Plot and analyze functions with key points and derivation
Select Function Type
y = ax + b
y =x +
y = ax^2 + bx + c
y =x^2 +x +
y = ax^3
y =x^3
y = a*sin(bx)
y =sin(x)
Result
Derivation
Graphing Rules
Linear: y=ax+b, slope a, y-intercept (0,b)
Quadratic: vertex at x=-b/(2a), parabola
Cubic: passing through origin, odd function
Sine: amplitude |a|, period 2pi/|b|
Function graphing reveals the relationship between input and output visually. Key points like intercepts, vertices, and asymptotes define the graph shape and aid in analysis.
⚠Always find domain first before graphing. Some functions have restricted domains that limit the visible graph area.
What Is Function Graphing?
Function graphing plots pairs (x, f(x)) on a coordinate plane. Each function type has a characteristic shape. Linear functions are straight lines, quadratics are parabolas, cubics have S-shapes, and sine waves oscillate periodically.
Linear
y=ax+b. One x-intercept at x=-b/a, one y-intercept at (0,b). Slope a = rate of change.
Quadratic
y=ax^2+bx+c. Vertex is the minimum (a>0) or maximum (a<0). Axis of symmetry at x=-b/(2a).
Cubic
y=ax^3. Odd function: f(-x)=-f(x). Passes through origin. Increasing when a>0, decreasing when a<0.
Sine
y=a*sin(bx). Amplitude = |a|, period = 2pi/|b|. Range = [-|a|, |a|]. Zeros at x = n*pi/b.
Teaching Example: y=2x+1. Slope=2, y-intercept=(0,1). x-intercept: set 0=2x+1 -> x=-0.5. Two key points: (0,1) and (-0.5,0). Draw line through both.
Applications
AlgebraCalculusPhysicsEngineeringEconomics
Frequently Asked Questions
How to graph linear functions?▼
y=ax+b. Plot y-intercept (0,b). Use slope a to find second point. Draw line through both points.
How to find graph key points?▼
Intercepts (x=0, y=0), vertex for quadratics, amplitude/period for sine. These define the shape.
Quadratic graph shape?▼
Parabola. Opens up if a>0, down if a<0. Vertex at x=-b/(2a) is min (a>0) or max (a<0).
Sine graph properties?▼
y=a*sin(bx). Amplitude=|a|, period=2pi/|b|. Oscillates between -|a| and |a|. Zeros at multiples of pi/b.
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