Algbra 2 Graphs

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26 Terms

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y = x

Linear function represented by a straight diagonal line through the origin.

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y = |x|

Absolute Value function, shaped like a V with vertex at (0,0).

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y = x²

Quadratic function, represented by a U-shaped parabola opening upwards.

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y = x³

Cubic function, an S-curve passing through the origin, increasing in both directions.

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y = x⁴

Quartic function, resembling a tighter U than the quadratic function, symmetric.

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y = x⁵

Quintic function, an S-curve that is steeper than the cubic function.

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y = √x

Square Root function, starts at (0,0) curving right and upwards.

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y = ∛x

Cube Root function, an S-curve sideways through the origin.

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y = 1/x

Rational function consisting of two curves in opposite quadrants and having vertical and horizontal asymptotes.

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y = b^x

Exponential Growth function characterized by a horizontal asymptote at y = 0 and an increasing curve.

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y = (1/b)^x

Exponential Decay function, showing a decreasing curve with a horizontal asymptote at y = 0.

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y = log_b(x)

Logarithmic Growth function, featuring a vertical asymptote at x = 0 and a slow increase to the right.

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y = log_{1/b}(x)

Logarithmic Decay function, which decreases to the right and has a vertical asymptote at x = 0.

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Vertex of Absolute Value and Quadratic

For y = |x| and y = x², the vertex is at (h, k).

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Inflection point for Cubic and Cube Root Functions

The inflection point or center of rotation is observed in y = x³ and y = ∛x.

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Asymptotes in Exponential, Rational, Logarithmic functions

Exponential, Rational, and Logarithmic functions have asymptotes that are not points.

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Only 1 y-intercept behavior

Linear, Quadratic, Absolute Value, Exponential, and Logarithmic functions always have one y-intercept.

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No x-intercepts condition

Exponential function may have none; Rational function depends on asymptotes.

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Domain of all real numbers

Linear, Cubic, Cube Root, Exponential, and Logarithmic functions have a domain of all real numbers.

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Range behavior for Linear, Cubic, Cube Root, Logarithmic

The range is all real numbers for Linear, Cubic, Cube Root, and Logarithmic functions.

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Finding solutions for Absolute Value

To solve absolute value equations, isolate the absolute value and split into two equations.

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Rational function solution approach

For rational functions, find common denominators, cross-multiply, and check for extraneous solutions.

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Find the number of extraneous solutions

Extraneous solutions are found in Rational, Square Root, and Logarithmic functions due to their restricted domains.

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Imaginary solutions in Quadratics

Quadratics yield imaginary solutions when the discriminant is negative.

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Conditions for exactly one real solution

Linear functions have exactly one real solution.

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No solution scenarios

No solutions occur when the absolute value is set to a negative number or in logarithmic functions with negative inputs.