Reflections and Stretches

17 min read

Textbook Reference

Both sources are free and openly licensed. OpenStax Calculus Volume 1 Section 1.3 covers reflections and scalings as the second class of transformation from an existing function.

Try This First

Here is a table of values for a function $f$.

Before reading further, predict the following:

1. Write your prediction for the table of $g(x) = -f(x)$. What do you expect happens to each output value?
2. Write your prediction for the table of $h(x) = f(-x)$. What do you expect happens to each input value?
3. Write your prediction for the table of $p(x) = 2f(x)$. What do you expect happens to each output value?

Work through the three predictions on paper, then build the actual tables to compare.

These three transformations differ in what they change: flipping outputs, flipping inputs, and scaling outputs. Notice which ones require knowing values that are outside the original table.

Key idea

The previous skill (Vertical and Horizontal Shifts) dealt with adding constants. This skill deals with multiplying by constants. The two types of transformation are genuinely different in how they affect a graph.

Adding to the output: moves the graph up or down without changing its shape.

Multiplying the output: stretches or compresses the graph vertically, and can flip it upside down.

Adding to the input: moves the graph left or right without changing its shape.

Multiplying the input: stretches or compresses the graph horizontally, and can flip it sideways.

The most common confusion is treating multiplication as if it were addition -- expecting a factor of 2 to shift the graph rather than to stretch it. That confusion has a name (see the Named Misconception section), and recognizing it is half the work.

Prerequisite Check

Before this lesson, make sure you can do all of the following:

• State the effect of $f(x) + c$ (vertical shift) and $f(x + c)$ (horizontal shift)
• Evaluate $f(-3)$, $-f(3)$, and $f(3)$ from a table and distinguish among them
• Multiply negative numbers: $(-1)(-4) = 4$, $(-1)(5) = -5$
• Identify the $x$-intercepts and $y$-intercept of a graph
• Recognize the shape of $y = x^2$ and $y = \sqrt{x}$

If any of these is uncertain, review Vertical and Horizontal Shifts first.

Quick Reference

Reflections.

Vertical scaling (outside the function).

Horizontal scaling (inside the function).

Memory device: Outside affects output (vertical). Inside affects input (horizontal). Both multiplications are opposite in the horizontal case: $c > 1$ inside compresses (narrows) the graph.

Key Concepts

1. Reflecting Over the $x$-Axis: Negating the Output

The transformation $g(x) = -f(x)$ multiplies every output by $-1$. Every positive output becomes negative; every negative output becomes positive; zero stays at zero.

Why this is true. The graph of $f$ consists of points $(x, f(x))$. The graph of $g(x) = -f(x)$ consists of points $(x, -f(x))$. For each $x$, the $x$-coordinate is unchanged and the $y$-coordinate is negated. Negating the $y$-coordinate is a reflection over the $x$-axis.

Numerical representation. Using the Try This First table:

Every output is negated. Points above the $x$-axis move below it, and points below move above. Points on the $x$-axis (where $f(x) = 0$) stay fixed.

Translation prompt. Look at the table. Which output values are positive in $f$ and negative in $g$? Which are negative in $f$ and positive in $g$? Can you describe in words, without using the formula, how the two graphs are related? Then write the formula that produces that relationship.

Graphical meaning. If the graph of $f$ has a peak at $(2, 5)$, the graph of $g = -f$ has a trough at $(2, -5)$. Every peak becomes a trough and vice versa. The graph is flipped upside down about the $x$-axis.

Ask why. If $f$ is an even function satisfying $f(-x) = f(x)$, is $g(x) = -f(x)$ also even? Why? If $f$ has a zero at $x = a$, what can you say about $g$ at $x = a$?

2. Reflecting Over the $y$-Axis: Negating the Input

The transformation $g(x) = f(-x)$ replaces the input $x$ with $-x$ before passing it to $f$. The output at position $x$ is the same as the output of $f$ at position $-x$.

Why this is true. The point $(x, f(x))$ on the graph of $f$ corresponds to the point $(-x, f(x))$ on the graph of $g$, because $g(-x) = f(-(-x)) = f(x)$. The $y$-coordinate is unchanged; the $x$-coordinate is negated. Negating the $x$-coordinate is a reflection over the $y$-axis.

Numerical representation. Using the Try This First table:

Note that $g$ is now defined at the inputs $-4, -2, 0, 1, 3$, which are the negatives of the original inputs. The output at each new input is the original output at the mirrored input.

