Graphing Functions Study Guide

Absolute Value

Parentf(x) = |x|

Generalf(x) = a|x − h| + k

Parameter Effects

a > 0 opens upward; a < 0 opens downward
|a| > 1 narrows; 0 < |a| < 1 widens
h shifts the vertex horizontally (right if positive)
k shifts the vertex vertically (up if positive)
Vertex is at (h, k)
Domain: all real numbers
Range: [k, ∞) if a > 0; (−∞, k] if a < 0

Parent Graph: f(x) = |x|

Parameter	Effect on Graph
a = 2	Narrower V (steeper)
a = ½	Wider V (shallower)
a = −1	Flips upside down ∧
h = 3	Vertex shifts right 3
k = −2	Vertex shifts down 2

Worked Examples

Example 1

f(x) = 2|x − 3| + 1

1Identify a = 2, h = 3, k = 1

2Vertex: (3, 1) — shift right 3, up 1

3a = 2 > 0 → opens upward, narrower

4Range: [1, ∞); Domain: all reals

Example 2

f(x) = −|x + 2| − 1

1Identify a = −1, h = −2, k = −1

2Vertex: (−2, −1) — shift left 2, down 1

3a = −1 < 0 → opens downward (∧ shape)

4Range: (−∞, −1]; Domain: all reals

Square Root

Parentf(x) = √x

Generalf(x) = a√(x − h) + k

Parameter Effects

a > 0 curves upward; a < 0 reflects over x-axis
|a| > 1 stretches vertically; 0 < |a| < 1 compresses
h shifts the starting point horizontally
k shifts the starting point vertically
Starting point (endpoint): (h, k)
Domain: x ≥ h (requires x − h ≥ 0)
Range: [k, ∞) if a > 0; (−∞, k] if a < 0

Parent Graph: f(x) = √x

Parameter	Effect on Graph
a = 3	Stretch — rises faster
a = ¼	Compress — rises slower
a = −1	Reflects below x-axis
h = 4	Start point shifts right 4
k = −2	Start point shifts down 2

Worked Examples

Example 1

f(x) = √(x − 4) + 2

1a = 1, h = 4, k = 2

2Starting point: (4, 2)

3Domain: x ≥ 4 · Range: y ≥ 2

4No stretch/compress; opens normally

Example 2

f(x) = −2√(x + 1) + 3

1a = −2, h = −1, k = 3

2Starting point: (−1, 3)

3a < 0 → reflects down; |a|=2 → steeper

4Domain: x ≥ −1 · Range: y ≤ 3

Exponential

Parentf(x) = bˣ (b > 0, b ≠ 1)

Generalf(x) = a · b^(x−h) + k

Parameter Effects

b > 1 → exponential growth (rises right)
0 < b < 1 → exponential decay (falls right)
a > 0 same side as parent; a < 0 reflects over asymptote
h shifts the graph horizontally
k sets the horizontal asymptote y = k
Domain: all real numbers
Range: (k, ∞) if a > 0; (−∞, k) if a < 0

Parent Graph: f(x) = 2ˣ

Parameter	Effect on Graph
b = 3	Grows faster than b=2
b = ½	Decay — falls to the right
k = 2	Asymptote rises to y = 2
h = −3	Shifts entire graph left 3
a = −1	Reflects over asymptote

Worked Examples

Example 1

f(x) = 2^(x−1) + 3

1a = 1, b = 2, h = 1, k = 3

2Asymptote: y = 3

3Key point: (1, 4) since 2⁰ + 3 = 4

4Growth; Domain all reals; Range (3, ∞)

Example 2

f(x) = 3·(½)^x − 2

1a = 3, b = ½, h = 0, k = −2

2Asymptote: y = −2 (decay toward it)

3y-intercept: (0, 1) since 3·1 − 2 = 1

4Decay; Domain all reals; Range (−2, ∞)

Quadratic

Parentf(x) = x²

Generalf(x) = a(x − h)² + k

Parameter Effects

a > 0 opens upward (∪); a < 0 opens downward (∩)
|a| > 1 narrows the parabola; 0 < |a| < 1 widens it
h shifts the vertex horizontally
k shifts the vertex vertically
Vertex: (h, k) — maximum or minimum
Axis of symmetry: x = h
Domain: all real numbers
Range: [k, ∞) if a > 0; (−∞, k] if a < 0

Parent Graph: f(x) = x²

Parameter	Effect on Graph
a = 3	Narrower parabola
a = ⅓	Wider parabola
a = −2	Opens downward, narrow
h = −4	Vertex shifts left 4
k = 5	Vertex shifts up 5

Worked Examples

Example 1

f(x) = (x − 2)² − 3

1a = 1, h = 2, k = −3

2Vertex: (2, −3); Axis: x = 2

3a = 1 > 0 → opens upward, standard width

4Range: [−3, ∞); Domain: all reals

Example 2

f(x) = −½(x + 1)² + 4

1a = −½, h = −1, k = 4

2Vertex: (−1, 4); Axis: x = −1

3a < 0 → opens downward; |a| < 1 → wider

4Range: (−∞, 4]; Domain: all reals

Cube Root

Parentf(x) = ∛x

Generalf(x) = a·∛(x − h) + k

Parameter Effects

a > 0 rises left to right; a < 0 reflects (falls left to right)
|a| > 1 stretches vertically; 0 < |a| < 1 compresses
h shifts the inflection point horizontally
k shifts the inflection point vertically
Inflection/center point: (h, k)
Domain: all real numbers (can take cube root of negatives!)
Range: all real numbers

Parent Graph: f(x) = ∛x

Parameter	Effect on Graph
a = 2	Steeper S-curve
a = ½	Shallower S-curve
a = −1	Reflects; falls left→right
h = 5	Center shifts right 5
k = −3	Center shifts down 3

Worked Examples

Example 1

f(x) = ∛(x + 3) − 2

1a = 1, h = −3, k = −2

2Inflection point: (−3, −2)

3No stretch; standard S-shape, shifted

4Domain: all reals · Range: all reals

Example 2

f(x) = −2·∛(x − 1) + 1

1a = −2, h = 1, k = 1

2Inflection point: (1, 1)

3a < 0 → reflects (falls left→right); |a|=2 → steeper

4Domain: all reals · Range: all reals