AMC 10 · 2024 · #13

Grade 11 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

exponential-functionline-symmetryreflection-symmetryexponentsfunction-evaluation convert-to-algebrasymmetry-argument ↑ Prerequisites: exponentscoordinate-geometrylinear-equations-one-var

📏 Medium solution 💡 3 insights

Problem

The curve y = e^x+1 + e^-x - 2 is claimed to have an axis of symmetry: a line you could fold the paper along so the curve lands exactly on itself. Find that line, then find where the point (-1,1/2) lands when it is folded across the same line.

Pick an answer.

(A)

$\left(-1,-\frac{3}{2}\right)$

(B)

(-1,0)

(C)

$\left(-1,\frac{1}{2}\right)$

(D)

$\left(0,\frac{1}{2}\right)$

(E)

$\left(3,\frac{1}{2}\right)$

How to solve

Strategy Introduce a Variable

The tempting move is to stare at e^x+1 + e^-x and guess the axis off the exponents. The +1 makes x = -1 look special, and the e^-x makes x = 0 look special, and both guesses are wrong. So do not guess: give the axis a name. Let the axis be the unknown vertical line x = a, and let t be how far you step sideways from it. Symmetry then has an exact meaning, f(a+t) = f(a-t) for every t, and that meaning is an equation. Once the demand is an equation, the exponent rules turn the difference f(a+t) - f(a-t) into a product of two factors, and one of those factors is obviously non-zero, which pins a down with no guessing at all. Reflecting the point afterwards is the easy half: over a vertical line the height never moves and the horizontal distance just flips sides.

1STEP 1

Name the axis, then write what symmetry means

The condition is that both sides agree for every t.

f(x) = e^x+1 + e^-x - 2; x = a is an axis of symmetry ⇔ f(a+t) = f(a-t) for every real t

2STEP 2

Split the exponentials and factor

The difference folds into a product of two factors.

f(a+t) - f(a-t) = (e^a+1e^t + e^-ae^-t) - (e^a+1e^-t + e^-ae^t) = (e^a+1 - e^-a)(e^t - e^-t)

3STEP 3

Only one factor is allowed to vanish

The t factor is nonzero, so the other factor vanishes.

(e^a+1 - e^-a)(e^t - e^-t) = 0 for all t; e¹ - e⁻¹ ≠ 0 ⟹ e^a+1 - e^-a = 0

4STEP 4

Equal powers force equal exponents

Matching exponents gives the axis x = -1/2.

e^a+1 = e^-a ⟹ a + 1 = -a ⟹ 2a = -1 ⟹ a = -1/2

5STEP 5

Recentre and watch the symmetry appear

Recentred, it is visibly even in u.

u = x + 1/2: e^x+1 + e^-x - 2 = e¹/2e^u + e¹/2e^-u - 2 = √(e)(e^u + e^-u) - 2

6STEP 6

Mirror the point across the line

x goes to 0 while y stays 1/2.

x' = 2a - x = 2(-1/2) - (-1) = -1 + 1 = 0, y' = 1/2; (-1 + 0)/2 = -1/2 = a ✓

7STEP 7

Check the near misses

Only (0,1/2) survives.

(-1,-3/2), (-1,0): y changed ×; (-1,1/2) → a = -1 ×; (3,1/2) → a = 1 ×; a = -1/2 → (0,1/2)

Answer

(0,1/2)

Test the claimed axis numerically by stepping equal distances either side of x = -1/2. With t = 0.1: f(-0.4) = 1.31394 and f(-0.6) = 1.31394. With t = 0.5: f(0) = 1.71828 and f(-1) = 1.71828, both equal to e - 1 exactly. With t = 1: f(0.5) = 3.08822 and f(-1.5) = 3.08822. Even far out at t = 10: f(9.5) = 36313.503 and f(-10.5) = 36313.503. The match holds at every step size, which is what an axis of symmetry demands. The rejected guesses fail the same test at once: about x = -1, f(0) = 1.718 but f(-2) = 5.757, so that fold does not close. The axis also lands where a fold line should, at the bottom of the curve: f(-1/2) = 2√(e) - 2 ≈ 1.29744, while f(-0.51) and f(-0.49) are both ≈ 1.29761, higher and equal to each other. Finally check the reflected point directly. The segment from (-1,1/2) to (0,1/2) is horizontal, so it crosses the vertical line x = -1/2 at a right angle, and its midpoint is (-1/2,1/2), which lies on that line. Perpendicular and bisected are exactly the two conditions a reflection has to satisfy.

💡Key takeaway

Do not guess a mirror line from how the formula looks; call it x = a, write down what mirroring demands, f(a+t) = f(a-t), and let the algebra tell you what a is.

Name the axis, then write what symmetry means
Split the exponentials and factor
Only one factor is allowed to vanish
Equal powers force equal exponents
Recentre and watch the symmetry appear
Mirror the point across the line
Check the near misses