Translation prompt. The table for $f$ is defined on $\{-3, -1, 0, 2, 4\}$. The table for $g = f(-x)$ is defined on $\{-4, -2, 0, 1, 3\}$. Describe in words what happened to the domain. Why did the domain change? If the original domain of $f$ is $[a, b]$, what is the domain of $g(x) = f(-x)$?

Graphical meaning. The graph of $f(-x)$ is the mirror image of the graph of $f$ across the $y$-axis. A point to the right of the $y$-axis appears on the left, and vice versa, at the same height.

Ask why. For which functions does reflecting over the $y$-axis produce the same graph? That is, for which $f$ is it true that $f(-x) = f(x)$ for all $x$? (These are called even functions.) Give two examples and one non-example.

3. Vertical Scaling: Multiplying the Output

The transformation $g(x) = c \cdot f(x)$ multiplies every output by the constant $c$.

• If $c > 1$: the graph is stretched vertically (pulled away from the $x$-axis).
• If $0 < c < 1$: the graph is compressed vertically (pushed toward the $x$-axis).
• If $c = -1$: reflection over the $x$-axis (covered in Section 1).
• If $c < 0$: combination of vertical scaling and reflection.

Numerical representation. Let $f(x) = x^2$.

Every non-zero output in $g$ is 3 times the corresponding output in $f$: the parabola is taller and narrower in appearance. Every output in $h$ is half: the parabola is shorter and wider in appearance. Points on the $x$-axis ($f(x) = 0$) are fixed by any vertical scaling.

Translation prompt. In the table, which outputs change under vertical scaling? Which stay the same? Write a sentence describing, in terms of a graph, which points are fixed under $g(x) = c \cdot f(x)$ for any $c$.

Graphical interpretation. Vertical scaling stretches or compresses the graph in the $y$-direction only. The $x$-intercepts (zeros) do not move. The $y$-intercept is scaled by $c$.

4. Horizontal Scaling: Multiplying the Input

The transformation $g(x) = f(c \cdot x)$ multiplies the input by $c$ before passing it to $f$.

• If $c > 1$: the graph is compressed horizontally (features move toward the $y$-axis).
• If $0 < c < 1$: the graph is stretched horizontally (features move away from the $y$-axis).
• If $c = -1$: reflection over the $y$-axis (covered in Section 2).

The direction is opposite to the number: a factor greater than 1 inside the argument compresses the graph. This parallels the horizontal shift surprise: inside changes move the graph in the opposite direction from what the number suggests.

Why this is true. Suppose $f$ achieves output $y_0$ at $x = a$. For $g(x) = f(cx)$, the same output occurs when $cx = a$, that is, $x = a/c$. Every feature of $f$ at $x = a$ appears in $g$ at $x = a/c$.

• $c > 1$: $a/c < a$, so features move toward the $y$-axis (compressed).
• $0 < c < 1$: $a/c > a$, so features move away from the $y$-axis (stretched).

Numerical representation. Let $f(x) = x^2$ with vertex at $(0,0)$ and points $(1,1)$, $(2,4)$, $(3,9)$.

For $g(x) = f(2x) = (2x)^2 = 4x^2$: the point that was at $(3, 9)$ in $f$ now appears at $(3/2, 9)$ in $g$. Check: $g(3/2) = f(2 \cdot 3/2) = f(3) = 9$.

For $h(x) = f(x/2) = (x/2)^2 = x^2/4$: the point that was at $(3, 9)$ in $f$ now appears at $(6, 9)$ in $h$. Check: $h(6) = f(6/2) = f(3) = 9$.

Translation prompt. From the paragraph above, $g(x) = 4x^2$ is both a horizontal compression by factor 2 and a vertical stretch by factor 4. Can you write two different formulas for the same function? Does every horizontal scaling have an equivalent vertical scaling? (Think about $g(x) = f(cx)$ and what it equals for the specific case $f(x) = x^2$. Then think about whether the same equivalence holds for $f(x) = |x|$ or $f(x) = \sin x$.)

Ask why. Why does a factor $c > 1$ inside the argument compress (rather than stretch) the graph horizontally? Explain using the argument about where the same output reappears.

Named Misconception: multiplicative-not-additive

The tempting reasoning. Students who have just learned vertical shifts sometimes treat a vertical scale factor as a vertical shift. Specifically, they may reason: “The graph of $g(x) = 2f(x)$ is the graph of $f$ shifted up by 2 units, because the 2 is added to the output.”

A related error is treating a factor as a shift in the other direction: “The graph of $g(x) = f(2x)$ is the graph of $f$ shifted right by 2 units, because $2x$ is in the input.”

