Math TAKS Review
Formula Charts
Graphing Quadrants
III
III IV
The Quadrants on a Graph go Counter-Clockwise
Plotting Points on the Coordinate Plane
Remember in ordered pairs, x comes before y.
(x, y) First go left or right to plot x, then up or
down to plot y.
Parent FunctionsA Parent Function is the original graph of a
function before it has been changed in any way.Linear Function Quadratic Function
y=x y=x2
Determining Functions
When given ordered pairs, table values, mappings, or graphs, the x-value cannot be repeated. In other words, the input value cannot have more than one output value.
(3,4)
X and Y Terminology The following words are used during the
test but have the same meaning as x and y values.
( x-value , y-value ) (independent, dependent) ( domain , range ) ( input , output )
Interpreting Information Drawing conclusions to given scenarios Note: Dummy variables, terminology (per,
more than, less than, etc…
Systems and Solutions (X , Y) In order for a value to be a solution to an
equation or a system of equations then it implies that it is in fact a true statement. When graphing, the solution is the intersection point.
You will have to write equations for systems. Remember that ordered pairs represent x and
y values whereas set notation represent simply all x-values (domain).
y = mx + b Be able to identify the slope (also known as rate of
change = m) and the y-intercept (b). It is essential to understand what these values
represent in any given equation. M = rate of change → this implies for each
change in the x-value, there is a corresponding change in the y-value.
M= y2-y1 ∕ x2-x1
Parallel Lines vs. Perpendicular Lines
If two lines are parallel then they have the same slope . This implies they never cross one another therefore have no point of intersection (no solution).
If two lines are If two lines are perpendicular then they have perpendicular then they have opposite reciprocal slopes . This opposite reciprocal slopes . This implies the two lines have implies the two lines have exactly one point of intersection exactly one point of intersection and the intersection forms 90 and the intersection forms 90 degree angles (one solution).degree angles (one solution).
Y= 1/3x+2.5
Y=1/3x-2
Y=3/4x
y=-4/3x
Midpoint and Distance Formulas Note key words. Midpoint implies CENTER or MIDDLE
position. Distance implies LENGTH between two
places or positions
Positive Correlation As you move from left
to right the data plots move up.
0
1
2
3
4
5
6
0 5 10
PositiveCorrelation
Negative Correlation As you move from left
to right the data plots move down.
0
1
2
3
4
5
6
0 5 10
NegativeCorrelation
No Correlation As you move from left
to right the data plots show no pattern.
0
1
2
3
4
5
6
0 5 10
NoCorrelation
Undefined Correlation As you move from left
to right the plots represent a vertical or horizontal pattern.
0
2
4
6
0 5
X = 3
X = 3
X = 3
X = 3
X = 3
0
1
2
3
Y = 2
Effects of Change and Comparison of Quadratic Equations Parabolas can be
graphed in the calculator to determine change
The larger the a value the narrower the graph.
The smaller the a value the wider the graph
y=x2
Roots A root to an equation
simply implies where the function crosses the x-axis on a graph.
Also known as x- intercepts, solutions and zeros.
The answer is usually given in set notation form. It is very important to read these types of questions very carefully because the question could ask either how many solutions or roots the equation has or to determine what in fact the actual solution or root is (its specific point on the x-axis).
Do not get confused when given two values in a { } answer and assume that it represents the same thing as (x , y). Remember that only ordered pairs represent these values while { } represent all the roots of the given equation.
Pythagorean Theorem This formula is used when
given two sides of a right triangle and solving for the third side (remember the side representing c is always the hypotenuse). Usually there will be more steps to solving the given question when using this theorem than simply solving for the unknown sides. It is imperative to read the question thoroughly
A
B
C
A2+B2=C2
Special Right Triangles These types of triangles are used when
given only one side and one angle of a right triangle.
45-45-90 30-60-90
X
X√3
2X
Arc or Sector Lengths and Arc or Sector Areas
These questions are asked when given circle graphs and a triangle is inscribed inside the circle. The information needed to solve these types of questions is the angle measurement and radius. These formulas must be memorized.
Arc or Sector Length: Always use circumference)
(angle degree ∕ 360)(2πr) Arc or Sector Area: (Always use area)
(angle degree ∕ 360)πr2
Probability The answer can be represented as a percent or fraction. The
formula used to find the probability of an event occurring is found by . Remember that probability always implies multiplication (if given more than one scenario, you should multiply the choices together). Remember the marble or spinning wheel examples.
Theoretical: getting heads if flipping a coin Experimental: doing the process ( taking marbles out of the
jar)
FAVORABLEOUTCOME ∕ TOTALCHOICES
Combinations The total combinations you can have when
given a certain scenario. You always multiply the different number of choices given with the total number of combinations in each distinct choice.
Angle Terminology Complimentary- add to equal 90 degrees Supplementary- add to equal 180 degrees Interior Angles- sum of inside angles equal
(n-2)180, where n represents the # of sides.
Vertical Angles- directly across from each other and are congruent.
Perimeter vs. Area vs. Volume The formulas for each are given in the formula charts
but it is essential to understand when to use the formulas. Perimeter: the distance around a given object or
figure (ex. fence): P=2l+2w Area: the amount of material needed to cover an
object or figure (ex. wrapping a present) A=LW Surface Area: the area of a 3-deminsional figure (remember
the SA of a rectangular prism is the area of each side including the top and bottom added together.
