Algbra 2 Graphs

9.1 Parent Functions (Name + Sketch Summary)

Here are the parent functions you’re expected to recognize:

a) y = x — Linear

Straight diagonal line through origin

b) y = |x| — Absolute Value

V-shape with vertex at (0,0)

c) y = x² — Quadratic

U-shaped parabola opening up

d) y = x³ — Cubic

S-curve through origin, increases both directions

e) y = x⁴ — Quartic

Like a tighter U than x²; symmetric

f) y = x⁵ — Quintic

S-curve, steeper than x³

g) y = √x — Square Root

Starts at (0,0), curves right and up

h) y = ∛x — Cube Root

S-curve sideways through origin

i) y = 1/x — Rational

Two curves in opposite quadrants; vertical & horizontal asymptotes

j) y = b^x — Exponential Growth

Horizontal asymptote at y = 0; increasing curve

k) y = (1/b)^x — Exponential Decay

Decreasing curve, still asymptote at y = 0

l) y = log_b(x) — Logarithmic Growth

Vertical asymptote at x = 0; slow increase to the right

m) y = log_{1/b}(x) — Logarithmic Decay

Decreasing to the right; vertical asymptote x = 0

9.2 Analyzing (h, k)

a) Vertex:

• Absolute Value (y = |x|)

• Quadratic (y = x²)

• Vertex is at (h, k)

b) Inflection point/center of rotation:

• Cubic (y = x³)

• Cube Root (y = ∛x)

c) (h, k) at (0, 0):

• All except rational & logarithmic functions

d) Asymptotes (not a point):

• Exponential, Rational, Logarithmic

9.2 Analyzing Intercepts

a) Always 1 y-intercept:

• Linear, Quadratic, Absolute Value, Exponential, Logarithmic

b) May have none:

• Rational (depends on asymptotes)

x-intercepts (sketch possible numbers):

• Quadratic: 0, 1, or 2

• Absolute Value: 0 or 1

• Exponential: maybe none

• Cube Root: always 1

• Logarithmic: always 1

• Square Root: 1 or none

9.2 Analyzing Asymptotes

a) Horizontal asymptote:

• Exponential functions (y = b^x)

b) Vertical asymptote:

• Logarithmic (y = log_b(x))

• Rational (y = 1/x)

c) Both horizontal and vertical:

• Rational (y = 1/x)

9.2 Analyzing Domain and Range

a) Domain = all real numbers:

• Linear, Cubic, Cube Root, Exponential, Logarithmic

b) Range = all real numbers:

• Linear, Cubic, Cube Root, Logarithmic

9.2 Analyzing Equations

How to solve each type:

a) Absolute Value:

Isolate absolute value, split into two equations (positive and negative)

b) Quadratic:

Factor, complete the square, or use quadratic formula

c) Square Root:

Isolate radical, square both sides

d) Rational:

Find common denominators, cross-multiply, check for extraneous solutions

e) Logarithmic:

Rewrite in exponential form, solve normally

f) Exponential:

Take log of both sides, then solve

g) Linear:

Isolate x using inverse operations

9.2 Analyzing Equations (Special Types)

a) Extraneous solutions:

• Rational, Square Root, Logarithmic — due to restricted domains

b) Imaginary solutions:

• Quadratic — when discriminant is negative

c) Exactly one real solution:

• Linear

d) No solution:

• Absolute value = negative number

• Logarithmic with negative input

• Square roots of negative numbers