Curve Quest Worksheet
Calculus & Algebra • Graphing Analysis
Curve Quest Worksheet
Master the systematic process of analyzing and sketching functions.
Name:
Date: Period:
1 Intercepts
Find \(y\)-int (\(x=0\)) and roots \(f(x)=0\).
2 Asymptotes
Check domain, VA (zeros of denom) & HA (\(\lim_{x \to \pm\infty}\)).
3 First Derivative
Find \(f'(x)=0\) for extrema & inc/dec sign intervals.
4 Second Deriv.
Find \(f''(x)=0\) for inflection points & concavity.
Problem 1 Polynomial Curve: \(f(x) = x^3 - 3x^2 + 2\)
Determine all key features, then sketch the graph.
A. Intercepts & End Behavior:
\(y\)-intercept: \((0, \_\_\_\_ )\)
End behavior: as \(x \to \infty\), \(y \to \_\_\_\)
B. First Derivative & Critical Points:
\(f'(x) =\) ________________________ \(\implies\) Critical \(x\)-values: \(x =\) ________
Local Max: \(( \_\_\_\_, \_\_\_\_ )\) | Local Min: \(( \_\_\_\_, \_\_\_\_ )\)
C. Concavity & Inflection Point:
\(f''(x) =\) ________________________ \(\implies\) Possible inflection: \(x =\) ________
Point of Inflection: \(( \_\_\_\_, \_\_\_\_ )\)
-2 -1 1 2 3 4 3 2 1 -1 -2 -3 x y
Graph: \(f(x) = x^3 - 3x^2 + 2\)
Problem 2 Rational Curve: \(g(x) = \frac{2x}{x - 2}\)
Include all asymptotes as dashed lines.
A. Domain & Asymptotes:
Domain: \(x \in\) _______________________________________
Vertical Asymptote: \(x =\) ________ Horizontal Asymptote: \(y =\) ________
B. Intercepts & Test Points:
\(x\)-intercept: \(( \_\_\_\_, 0 )\)
\(y\)-intercept: \(( 0, \_\_\_\_ )\)
Test point at \(x = 3\): \(g(3) =\) ________ \(\implies (3, \_\_\_\_ )\)
C. First Derivative Behavior:
\(g'(x) = \frac{(x-2)(2) - (2x)(1)}{(x-2)^2} = \frac{-4}{(x-2)^2}\)
Notice: \(g'(x) < 0\) for all \(x \neq 2\). The curve is always decreasing on its domain.
-2 -1 1 2 3 4 5 4 2 -2 -4 -6 x y
Graph: \(g(x) = \frac{2x}{x - 2}\)
Advanced Graphing Analysis
Extrema, Concavity & Mystery Curves
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Problem 3 Quartic Function: \(h(x) = \frac{1}{4}x^4 - 2x^2\)
Check for symmetry (even/odd) first!
A. Symmetry & \(x\)-Intercepts:
Symmetry: \(h(-x) =\) __________________ \(\implies\) [ Even / Odd / Neither ]
Factor: \(\frac{1}{4}x^2(x^2 - 8) = 0 \implies x =\) ______________________
B. Critical Numbers & Extrema:
\(h'(x) = x^3 - 4x = x(x-2)(x+2) = 0\)
Local Max: \(( \_\_\_\_, \_\_\_\_ )\)
Local Mins: \(( \_\_\_\_, \_\_\_\_ )\), \(( \_\_\_\_, \_\_\_\_ )\)
C. Inflection Points:
\(h''(x) = 3x^2 - 4 = 0 \implies x = \pm\sqrt{\frac{4}{3}} \approx \pm 1.15\)
Concave Up on: _____________________ Concave Down on: ____________________
-3 -2 -1 1 2 3 2 -2 -4 -6 x y
Graph: \(h(x) = \frac{1}{4}x^4 - 2x^2\)
Problem 4 The Mystery Function: Graph from Given Properties
Synthesize derivative criteria into one sketch.
Given Conditions for continuous curve \(y = F(x)\):
- Coordinates: \(F(-3) = 0\), \(F(0) = 2\), \(F(3) = 0\)
- Stationary Points: \(F'(-1) = 0\) with \(F(-1) = 4\); and \(F'(2) = 0\) with \(F(2) = -1\)
- Increasing on: \((-\infty, -1)\) and \((2, \infty)\)
- Decreasing on: \((-1, 2)\)
- Concavity: \(F''(x) < 0\) on \((-\infty, 0.5)\) and \(F''(x) > 0\) on \((0.5, \infty)\)
Deductions:
Local maximum at: \(( \_\_\_\_, \_\_\_\_ )\)
Local minimum at: \(( \_\_\_\_, \_\_\_\_ )\)
Point of Inflection occurs around \(x \approx 0.5\).
-3 -2 -1 1 2 3 4 2 -2 -4 x y
Graph: Mystery Function \(y = F(x)\)
Calculus Curriculum • Curve Sketching & Derivative Tests Remember to plot intercepts and stationary points before connecting the smooth curve!