Why this breaks. Test the first error on $f(x) = x^2$. The table for $f$ at $x = 0$ gives $f(0) = 0$. A shift up by 2 would give $g(0) = 2$. But $g(x) = 2f(x) = 2 \cdot 0 = 0$ at $x = 0$. The graph does not move up at $x = 0$ -- it stays at the $x$-axis. Points on the $x$-axis are unchanged under vertical scaling. A shift would move every point, including those on the axis. Scaling does not.

Test the second error: if $g(x) = f(2x)$ were a shift right by 2, the zero of $g$ would be at $x = 2$. But $g(0) = f(0) = 0$: the zero is still at $x = 0$. The zero (and any other point on the $y$-axis) does not move under $f(cx)$.

The repair. Multiplication by a constant outside the function scales every output proportionally. It does not shift the graph; it stretches or compresses it. Points at zero output stay at zero. Points at large positive output grow larger. Multiplication by a constant inside compresses or stretches horizontally, moving features closer to or farther from the $y$-axis.

The simplest diagnostic: check whether the $y$-intercept ($x = 0$) and the $x$-intercepts (where $f = 0$) move. Under a vertical shift, the $y$-intercept moves. Under a vertical scale, the zeros are fixed and the $y$-intercept scales. These are genuinely different effects.

Worked Examples

Worked Example 1: Building Tables for Reflections

Problem. The function $f$ has the table of values:

Build tables for (a) $g(x) = -f(x)$ and (b) $h(x) = f(-x)$.

Predict first. For (a): what happens to the sign of each output? For (b): which inputs produce defined outputs? (The inputs of $h$ are the negatives of the inputs of $f$.)

Solution.

(a) $g(x) = -f(x)$: negate each output, keep each input.

Note that $x = -1$ gives $g(-1) = -f(-1) = 0$: it is still a zero. The $x$-intercept did not move; it is fixed under reflection over the $x$-axis.

(b) $h(x) = f(-x)$: for each $x$, compute $f(-x)$.

$h(-3) = f(3) = 6$, $h(-1) = f(1) = 1$, $h(0) = f(0) = -2$, $h(1) = f(-1) = 0$, $h(2) = f(-2) = 4$.

The domain of $h$ is $\{-3, -1, 0, 1, 2\}$, which is the mirror image of the domain of $f$ across the $y$-axis. The output at $x = 0$ is unchanged (since $f(-0) = f(0)$).

Worked Example 2: Identifying Transformations from Key Points

Problem. The graph of $f$ has vertex (minimum) at $(1, -3)$ and passes through $(3, 1)$.

Describe the graph of each:

(a) $g(x) = -f(x)$ (b) $h(x) = 3f(x)$ (c) $p(x) = f(2x)$

Predict first. For each transformation, predict: where does the vertex go? Does the vertex stay a minimum or become a maximum?

Solution.

(a) $g(x) = -f(x)$: reflection over $x$-axis. Negate all outputs.

Vertex: $(1, -3)$ becomes $(1, 3)$. The minimum of $f$ becomes a maximum of $g$.

Point $(3, 1)$: $g(3) = -f(3) = -1$. New point: $(3, -1)$.

The vertex is now a maximum at $(1, 3)$.

(b) $h(x) = 3f(x)$: vertical stretch by factor 3.

Vertex: $(1, -3)$ becomes $(1, -9)$ (multiply $y$-coordinate by 3). Still a minimum.

Point $(3, 1)$: $h(3) = 3 \cdot 1 = 3$. New point: $(3, 3)$.

The parabola is taller and the minimum is deeper at $(1, -9)$.

(c) $p(x) = f(2x)$: horizontal compression by factor 2. Each $x$-coordinate of a key feature is divided by 2.

Vertex: $x$-coordinate $1$ becomes $1/2$; $y$-coordinate unchanged at $-3$. New vertex: $(1/2, -3)$.

Check: $p(1/2) = f(2 \cdot 1/2) = f(1) = -3$. Correct.

Point $(3, 1)$: the input $x = 3$ in $f$ appears in $p$ at $x = 3/2$. $p(3/2) = f(2 \cdot 3/2) = f(3) = 1$. New point: $(3/2, 1)$.

The parabola is compressed horizontally; the vertex moved from $x = 1$ to $x = 1/2$.

Worked Example 3: Determining the Transformation from Two Graphs

Problem. The graph of $f$ passes through $(0, 0)$, $(1, 1)$, $(2, 4)$. The graph of $g$ passes through $(0, 0)$, $(1, -1)$, $(2, -4)$.

