Summer 2012 Math151-903 Prof. Lloyd : Test 1 Material

Front 1

Definition: Relation

Back 1

Set of ordered pairs. Note: (a,b) ≠ (b,a)

Front 2

Definition: Function

Back 2

A relation in which each x-coordinate is matched (mapped) with only one y-coordinate. Each x has exactly one y.

Front 3

Definition: Vertical Line Test

Back 3

If the graph/data is a function, then no vertical line will ever intersect points on an x-coordinate twice.

Front 4

Definition: ( )

Example: (a, b)

Back 4

a and b are not exactly at a or b, by either being noted as an open circle, or where no value can be reached (infinity). Described by professor as "Up to, but not equal to"

Front 5

Definition: [ ]

Example: [a, b]

Back 5

a and b are exact end points.

Front 6

Definition: Domain

Back 6

Set of x coordinates of a function. Written left to right.

Front 7

Definition: Range

Back 7

Set of y coordinates of a function. Written bottom to top.

Front 8

Definition/Symbol: U

Back 8

Union. X-coordinates which are only found in both sets, if an x-coordinate only exists in a single set then it will NOT be represented in the union.

Front 9

Definition/Symbol: ∩

Back 9

Intersection. Includes points from multiple sources which have been combined into a single source (most likely, with gaps where there are no values for y coordinates.

Front 10

Definition/Symbol: Empty circle o

Back 10

A graphed line can get infinitely closer to this point, but will never intersect this point.

Front 11

Formula: Slope/Intercept

Back 11

Formula: y = mx + b.
b is the y intercept where x=0 (in other words, where the line crosses from positive to negative, or vice versa. Where it crosses the vertical line).
The slope Formula: m = y1 - y2 / x1 - x2 (one point minus the other point)

Front 12

Definition: Dependent Variable

Back 12

y. y is dependent because the value of y depends on the value of x in a function.

Front 13

Definition: Independent Variable

Back 13

x. x is independent because it does not depend on anything, things depend on it in a function.

Front 14

Definition: f()

Back 14

Indicates a process. Used to figure out exactly how y depends on x for it's value.

Front 15

Usage: Inequalities

Back 15

Negatives: Dividing by a negative, you change the value of the numbers from negative to positive and vice verse, and if you do then you must flip the inequality sign itself.
Example: -9 > -3x (divide by -3) = 3 < x
Graphing: If only showing < or >, then the graph will show a open circle at that point. If < or > then include a closed circle because it does include that exact point.

Front 16

Identification/Usage: Parabola

Back 16

Parabola graphs will have one of two forms.
In this course, so far, we have used the form y = ax2 + bx + c.
The other form is y = a(x-h)2 + k.
It is possible to convert between the two forms. The vertex is located at (h,k) so being able to convert between the two through factoring is useful (c would need to be any number that would allow the first two terms to factor, anything left over (or negative) would become k.
If ax2 is positive, then the parabola opens upward.
if ax2 is negative, then the parabola opens downward.
For a sideways parabola, x and y are reversed.
If ay2 is positive, then the parabola opens right.
if ay2 is negative, then the parabola opens left.

Front 17

Usage: x and y intercepts

Back 17

y-intercept - solve for f(0). (in other words, let x = 0 and solve for y)
x-intercept - solve for f(x) = 0. (in other words, let y = 0 and solve for x)
If you get a quadratic (2 numbers), then there are 2 intercepts. Factor the quadratic and set each factor = 0 and solve for x or y (whichever intercept you are solving for).

Front 18

Definition: Piece Wise

Back 18

Several functions combined (several lines on a single graph)

Front 19

Usage: Arithmetic of functions

Back 19

(f+g)(x) = f(x) + g(x)
(f-g)(x) = f(x) - g(x)
(f*g)(x) = f(x)*g(x)
(f/g)(x) = f(x)/g(x) WARNING! See Note.
Note: Find domain of denominator before simplifying or the correct domain will most likely be masked or altered.

Front 20

Formula: Difference Quotient

Back 20

Finds the derivative of a function. Assume h is = 0 (do not substitute 0 for h), therefore the h cannot exist in the denominator and must be simplified out if possible.
Formula:
( f(x+h) - f(x) ) \ h

Front 21

Usage: Factoring (a+b)^2

Back 21

Equal to a^2 + ab + b^2 in most cases (unless both a and b are perfect squares, in which there will only be 2 terms)
Perfect Squares: Equal to ( √a + √b) ( √a - √b). Example: x^2 + 4 = (x + 2)(x - 2)

Front 22

Usage: Rationalizing fractions

Back 22

Although not required when using the difference quotient (because it just returns h back into the denominator again), when there is a square root in the denominator you need to multiply by itself over itself, or if there are multiple terms then multiply by the conjugate of the denominator over the conjugate of the denominator.
1 / (√2) + 2
=
1/(√2) + 2 * (√2) - 2 / (√2) - 2

Front 23

Determine: Even Function

Back 23

To find an even function, solve for f(-x). (in other words, replace every x with -x). If equal to f(x), then the the function is even.
f(x) = 1 / 4 - x^2 is this even?
f(-x) = 1 / 4 - (-x)^2 (note: any negative number squared is positive, therefore -x^2 = x^2)
f(-x) = 1 / 4 - x^2 (This is equal to the original, therefore this function is even. If this is true, you can stop here because it cannot also be odd.)

