Algebra II Midterm

Odd Function

f(-x) = -f(x); y = f(x) = x³ (all signs change)

Even Function

f(-x) = f(x); y = f(x) = x² (all signs stay the same)

Even functions are...

Symmetric about the y-axis

Odd functions are...

Symmetric about the origin

f(x) = 3x⁴ - 2x² + 5

Even

f(x) = 2x⁵ - 7x³ + 4x

Odd

f(x) = x³ + x²

Neither

When a function is neither odd nor even...

Only one or two/some signs change, they don't stay all the same or all change

f(x) = |x|

Even

To shift the graph of f upwards c units, use...

y = f(x) + c with c > 0 (add c to the y-coordinate)

To shift the graph of f downwards c units, use...

y = f(x) - c with c > 0 (subtract c from the y-coordinate)

y = f(x) + c and y = f(x) - c are...

Vertical shifts

To shift a graph right c units, use...

y = g(x) = f(x - c) with c > 0

To shift a graph left c units, use...

y = h(x) = f(x + c) with c > 0

y = f(x - c) & y = f(x + c) are...

Horizontal shifts

To stretch the graph of f, use...

y = cf(x) with c > 1 (multiply y-coordinates by #s greater than 1)

To compress the graph of f, use...

y = cf(x) with 0 < c < 1 (multiply y-coordinates by fractions/decimals b/t 0 - 1)

y = cf(x) with c > 1 & y = cf(x) with 0 < c < 1 are...

Vertical stretches and vertical compressions, respectively

the ordered pairs to graph y = x² are...

(-3, 9), (-2, 4), (-1, 1) (0, 0), (1, 1), (2, 4), & (3, 9)

|x| < 1 is the same as...

-1 < x < 1

When finished sketching the graph of a piecewise function, check using...

The vertical line test

Semicircles and circles are...

Neither odd nor even functions, circles are not functions at all because they don't pass the vlt

To determine the increasing and decreasing intervals of a graph...

Look from left to right across the graph

The difference between x² and -x²...

x² is a parabola that opens up and -x² is a parabola that opens down

rise/run = x/y

up x units, right y units or down x units, left y units

Vertical shifts

(x, y + c)

Horizontal shifts

(x + c, y)

Vertical stretches/compressions

(x, cy)

Quadratics are

Even functions

Quadratics are in the form...

f(x) = ax² + bx + c

2x² - 8x + 3

Quadratic/even function

If there is a square root in the numerator, then...

use x ≥ 0

If there is a square root in the denominator, then...

use x > 0 (because x cannot be 0 if it's in the denominator)

If m > 0 on (a , b), then...

The graph is increasing

If m < 0 on (a, b), then...

The graph is decreasing

If m = 0, then

The function is constant

Domain is ___, range ___, and increasing, decreasing, and constant refer to ___

x, y, x

Linear functions are in the form...

y = mx + b

Difference of squares

a2 - b2 = (a + b) (a - b)

Difference of cubes

a3 - b3 = (a2 + ab + b2)

Sum of cubes

a3 + b3 = (a2 - ab + b2)

(a + b)² =

a2 + 2ab + b2

(a - b)² =

a2 - 2ab + b2

(a + b)³ =

a3 + 3a2b + 3ab2 + b3

(a - b)³ =

a3 - 3a2b + 3ab2 - b3

a sqrt -m/b sqrt -n =

b sqrt n/ a sqrt m