What transformation maps $f$ to $g$?

Predict first. Compare the $y$-values of $f$ and $g$ at each common $x$-value. What pattern do you see?

Solution.

At $x = 0$: $f(0) = 0$, $g(0) = 0$. Ratio: $g/f$ is $0/0$, undefined; but both are zero.

At $x = 1$: $f(1) = 1$, $g(1) = -1$. Every output is negated.

At $x = 2$: $f(2) = 4$, $g(2) = -4$. Confirmed: output negated.

The inputs are identical and each output is the negative of $f$’s output. This is a reflection over the $x$-axis: $g(x) = -f(x)$.

Check. If $f(x) = x^2$, then $g(x) = -x^2$. $g(1) = -1$, $g(2) = -4$. Matches. The transformation is confirmed.

One way to see this / Another way to see this: One could also ask: are the inputs different? No. Are the outputs scaled by a constant other than $-1$? No. So it is specifically the $-1$ multiple: a reflection, not a general stretch.

Worked Example 4: LFHC -- Identifying the Transformation Algebraically

Problem. Suppose $f(x) = \sqrt{x}$ and $g(x) = \sqrt{9x}$.

(a) Simplify $g(x)$ and identify it as a vertical or horizontal scaling of $f$.

(b) For $f(x) = x^2$, does the same argument work: is every horizontal scaling equivalent to a vertical scaling?

(c) For $f(x) = |x|$, is $g(x) = |2x|$ a vertical or horizontal scaling, or both?

Solution.

(a) $g(x) = \sqrt{9x} = \sqrt{9} \cdot \sqrt{x} = 3\sqrt{x} = 3f(x)$. So $g$ is both the horizontal compression $f(9x)$ (with factor 9 inside) and the vertical stretch $3f(x)$ (with factor 3 outside). For the square root function, a horizontal compression by factor $c^2$ is the same as a vertical stretch by factor $c$.

(b) For $f(x) = x^2$: $g(x) = f(cx) = (cx)^2 = c^2 x^2 = c^2 f(x)$. Yes: for $f(x) = x^2$, every horizontal scaling by $c$ is equivalent to a vertical scaling by $c^2$. The two transformations produce the same function.

(c) For $f(x) = |x|$: $g(x) = |2x| = 2|x| = 2f(x)$. A horizontal compression by factor 2 is the same as a vertical stretch by factor 2 for the absolute value function. Both give $g(x) = 2|x|$.

The deeper point. For $f(x) = x^2$, $f(x) = |x|$, and $f(x) = \sqrt{x}$, a horizontal scaling is algebraically equivalent to a vertical scaling, because the function’s structure allows factoring out the constant. For $f(x) = \sin x$, this equivalence fails: $\sin(2x) \neq c \cdot \sin(x)$ for any constant $c$. The two types of scaling are genuinely independent for most functions.

Common Errors

Common Misconceptions

Leveled Practice

Level 1: Direct Application

Problem 1. The table gives values of $f$. Build the table for each transformation.

(a) $g(x) = -f(x)$ (b) $h(x) = f(-x)$ (c) $p(x) = 4f(x)$ (d) $q(x) = f(4x)$

Problem 2. Describe the transformation and identify the key change to the graph of $f(x) = x^2$ in each case.

(a) $y = -x^2$ (b) $y = x^2$ replaced by $y = (-x)^2$ (c) $y = 5x^2$ (d) $y = (1/3)x^2$

Level 2: Translating Between Representations

Problem 3. The graph of $f$ has $x$-intercepts at $x = -1$ and $x = 4$, a $y$-intercept at $(0, 3)$, and a maximum at $(3/2, 5)$.

Find the $x$-intercepts, $y$-intercept, and maximum of each transformed graph.

(a) $g(x) = -f(x)$ (b) $h(x) = f(-x)$ (c) $p(x) = 2f(x)$

Problem 4. Explain why the following statement is incorrect and give a specific numerical example that demonstrates the error.

“The graph of $y = 3f(x)$ is the graph of $f$ shifted up by 3 units.”

Level 3: Extension and Justification

Problem 5. The function $f$ is defined on $[0, 4]$ with maximum value 10.

(a) What is the domain of $g(x) = f(-x)$?

(b) What is the maximum value of $g(x) = -2f(x)$? Is it a maximum or minimum?

(c) What is the domain of $h(x) = f(3x)$?

(d) The graph of $h(x) = f(3x)$ is described as “horizontally compressed by factor 3.” Verify this: find the $x$-value where $h$ achieves the maximum of $f$, given that $f$ achieves its maximum at $x = 2$.

