Graph Shapes Flashcards

Study Notes

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.

Description

This set of flashcards covers various graph shapes defined by different mathematical functions. From linear equations to exponential and logarithmic functions, test your knowledge of how these graphs behave. Perfect for students looking to sharpen their understanding of function graphs.