Graph Shapes Flashcards

Linear and Polynomial Functions

• y = x: Represents a linear function with a slope of 1, crossing the origin.
• y = x²: A quadratic function forming a parabola that opens upwards, vertex at the origin, symmetric about the y-axis.
• y = x³: A cubic function that has an S-shaped curve, passing through the origin, with points of inflection.

Absolute and Square Root Functions

• y = |x|: Absolute value function, producing a V-shaped graph that reflects negative values across the x-axis; vertex at the origin.
• y = √x: Square root function, defined for x ≥ 0; graphs as a curve starting at the origin and increasing gradually.

Rational Functions

• y = 1/x: Hyperbolic function with asymptotes along the axes; grows positively in quadrant I and negatively in quadrant III.
• y = 1/x²: Similar to 1/x but only exists in quadrants I and II due to the squared denominator; symmetric across the y-axis.

Exponential and Logarithmic Functions

• y = e^x: Exponential function that rapidly increases as x becomes positive; horizontal asymptote at y = 0 as x approaches negative infinity.
• y = lnx: Natural logarithmic function, defined for x > 0; increases, passing through (1,0) and approaching negative infinity as x approaches 0.

Trigonometric Functions

• y = sinx: Sine function oscillating between -1 and 1; periodic with a period of 2π, crosses the origin at multiples of π.
• y = cosx: Cosine function also oscillates between -1 and 1; periodic with a period of 2π, crosses the y-axis at (0,1).
• y = tanx: Tangent function has vertical asymptotes at odd multiples of π/2, periodic with a period of π; oscillates between negative and positive infinity.

Circular Functions

• y = √(a² - x²): Represents the upper half of a circle with radius a, centered at the origin; only plotting points above the x-axis.