Problem 6 (LFHC Extension). Given two functions $f$ and $g$ from the same table of values, determine what transformation maps $f$ to $g$.

(a) Compare $g(x)$ to $f(x)$ at each $x$. What is the ratio $g(x)/f(x)$ at the non-zero outputs?

(b) Conjecture the formula for $g$ in terms of $f$.

(c) Verify your conjecture by checking all five points.

(d) Now consider a second transformed function $h$:

What transformation maps $f$ to $h$? Express $h$ in terms of $f$.

Mastery Checklist

You have mastered this skill when you can do all of the following without referring to notes:

• State the effect of $-f(x)$ on the graph (reflection over $x$-axis)
• State the effect of $f(-x)$ on the graph (reflection over $y$-axis)
• State the effect of $c \cdot f(x)$ for $c > 1$, $0 < c < 1$, and $c < 0$
• State the effect of $f(cx)$ for $c > 1$ and $0 < c < 1$, and why the direction is opposite
• Given a table for $f$, build the table for each of the four transformation types
• Identify which points are fixed under vertical scaling, and which are fixed under $x$-axis reflection
• Identify the transformation from two tables or two key-point lists
• Explain the difference between $2f(x)$ (vertical stretch) and $f(x) + 2$ (vertical shift)

Mental Model

Vertical transformations (outside the function) act on the output, after the function has finished its process. Think of them as adjustments to a ruler held vertically: stretching the ruler scales every measurement; flipping the ruler negates every measurement.

Horizontal transformations (inside the function) act on the input, before the function runs. Think of them as adjustments to the clock that feeds time into a machine: running the clock twice as fast compresses all the events into half the time (horizontal compression); running it at half speed stretches them out.

The key distinctions:

1. Reflection over $x$-axis vs. $y$-axis. Ask: does the formula change the sign of $x$ (inside) or the sign of $f(x)$ (outside)? Inside = $y$-axis. Outside = $x$-axis.

2. Scale vs. shift. Multiplying and adding are different. Zeros of $f$ are fixed under vertical scaling but move under vertical shifting. Check the zero to distinguish them.

3. Inside compression is backward. A factor $c > 1$ inside compresses (features move toward the $y$-axis). The feature that was at $x = a$ moves to $x = a/c$, which is closer to zero.

Connections

Within MATH161

• Vertical and Horizontal Shifts: Shifts are the additive partner of scalings. Both types of transformation are combined in the standard form $y = Af(B(x - h)) + k$.
• Even and Odd Functions: A function is even if $f(-x) = f(x)$ (the $y$-axis reflection leaves it unchanged) and odd if $f(-x) = -f(x)$ (the $y$-axis reflection is the same as the $x$-axis reflection). Both properties are defined directly in terms of the reflections in this lesson.
• Combining Transformations: The next skill applies vertical and horizontal scalings together with shifts and asks about order of application.

Toward Later MATH161

• Trigonometric functions: The amplitude of a sinusoidal function is a vertical scaling: $y = A\sin(x)$ has amplitude $|A|$. The period compression $y = \sin(Bx)$ is a horizontal scaling.
• Derivatives: If $g(x) = c \cdot f(x)$, then $g'(x) = c \cdot f'(x)$ (the Constant Multiple Rule for derivatives). Vertical scaling commutes with differentiation.
• Chain rule: If $g(x) = f(cx)$, then $g'(x) = c \cdot f'(cx)$. A horizontal compression introduces a factor of $c$ in the derivative -- another reason the two types of scaling are genuinely different.

Audience Notes

For students who confuse reflections: A helpful physical image: reflecting over the $x$-axis flips the page upside down (the horizontal axis stays in place; the vertical positions flip). Reflecting over the $y$-axis flips the page left-to-right (the vertical axis stays in place; horizontal positions flip). The axis you reflect over is the axis that stays fixed.

For students interested in proof: The full set of transformations in this lesson and the previous one (shifts and scalings) generates a group under composition, called the affine group of the line. The transformations preserve collinearity (lines map to lines) and parallelism, but not distances.

For students interested in applications: Amplitude and period in sound and signal processing are vertical and horizontal scalings of sinusoidal functions. A louder signal has larger amplitude (vertical stretch). A higher-frequency signal has shorter period (horizontal compression). These are exactly $A\sin(Bx)$.

Back to Ch. 1 Sec. 3 Skills | Previous: Vertical and Horizontal Shifts | Next: Combining Transformations