points. The sign is multiplied by \(A\): positive \(A\) preserves the concavity type, while negative \(A\) reverses it.

Deriving the Inflection-Point Coordinates

Suppose \((u,f(u))\) is an inflection point of \(f\). To find the corresponding horizontal coordinate on the transformed graph, set the input to \(f\) equal to \(u\): \(B(x-H)=u\). Since \(B\ne 0\), this equation has the unique solution \(x=H+u/B\). The transformed vertical coordinate is obtained by applying the vertical transformation to \(f(u)\).

This point is an inflection point of \(g\). To see why, note that the horizontal change of variables \(u=B(x-H)\) maps neighborhoods of the original point to neighborhoods of the transformed point. If \(B<0\), it reverses their left-right order; it does not remove the change in concavity. Multiplication by \(A>0\) preserves concavity, while multiplication by \(A<0\) exchanges concave up and concave down. Either way, a change in concavity remains a change in concavity. The transformed graph is therefore an inflection-point image, not merely a point where its second derivative might be zero.

The argument also explains why a horizontal reflection can swap which side is concave up without changing the concavity type at corresponding locations. The two sides are exchanged, while the concavity on each matching piece remains the same if \(A>0\).

Worked Examples

A Common Mistake: Giving a Horizontal Reflection a Negative Second-Derivative Factor

A common error is to assume that a negative horizontal scale reverses concavity. For example, one might incorrectly differentiate \(g(x)=f(-2x)\) twice and claim that \(g''(x)=-4f''(-2x)\). That is not the chain rule: the inner derivative \(-2\) appears twice, so the product is \((-2)(-2)=4\), not \(-4\).

For \(f(x)=x^3\), this gives \(g''(x)=4\cdot 6(-2x)=-48x\), not \(48x\). A horizontal reflection changes which original side corresponds to each new side. It does not, by itself, flip concave up to concave down at the corresponding locations. A negative vertical factor is what reverses the concavity type.

Using the Transformation Rules Reliably

For a transformed function \(g(x)=A f(B(x-H))+K\), identify \(A\), \(B\), \(H\), and \(K\) before drawing conclusions. Then use \(AB^2 f''(B(x-H))\) for the second derivative and solve \(B(x-H)=u\) to map an original inflection coordinate \(u\). This keeps the concavity calculation and the coordinate calculation separate, reducing sign and placement errors.

The nonzero conditions matter. If \(B=0\), the input to \(f\) is constant, so the graph is constant wherever defined and the horizontal coordinate map cannot be inverted. If \(A=0\), the transformed graph is also constant, regardless of the shape of \(f\). Neither case is an invertible scaling of the graph, so the transformation and inflection correspondence above do not apply.

These rules are useful when comparing families of graphs or interpreting a function after a change of scale or origin. They also provide a quick check on a sketch: a vertical reflection must exchange concavity types, while a horizontal reflection must exchange left and right correspondence. Translations change locations, but do not alone alter concavity.