01.01-04.05 summaries

04.04

Linear Function A function whose graph is represented by a straight line. Line A collection of points that goes on forever in opposite directions. Linear function A function whose graph is represented by a straight line; equation can be in the form y = mx + b. Linear functions have exponents of only 1 or 0 on each variable.

04.05

Linear function: A function that shows a constant rate of change; a straight line when graphed. Nonlinear function: Any function that is not a linear function; a constant change in x does not result in a constant change in y. Increasing interval: An area on a graph where a function is moving in an upward direction from left to right; a positive rate of change. Decreasing interval: An area on a graph where a function is moving in a downward direction from left to right; a negative rate of change. Constant interval: An area on a graph where a function is horizontal, not moving up or down; a zero rate of change.

02.06

Adjacent Angles: Pairs of adjacent angles are created by transversals. Adjacent angles are angles that share a vertex and exactly one side, but do not overlap. In other words, they are stuck together. Adjacent angles: Angles that share a vertex and exactly one side, but do not overlap. Remember! Supplementary angles are angles that add up to 180 degrees. Vertical angles: Angles opposite each other when two lines cross. Alternate exterior angles: The pairs of congruent angles on opposite sides of the transversal but outside the two parallel lines. Alternate interior angles: The pairs of congruent angles on opposite sides of the transversal but inside the two parallel lines. Corresponding angles: Congruent angles that lie in the same position relative to the intersection of one parallel line and the transversal. Go to page 4 to understand more.

03.05

Base The surface on which an object rests. Cylinder A three-dimensional object with a flat, circular base on top and bottom. Diameter A straight line going from one side of a circle to the other through the center. Height The vertical distance from top to bottom that creates a 90-degree angle with the base. Radius The distance from the center to the edge of a circle; half of the diameter. Formula Volume of a Cylinder: V = πr2h where V is volume, r is radius, and h is height. Formula Volume of a Cone: V= 13 πr2h where V is volume, r is radius, and h is height Formula Volume of a Sphere: V= 43 πr3 where V is volume and r is radius

03.03

Coordinate plane (or Cartesian plane) An x-axis and y-axis that meet at a right angle at a center point called the origin; a graph. Ordered pair: A coordinate or point written in the form of (x, y). Origin: The center of a coordinate plane, expressed with the ordered pair (0,0); where the x-axis and y-axis intersect. x-axis: The horizontal number line of the coordinate plane. y-axis: The vertical number line of the coordinate plane. So far, you have seen how to find the distance between two points that lie along the same horizontal or vertical line segment. But what if the segment is "slanted"? What if the two points don't lie along a vertical or horizontal line? Using the coordinate points, you can create a right triangle that has the line segment as the hypotenuse. Then you can apply the Pythagorean Theorem to find the distance. See to page 4 for more.

02.05

Dilation: A type of transformation that changes the size but not the shape of a figure. Scale factor: A ratio of two corresponding lengths that determines the change in size from a pre-image to an image. Similar: Having the same shape and angle measure but different sizes; expressed with the symbol ∼. Characteristics of Similar Figures So what is special about similarity? Well, when you know that two figures are similar, you know that: angles correspond to one another and are congruent sides correspond to one another and are proportional sides all have the same proportional relationship. Note! Congruent figures also qualify as similar. They have the same angle measurements and are the same size. When figures are marked as congruent, you can also assume that they are similar.

01.03

Negative Exponent Property Any whole number a (except zero) raised to the -nth power is the reciprocal of a raised to the nth power. Using algebraic symbols, we can write this rule as: a^−n= 1/a^n, when a ≠ 0. Also just because a power has a negative exponent, it does not necessarily mean the value of the power will be negative. When working with negative exponents, it is important to remember these things: Simplify the power using properties of exponents—product and quotient rules. Write the negative power as its reciprocal and drop the negative sign. Simplify again, if necessary. Power of a Power Property To find the power of a power, keep the base and multiply the exponents. Using algebraic symbols, we can write this rule as: (a^m)^n=a^m•^n Power of a Product or Quotient To find the power of a product or quotient, apply the power to each factor inside the parentheses, multiply exponents and simplify if needed. Simplify (3/7^4)^5. Apply the power to each factor. Apply the fifth power to each factor on the inside of the parentheses: ( 3/7^4)^5= (3^1)^5/(7^4)^5 Multiply the exponents. (3/7^4)^5 (3^1)^5/(7^4)^5= (3)^1*^5 (7)^4*^5= (3^5)/(7^20)

