Graphs of Common Functions

Parent Functions Reference

Constant Function: $f(x) = c$

• Horizontal line at $y = c$
• Domain: $(-\infty, \infty)$
• Range: $\{c\}$

Identity Function: $f(x) = x$

• Straight line through origin
• Slope $= 1$
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$

Linear Function: $f(x) = mx + b$

• Straight line
• Slope $= m$
• y-intercept $= b$
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$

Quadratic Function: $f(x) = x^2$

• U-shaped parabola
• Vertex at origin
• Opens upward
• Domain: $(-\infty, \infty)$
• Range: $[0, \infty)$
• Even function (symmetric about y-axis)

Cubic Function: $f(x) = x^3$

• S-shaped curve
• Passes through origin
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$
• Odd function (symmetric about origin)

Square Root Function: $f(x) = \sqrt{x}$

• Half of a sideways parabola
• Starts at origin
• Domain: $[0, \infty)$
• Range: $[0, \infty)$

Cube Root Function: $f(x) = \sqrt[3]{x}$

• S-shaped curve (gentler than cubic)
• Passes through origin
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$
• Odd function

Absolute Value Function: $f(x) = |x|$

• V-shaped graph
• Vertex at origin
• Domain: $(-\infty, \infty)$
• Range: $[0, \infty)$
• Even function

Reciprocal Function: $f(x) = \frac{1}{x}$

• Two branches (hyperbola)
• Vertical asymptote: $x = 0$
• Horizontal asymptote: $y = 0$
• Domain: $(-\infty, 0) \cup (0, \infty)$
• Range: $(-\infty, 0) \cup (0, \infty)$
• Odd function

Exponential Function: $f(x) = e^x$

• Rapidly increasing curve
• y-intercept at $(0, 1)$
• Horizontal asymptote: $y = 0$
• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$

Natural Log Function: $f(x) = \ln(x)$

• Slowly increasing curve
• x-intercept at $(1, 0)$
• Vertical asymptote: $x = 0$
• Domain: $(0, \infty)$
• Range: $(-\infty, \infty)$

Key Features to Identify

Feature	Description
Domain	All valid x-values
Range	All possible y-values
Intercepts	Where graph crosses axes
Asymptotes	Lines graph approaches
Symmetry	Even, odd, or neither
Increasing/Decreasing	Where function rises/falls
Maximum/Minimum	Highest/lowest points
End Behavior	What happens as $x \to \pm\infty$

Graph Recognition Tips

If you see...	It's likely...
Straight line	Linear
U-shape	Quadratic
S-curve through origin	Cubic
V-shape	Absolute value
Half parabola (right side)	Square root
Two branches in quadrants I & III	$\frac{1}{x}$
Rapid growth, never touches x-axis	Exponential
Slow growth, never touches y-axis	Logarithmic