Difficulty: Foundational to Intermediate | Prerequisites: All prior topics.
You need to sketch and identify the basic function families (linear, quadratic, polynomial, trig, exponential, logarithmic, absolute value) without a calculator. In calculus, reading a graph quickly tells you about derivatives, limits, and behaviour at infinity.
Parent function
The simplest form of a function family: y = x (linear), y = x² (quadratic), y = sin x (trig), y = eˣ (exponential), y = ln x (logarithmic), y = |x| (absolute value).
Transformation
A shift, stretch, compression, or reflection applied to a parent function. For example, y = (x - 1)² + 2 shifts the parabola right 1 and up 2.
Reflection across y = x
If f and g are inverse functions, then the graph of g is the reflection of the graph of f across the line y = x. This is why eˣ and ln x are mirror images.
Vertex form of a parabola
y = a(x - h)² + k. The vertex is at (h, k). If a > 0 it opens upward; if a < 0, downward.
On the interval [0, 1], higher powers produce smaller outputs for 0 < x < 1 (they "hug" the x-axis more). So x⁴ < x² for 0 < x < 1. Both pass through (0, 0) and (1, 1).
The curve closer to the x-axis (lower) on (0, 1) is the higher power. If curve A sits above curve B on this interval, A is x² and B is x⁴.
On [0, 1], higher roots produce larger outputs for 0 < x < 1: ⁴√x > √x for 0 < x < 1.
The curve that sits higher on (0, 1) is the higher root. Both still pass through (0, 0) and (1, 1).
y = eˣ grows rapidly, passes through (0, 1), and is always positive. It is concave up.
y = ln x passes through (1, 0), grows slowly, and is defined only for x > 0. It is concave down.
They are reflections of each other across y = x.
y = e^(x+1) is eˣ shifted one unit to the left, and intersects the y-axis at e¹ = e.
y = ln(x - 1) is ln x shifted one unit to the right.
y = 2ˣ is an increasing exponential curve. y = 2^(-x) is a decreasing exponential (it is the reflection of 2ˣ across the y-axis). They intersect at (0, 1).
The curve that increases to the right is 2ˣ.
sin x has period 2π. sin 2x has period π (it oscillates twice as fast).
The curve with more oscillations in the same window is sin 2x.
y = |x| - 1 has its vertex at (0, -1). y = |x - 2| has its vertex at (2, 0).
The vertex location tells the two apart: a vertical shift moves the V up or down, a horizontal shift moves it left or right.
A quadratic has at most one turning point (parabola shape). A cubic has at most two turning points and extends to -∞ in one direction and +∞ in the other.
A curve with an S-shape or two bends is the cubic.
y = tan x has vertical asymptotes at x = ±π/2 and passes through the origin with steep growth.
y = tan⁻¹x is bounded between -π/2 and π/2 (horizontal asymptotes) and passes through the origin with gentle growth.
The bounded, flattening curve is tan⁻¹x.
Students often assume the "higher" curve on [0, 1] is the higher-degree polynomial. The opposite is true: on (0, 1), x² > x⁴, so the higher curve is the lower-degree function.
Students often mix up eˣ and ln x. The exponential passes through (0, 1) and grows; the log passes through (1, 0) and flattens.
Students often confuse horizontal and vertical shifts. In y = f(x - h) + k, the horizontal shift is right by h (note the minus sign), and the vertical shift is up by k.
⚠️ Graph recognition is tested directly on no-calculator sections, where you must sketch or identify curves from their equations.
⚠️ Understanding concavity, increasing/decreasing behaviour, and asymptotes from a graph is essential for first and second derivative tests.
⚠️ Knowing the relationship between a function and its inverse (reflection across y = x) helps with inverse function differentiation.
On [0, 1], which sits higher: y = x² or y = x⁴? (Answer: x².)
On [0, 1], which sits higher: y = √x or y = ⁴√x? (Answer: ⁴√x.)
What point does every exponential y = bˣ (b > 0) pass through? (Answer: (0, 1).)
What point does every logarithm y = log_b(x) pass through? (Answer: (1, 0).)
Q: In a figure, curve A sits above curve B on (0, 1), and both pass through (0, 0) and (1, 1). One is x², one is x⁴. Which is A?
A: A is x². On (0, 1), lower powers produce larger values.
Q: One curve is eˣ and the other is ln x. Curve B passes through (0, 1) and grows steeply. Which is B?
A: B is eˣ. It passes through (0, 1) and grows exponentially.
Q: One curve is sin x and the other is sin 2x on [-2π, 2π]. Curve A completes two full cycles in [0, 2π]. Which is A?
A: A is sin 2x. It has period π, so two full cycles in [0, 2π], while sin x has only one.
Q: One curve is 2ˣ and the other is 2^(-x). Both pass through (0, 1). Curve B increases as x increases. Which is B?
A: B is 2ˣ. The function 2^(-x) decreases as x increases.
Graph recognition feeds directly into curve sketching in calculus, where you combine first and second derivative information with your knowledge of parent functions to draw accurate graphs. Understanding transformations helps you sketch derivatives and antiderivatives. The relationship between eˣ and ln x as inverse functions is used throughout the course.
Related Terms / Search Tags
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