Front 24

Determine: Odd Function

Back 24

To find an odd function, first solve to see if it is even and if it is not... then keep the simplified version of f(-x) available to use here. Now, you want to solve to solve for - ( f(x) ). (in other words, negate the entire expression for f(x) ). If f(-x) = - ( f(x) ), then the function is odd.
f(x) = x / x^2 + 1 is this even?
f(-x) = (-x) / (-x)^2 + 1 (note: any negative number squared is positive, therefore -x^2 = x^2)
f(-x) = (-x) / x^2 + 1 (This is not equal to the original, therefore this function is NOT even)
Since it is not even, is this function odd?
-( f(x) ) = - ( x / x^2 + 1 ) (Note: When distributing the negative, you can negate the numerator or denominator, but not both. Look to see if negating either side could make the function equal to f(-x). In this case, negating the top would be the best option. If you are unsure, negate both and work each individually).
-( f(x) ) = -x / x^2 + 1 (This is equal to to f(-x), therefore this function is odd.)

Front 25

Determine: Function neither even or odd

Back 25

If a function is not even or odd by the definitions above, then it is neither even or odd. This is the case for many functions.

Front 26

Tip part 1: Is -(x)^2 = (x)^2 ?
Tip part 2: Is -x^2 = x^2 ?
Tip part 3: Is (-x)^2 = x^2
Why: Where will you need to know this?

Back 26

1: No. Because x would be squared before the negative is applied (PEMDAS).
2: No. Because exponents would be applied before the negative (PEMDAS).
3: Yes. Because the negative is applied first, then the exponent, which would make the (-x)^2 = x^2.
This was used in EVERY example of determining an even or odd function in class.

Front 27

Definition: Increasing

Back 27

Rises as we go from left to right, or "Increasing interval i iff f(a) < f(b) for all a < b.
See: http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/11-graphing-increasing-decreasing-03.htm

Front 28

Definition: Decreasing

Back 28

Falls as we go from left to right, or "Decreasing interval i iff f(a) > f(b) for all a > b.
See: http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/11-graphing-increasing-decreasing-03.htm

Front 29

Definition: Local Extrema

Back 29

The highest(maximum) and lowest(minimum) point of every curvature of the graph. (there can be many minimum and many maximum values, or none)
A great video explanation: http://www.youtube.com/watch?v=R91CRGrwQFs

Front 30

Definition: Absolute Extrema

Back 30

The one and only highest(maximum) and the one and only lowest(minimum) point at the center of a curvature on the graph.
A great video explanation: http://www.youtube.com/watch?v=R91CRGrwQFs

Front 31

Definition: Asymptotes

Back 31

Graphs where the line will get infinitely close to the x or y axis, but never touch. Examples are y = 1/x or y = 1 / x^2.

Front 32

Definition: Concave up

Back 32

Looks like a U (opening up) with the left side decreasing and the right side increasing.
See: http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII_files/image002.gif

Front 33

Definition: Concave down

Back 33

Looks like a n (opening down) with the left side increasing and the right side decreasing.
See: http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII_files/image002.gif

Front 34

Definition: Rigid Transformation

Back 34

The graph has the exact same size/shape, it is just in a different place.
f(x) + k = shift up k spaces. (Note: Think "over and out!" If the graph goes up (over) then k is out (outside of the parentheses). Hopefully this will help to remember the other four)
f(x) - k = shift down k spaces.
f(x + k) = shift left k spaces.
f(x - k) = shift right k spaces.
When faced with multiple transformations, start with those inside the parenthasis first, then move outside.

Front 35

Definition: Reflections

Back 35

For x-axis reflections - multiply f(x) by -1
For y-axis reflections - replace x with (-x)

Front 36

Definition: Vertical Scaling

Back 36

Alters a graph like transformations do, however for scaling the graph will remain in the same place but it's size and shape will be altered in relation to the x-axis instead.
For scaling, you multiply f(x) by a number giving the formula cf(x). This will effect the y values of the graph. The input (x) will remain the same, but the output (y) will be different.
As with reflections, a negative will inverse the graph (if a parabola was opening up, it will now open down).
2*f(x) - x values will be the same, y values will be doubled (further away from the x axis)
(1/2)*f(x) - x values will be the same, y values will be halved (closer to the x axis)
See: http://www.youtube.com/watch?v=SMeelHMiLJI&feature=plcp

Front 37

Definition: Horizontal Scaling

Back 37

Multiply X (input variable) by d>0. So f(cx).
Where vertical scaling effects graphs in relation to the x axis, this will effect graphs in relation to the y axis. This will effect the x values of the graph. The input (x) will change, but the output (y) will be the same (opposite of vertical).
f(2x) - y values will be the same, x values will be halved (closer to y axis. Note: this is counter intuitive since the opposite happened in vertical scaling with a number larget than 1.)
f( (1/2)x ) - y values will be the same, x values will be doubled (further from y axis. Again, see note above.)
See: http://www.youtube.com/watch?v=Ox-XaIoPZys