01.05

Perfect Square A number that is made by squaring a number: a^2 = a • a Square Root: If a^2 = b, then a = ±√b When finding square roots, the radicand may not always be a perfect square. It may be a nonperfect square resulting in an irrational number. For example, find √10. Ten is not a perfect square, so what do we do now? Find √10. Which perfect squares are closest to √10? To estimate the square root of a nonperfect square, start by thinking about which perfect squares would be closest to that number. You may remember that 9 and 16 are both perfect squares. So, you can conclude that √10 is between √9 and √16. Or this can be written as √9<√10<√16. Decide which perfect square is closer to √10. Look at the perfect squares around your radicand. Which one is closer? In this problem, √9<√10<√16. Since 10 is closer to 9 than it is 16, then √10 is closer to √9 than it is to √16. Use that square root as an estimated value for your radical. Take the square root of the closest perfect square. That is an estimated answer for your square root. Since you know that √9 is 3, you can conclude that √10≈3. The ≈ symbol means "approximately equal to." This indicates that this is an estimated answer, not an exact solution. Perfect Cube A number that is made by cubing a number:a^3 = a • a • a Cube Root: if a^3 = b, then a=^3√b Solving for Volume Formula using cube (V = n^3)

04.02

Rate of Change A measurement of the change in variables over a specific period of time; a measure of output value changes in comparison to input value changes Rate of Change = change in y and change in x look at page 3

01.01

Rational and irrational numbers. Repeating/terminating decimals, and fractions. Rational numbers of a den. of 9 repeat a single digit, and so on. Ex. 0.134134134 = 134/999. You can also simplify this.

01.02

Recall that a power is an expression that has a base and an exponent, like 34 or 52. The exponent tells how many times the base is a factor. Product Property To multiply powers that have the same base, keep the base and add the exponents. Using algebraic symbols, we can write this rule as: a^m⋅a^n=a^m+^n. The rule for multiplying powers can be used only for powers that have the same base. You cannot use this rule to simplify 24 ⋅ 62 because these powers have different bases. Expanded form: 2^4 ⋅ 6^2 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 6 ⋅ 6 Notice how the 2s and 6s do not merge for a combined exponent; the bases must be the same in order to add the exponents. Quotient Property To divide powers that have the same base, keep the base and subtract the exponents. The zero power rule is the numbers that have a 0 as their exponents is always 1.

02.02

Reflection: A type of transformation in which one figure is a direct mirror image of another. Rigid transformation: Moving a figure so that it is in a new location but has the same size, shape, and area. Line of reflection: The line over which a pre-image is reflected to create a new image. Characteristics of a Reflection There are several important characteristics of a reflection: A reflection maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle. Each piece of the original pre-image is exactly the same in the image. A reflection preserves lengths of segments. The length of a line segment will be the same in a reflection. A reflection preserves degrees of angles. The angles of the image are congruent to the angles in the pre-image. Rotation: A transformation that turns a figure a given angle and direction around a fixed point. Go to textbook on page 3 to see the angle degrees examples. Rotations can be either clockwise or counterclockwise in direction. Center of rotation is the point around which a figure is rotated. Characteristics of a Rotation There are several important characteristics of a rotation: A rotation maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle. All parts of the original object are the same in the pre-image and the image. A rotation preserves lengths of segments. A rotation preserves degrees of angles. Rotations on the Coordinate Plane When a figure is placed on the coordinate plane, the center of rotation is often the origin.

