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Function Graphing Calculator

Plot and analyze functions with key points and derivation

Select Function Type

Linear y=ax+b Quadratic y=ax^2+bx+c Cubic y=ax^3 Sine y=a*sin(bx)

y = ax + b

y =x +

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

Algebra Calculus Physics Engineering Economics

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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