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 Functions A Parent Function is the original graph of a function before it has been changed in any way. Linear Function Quadratic Function y=xy=x 2

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= y 2 -y 1 ∕ x 2 -x 1

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 perpendicular then they have opposite reciprocal slopes. This implies the two lines have exactly one point of intersection and the intersection forms 90 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.

Negative Correlation As you move from left to right the data plots move down.

No Correlation As you move from left to right the data plots show no pattern.

Undefined Correlation As you move from left to right the plots represent a vertical or horizontal pattern.

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 A 2 +B 2 =C 2

Special Right Triangles These types of triangles are used when given only one side and one angle of a right triangle 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)πr 2

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 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 then a dotted line implies 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 Closed Circles: Strictly Inequalities: dotted and solid lines, shading above and below Independent and Dependent Variables

Solutions Roots Probability Special Right Triangles ( and ) 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 Range Remember Open and Closed Circles

Reading Graphs Be sure you know what you are comparing

Rate of Change Using the formula to determine future outcome…follow pattern

Solution to a System The intersection point of two lines (x,y)

X- Intercept Graph equation or plug value (x,y) into equation

Slope Solve equation…. y = mx + b Also determined by rise / run

Points on a line Plug into calculator and go to table…

Mid Point Might 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

Roots Also 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

Arc Area and Arc Length Memorize formulas…use area and circumference of a circle

Probability Favorable Outcome / Total Choices Multiply scenarios together

Angle Terminology Complementary and Supplementary

Apothem Distance from center to outside length

Geometric Formulas Add or Subtract values

Edges, Vertices, and Sides

Changes and alterations to geometric shapes

Ratios Comparisons

Inequalities Shading above and below Solid and dotted lines

Exponents Multiplying…add exponents Dividing…subtract exponents

Fractals…the n th stage Plug given equations into calculator