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Precalculus Test #1
Chapter 1.1-2.1
50
Mathematics
Undergraduate 1
02/13/2013
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Term
Pythagorean Theorem
 Definition
a^2 + b^2 = c^2

where c is the hypotenuse
Term
Distance formula
 Definition
√((x2-x1)^2+(y2-y1)^2)
Term
Midpoint formula
 Definition
midpoint = ((x1+x2)/2, (y1+y2)/2)
Term
To find x-intercepts
 Definition
ley y be zero and solve the equation for x
Term
To find y-intercepts
 Definition
let X be zero and solve the equation for y
Term
A graph is symmetric with respect to the x-axis if
 Definition
f(x) = -f(x)
Term
A graph is symmetric with respect to the y-axis if
 Definition
f(x) = f(-x)
Term
A graph is symmetric with respect to the origin if
 Definition
-f(x) = f(-x)
Term
Standard form of the equation of a circle
 Definition
√((x-h)^2 + (y-k)^2) = r

where r is the radius and (h,k) is the center
Term
Slope-intercept form of the equation of a line
 Definition
y = mx + b

where m is the slope and b is the y-intercept
Term
The slope of a line passing through two points
 Definition
m = (y2-y1)/(x2-x1)

where x1 ≠ x2
Term
Point- slope form of the equation of a line
 Definition
y - y1 = m(x - x1)

where m is the slope and (x1, y1) is a point on the graph
Term
Two lines are parallel if
 Definition
m1 = m2
Term
Two lines are perpendicular if
 Definition
m1 = -1/m2
Term
A graph is a function if
 Definition
for every x input values there is only one y output
Term
The vertical line test
 Definition
tests if the graph is a function or not (if a vertical line drawn anywhere on the graph hits the graph more than once it is not a function)
Term
When finding the domain and range of a function, what is the difference between ( ) and [ ]?
 Definition
Parentheses do not include that point, whereas brackets do.
Term
The zeros of a function f(x) are
 Definition
the x values for which f(x) = 0
Term
A function f(x) is even if
 Definition
for each x in the domain of f, f(-x) = f(x)

[symmetric with respect to the y-axis]
Term
A function f(x) is odd if
 Definition
for each X in the domain of f, f(-x) = -f(x)

[symmetric with respect to the origin]
Term
Vertical shift c units upward
 Definition
h(x) = f(x) + c
Term
Vertical shift c units downward
 Definition
h(x) = f(x) - c
Term
Horizontal shift c units to the right
 Definition
h(x) = f(x-c)
Term
Horizontal shift c units to the left
 Definition
h(x) = f(x+c)
Term
Reflection in the x-axis
 Definition
h(x) = -f(x)
Term
Reflection in the y-axis
 Definition
h(x) = f(-x)
Term
(f + g)(x) =
 Definition
f(x) + g(x)
Term
(f - g)(x) =
 Definition
f(x) - g(x)
Term
(fg)(x) =
 Definition
f(x)g(x)
Term
(f/g)(x) =
 Definition
f(x) / g(x), g(x) ≠ 0
Term
(f ∘ g)(x) =
 Definition
f(g(x))
Term
Is g(x) the inverse of f(x)?
 Definition
If f(g(x)) = x for every x in the domain of g AND g(f(x)) = x for every x in the domain of f.
Term
Only ? functions have an inverse
 Definition
One-to-one functions (for every x there is only one y)
Term
The horizontal line test
 Definition
tests whether the graph has an inverse (if a horizontal line drawn anywhere on the graph touches the graph more than once it does not have an inverse)
Term
To find the inverse function of f(x)
 Definition
1. First verify that there is an inverse with the horizontal line test
2. Replace f(x) by y
3. Interchange the roles of x and y and solve for y
4. Replace y with f^-1 in the new equation
5. Verify that f(f^-1(x)) = x = f^-1(f(x))
Term
Vertex/Standard form of a quadratic function
 Definition
f(x) = a(x - h)^2 + k, a ≠ 0

where (h,k) is the vertex and x = h is the axis of symmetry
Term
Vertical stretch by a factor of c
 Definition
h(x) = cf(x)
Term
Vertical compression by a factor of 1/c
 Definition
h(x) = (1/c)f(x)
Term
Horizontal stretch by a factor of 1/c if c > 0
 Definition
h(x) = f(cx)
Term
Horizontal compression by a factor of 1/c if 0 < c < 1
 Definition
h(x) = f(cx)
Term
Intercept form
 Definition
f(x) = ax^2 + bx + c

tells us the y-intercept
Term
Factored form
 Definition
f(x) = a(x - r)(x - s)

where r and s are the roots/zeros/x-intercepts
Term
Complete the square
 Definition
f(x) = x^2 + (bx/a) + (b/2a)^2 - (b/2a)^2 + (c/a)

f(x) = (x + (b/2a)^2) + ((4ac-b^2)/4a)
Term
Quadratic formula
 Definition
x = (-b±√(b^2-4ac)) / 2a
Term
If n in a polynomial is even,
 Definition
the ends of the graph will end going towards the same direction
Term
If n is odd in a polynomial,
 Definition
the ends of the graph will end going opposite directions
Term
The highest possible number of zeros in a polynomial is
 Definition
the highest power in the polynomial
Term
The graph of f has at most ? turning points
 Definition
n-1
Term
Multiplicity
 Definition
when a zero is repeated

example: in f(x) = (x - 2)^2(x + 1)^3, 2 has a multiplicity of 2 and -1 has a multiplicity of 3
Term
Piece-wise defined function
 Definition
When a graph is broken into pieces that have different equations
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