Relacionar la ecuación o ecuaciones con las características...

Relacionar la ecuación o ecuaciones con las características dadas. i) $y=3 x^3-3 x$ ii) $y=(x+3)^2$ iii) $y=3 x-3$ iv) $y=\sqrt[3]{x}$ v) $y=3 x^2+3$ vi) $y=\sqrt{x+3}$ a) Simétrica con respecto al eje $y$ b) Tres intersecciones con el eje $x$ c) Simétrica con respecto al eje $x$ d) $(-2,1)$ es un punto de la gráfica e) Simétrica con respecto al origen f) La gráfica pasa por el origen

Relacionar la ecuación o ecuaciones con las características dadas.
i) $y=3 x^3-3 x$
ii) $y=(x+3)^2$
iii) $y=3 x-3$
iv) $y=\sqrt[3]{x}$
v) $y=3 x^2+3$
vi) $y=\sqrt{x+3}$
a) Simétrica con respecto al eje $y$
b) Tres intersecciones con el eje $x$
c) Simétrica con respecto al eje $x$
d) $(-2,1)$ es un punto de la gráfica
e) Simétrica con respecto al origen
f) La gráfica pasa por el origen

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Key Concepts

Verification of Specific Points

Determining whether a given point lies on the graph of a function involves substituting the coordinates of the point into the function’s equation. This concept is fundamental when relating algebraic equations to specific graphical features.

Function Transformations

This includes the study of how various algebraic operations, such as translations, reflections, and stretches, alter the graph of a function. Recognizing translations (like horizontal or vertical shifts) and other modifications is essential for matching a given function with its graphical behavior.

Symmetry Properties of Functions

This concept involves determining whether a function is symmetric with respect to the y-axis (even functions), the origin (odd functions), or other lines. Recognizing symmetry can simplify analysis and helps in understanding the behavior of the function across the coordinate plane.

Intercepts and Zeros of Functions

Understanding how to find the points where the graph of a function crosses the x-axis or y-axis is crucial. For example, the x-intercepts (zeros) are obtained by setting the function equal to zero, and the y-intercept is found by substituting x = 0. This concept helps in identifying key characteristics of the graph.

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