Volume: the amount needed to fill an object or figure
up (water in swimming pool) V=L x W x H
Geometric Nets and Figures Flat views of figures Front, side, and top views of figures Sides, edges, and vertices
Scale Factor and Dilations The word scale factor implies the number of
times an object increased or decreased in size compared to its original status.
The word dilation implies the actual act of increasing or decreasing an object.
Similarity The symbol that is used to imply similarity is ~ . If two
objects are compared with this symbol or simply implied that they are in fact similar then this means they are proportional to one another and should be set up as ratios that are equal to one another which gives us a proportion. Let the unknown value represent x and cross multiply to solve.
Note… the actual word similar might not be used in the problem so if two items are compared to one another then they are in fact similar (remember the flag pole and shadow example).
Congruent Figures Shapes that are identical in every way. Congruent triangles are referred to as
equilateral triangles. This implies all sides and all angles are the
same.
Finding Unknown Values When given information and asked to find
percent or exact number, you should use the following formula: PART ∕ WHOLE= % ∕ 100
This formula is to be memorized and is extremely useful when asked to find three different types of information.
An extremely similar formula can be used to change the value of a given percent to an actual angle degree measure (key word is central angle).
X ∕ 360 = % ∕ 100
Inequalities An inequality is an equation that divides the coordinate
plane into shaded areas. To determine which region is being represented
(shaded) we must first solve the inequality for y and then graph the line. Remember when solving an inequality and division by a negative value occurs, the inequality symbol < or > must flip directions.
Less than < and < represent the shaded region to be shaded below
Greater than > and > represent the shaded region to be shaded above the given line.
Note…if the inequality is < or > then a dotted line implies and < and > imply solid lines.
Mean, Median, Mode, and Range It is imperative to know and understand the
definitions of these terms. Mean: implies the average value Median: implies the middle value when they
are placed in ascending order Mode: implies the most given value Range: the difference between the highest
and lowest values
Tessellation, Rotation, Reflection, and Translation
You should know what each term implies. Tessellation: placing an object side by side in
different arrangements without developing any gaps.
Rotation: Taking a figure and simply rotating it around while keeping an end point centered.
Reflection: A mirror image. Translation: Simply sliding a figure from it original
status but not picking it up or changing its direction.
Exponents When solving problems with exponents it is incorrect to have
a negative exponent value in the final answer. In order to change the value of the negative exponent to a
positive value, you simply move the base of the negative to either to top (numerator) or to the bottom (denominator) depending on its original location.
All whole numbers are multiplied or divided depending on the question being asked just as if they were regular values without exponents.
When multiplying exponents simply combine all the same bases and add the exponents and when dividing exponents you are to subtract the exponents of the same base. Note….the overall answer can in fact have a negative value (the whole number) but NOT negative exponents.
Fractals Writing equations from given patterns The use of dummy variables Plug the equations into the calculator and
test the picture or table
Things to Remember Parent Functions: Linear and Quadratic Domain and Range Open Circles: Strictly < or > Closed Circles: Strictly < or > Inequalities: dotted and solid lines,
shading above and below Independent and Dependent Variables
Solutions Roots Probability Special Right Triangles (45-45-90 and 30-60-90) Pythagorean Theorem Midpoint Formula (when is it used, may ask for
center) Distance Formula (when is it used may ask for
length) Similarity means Proportional ( symbol is ~)
B* ( area of base of object) Surface Area of a rectangular prism Arc Area and Arc Length Mean, Median, Mode, and Range Translation, Tessellation, Reflection, and
Rotation Scale Factor Dilation Exponents (how to eliminate negative
exponents)
Perimeter Area Volume Surface Area Parallel Lines: same slope Perpendicular Lines: negative reciprocal
slopes
Y = mx + b (rate of change and x & y intercepts) X intercept: let y = 0 and solve, must graph Y intercept: let x = 0 and solve, the b value Complementary angles: 2 angles that =90 Supplementary angles: 2 angles that = 180 Equilateral Triangle: all 3 sides and angles are
equal Isosceles Triangle: 2 equal sides and angles Scalene Triangle: no equal sides Acute angle: an angle less than 90 degrees Obtuse angle: angle greater than 90 but less than
180 degrees
Sample Problems
Parent Functions
Domain and RangeRemember Open and Closed
Circles
Reading GraphsBe sure you know what you are
comparing
Rate of ChangeUsing the formula to determine future outcome…follow pattern
Solution to a SystemThe intersection point of two lines
(x,y)
X- InterceptGraph equation or plug value (x,y)
into equation
Slope Solve equation…. y = mx + b
Also determined by
rise / run
Points on a linePlug into calculator and go to
table…
Mid PointMight have to work backwards.
The answer is always an ordered pair.
(x, y)
Parallel and Perpendicular Lines
Parallel lines- same slope
Perpendicular lines- opposite reciprocal slopes
Alterations to graphs
Test the given choices in the calculator
RootsAlso known as x- intercepts and
zeros
Pythagorean Theorem
Used when given two sides and determining a third
Special Right Triangles
Used when given one side and one angle.
45-45-90
30-60-90
Arc Area and Arc Length
Memorize formulas…use area and circumference of a circle
ProbabilityFavorable Outcome / Total Choices
Multiply scenarios together
Angle TerminologyComplementary and Supplementary
ApothemDistance from center to outside
length
Geometric FormulasAdd or Subtract values
Edges, Vertices, and Sides
Changes and alterations to
geometric shapes
RatiosComparisons
InequalitiesShading above and below
Solid and dotted lines
ExponentsMultiplying…add exponents
Dividing…subtract exponents
Fractals…the nth stage
Plug given equations into calculator