O’Brien Sp09
CA 3rd BPB
Recognizing Common Graphs and Equations
Equation
Graph
How to recognize it
Comment
x=a
vertical line
no y in equation, just x = #
vertical lines are not functions
y=b
horizontal line
no x in equation, just y = #
horizontal lines are constant (0 degree) functions
ax + by = c OR y = mx + b
oblique line
x & y in equation, both to 1st power
oblique lines are linear (1st degree) functions
slope = m; y-intercept = b
y = ax2 + bx + c OR y = a(x – h)2 + k
parabola
highest exponent on x is 2
quadratic (2nd degree) function; x of vertex = 
b
2a
vertex: (h, k); if a > 0, opens up
(x – h)2 + (y – k)2 = r2
circle
x & y are both squared
center: (h, k); radius: r; circles are not functions
y = ax3 + bx2 + cx + d OR y = a(x – h)3 + k
cubic
highest exponent on x is 3
cubics are 3rd degree functions
inflection point: (h, k); if a > 0, LH down, RH up
y = ax4 + bx3 + cx2 + dx + e OR y = a(x – h)4 + k
quartic
highest exponent on x is 4
quartics are 4th degree functions
vertex: (h, k); if a > 0, opens up
y  a x h k
half parabola
square root with x to 1st power
half horizontal parabola with “key” point at (h, k)
y  a2  x 2
half circle
square root with x to 2nd power
half circle with center at origin & radius = a
y  a 3 (x  h)  k
cube root
cube root
looks like horizontal cubic with I.P. at (h, k)
y = a |x – h| + k
absolute value
absolute value bars
v-shape with vertex at (h, k); if a > 0, opens up
ax  b, for x  #
y
cx  d, for x  #
piecewise
equation is defined in pieces
piece with  or  has solid circle at #
y = ax
exponential
variable is in the exponent
base > 0 & base ≠ 0; J or banana shape
y  log a x
logarithm
only equation with log
r shape; loge x is written ln x