01.07

Steps for Multiplying and Dividing in Scientific Notation Group the numerical factors together—multiply or divide. Use the properties of exponents to simplify powers of 10—product or quotient rule. Convert to scientific notation if needed. Example 1: To multiply two numbers in scientific notation, use properties to rearrange the terms. Then use the exponent rules to simplify. (2.1 • 10^3) • (3 • 10^18) 1. Group the numerical factors together—multiply or divide. (2.1•10^3)•(3•10^18)(2.1•^3)•(10^3•10^18)6.3•(10^3•10^18) 2. Use the properties of exponents to simplify powers of 10—product or quotient rule. 6.3•(10^3•10^18) Using the product rule, add the exponents.6.3•10^3+186.3•10^21 Example 2: You divide numbers in scientific notation in a similar way. To divide two numbers in scientific notation, use properties to rearrange the terms. Then use the exponent rules to simplify. 4.8 • 10^9 ÷ 1.2 • 10^6 1. Group the numerical factors together—multiply or divide. 4.8•109÷1.2•106(4.8÷1.2)•(109÷106) 4.8/1.2 • 10^9/10^6 4 • 10^9/10^6 2. Use the properties of exponents to simplify powers of 10—product or quotient rule. 4•109106 Using the quotient rule, subtract the exponents.4 • 10^9−6 4 • 10^3 The answer is in scientific notation, so Step 3 is not necessary. Steps for Adding or Subtracting in Scientific Notation Determine the smallest power of 10. Decrease the larger power of 10 until it matches the smaller one found in Step 1—move the decimal point of the number with the larger power of 10 to the right until it matches the smaller power. Add or subtract the new number factors and keep the power of 10. Convert to scientific notation if needed. Example 1: (2.456•10^5)+(6.0034•10^8) 1. Determine the smallest power of 10. 105 is the smaller power of 10. That means that we have to decrease 10^8 to match 10^5 so we can add the bases together. 2. Decrease the larger power of 10 until it matches the smaller one found in Step 1. We have to decrease the exponent on 10^8 by 3 in order for it to match 10^8 (^8 − ^5 = 3). In order to do this, we will need to move the decimal point three spaces to the right: 6.0034•108=6003.4•105. 3. Add or subtract the new number factors and keep the power of 10. (2.456•10^5)+(6003.4•10^5)(2.456+6003.4)•1056005.856•10^5 4. Convert to scientific notation if needed. 6005.856 • 10^5 is not in scientific notation because the numerical factor is not a number greater than or equal to 1 and less than 10. The decimal point must be moved three places to the left, so the exponent will increase to 8. 6005.856•10^5=6.005856•10^8 Example 2: 480,100−(2.21•10^6) 1. Determine the smallest power of 10. First, we have to write 480,100 in scientific notation: 4.801 • 10^5 Now we can compare the powers of 10. 10^5 is the smallest power of 10. That means that we have to decrease 10^6 to match 10^5. 2. Decrease the larger power of 10 until it matches the smaller one found in Step 1. We have to decrease the exponent on 10^6 by one in order for it to match 10^5 (^6 − ^5 = 1). Move the decimal point one space to the right: 2.21•10^6=22.1•10^5. 3. Add or subtract the new number factors and keep the power of 10. (4.801•10^5)−(22.1•10^5)(4.801−22.1)•105−17.299•10^5 4. Convert to scientific notation if needed. −17.299 •10^5 is not in scientific notation because the absolute value of the numerical factor is not a number greater than or equal to 1 and less than 10. The decimal point must be moved one place to the left, so the exponent will increase to 6. −17.299•10^5=−1.7299•10^6

03.01

The Pythagorean Theorem A triangle is a polygon with three sides and three angles. A right triangle is a triangle that has one right angle. In a right triangle, you can assign special names to the three sides. The two sides that meet at a right angle are called the legs. The third side is called the hypotenuse and is the side opposite the right angle. The hypotenuse is always the longest side of a right triangle and is usually labeled c in the Pythagorean Theorem. Hypotenuse: The longest side of a right triangle; side opposite from the right angle. Legs: The two shorter sides of a right triangle; the sides that meet at 90 degrees. Right triangle: A triangle with a 90-degree angle. a^2 + b^2 = c^2 The Pythagorean Theorem Application 1: In a triangle with sides a, b, and c, if a2 + b2 = c2, then the triangle is a right triangle. Application 2: In a right triangle with sides a and b and with hypotenuse c, a2 + b2 = c2. Pythagorean triple: A set of positive integers that fits the rule:a2 + b2 = c2. Remember! Here is a list of some commonly used Pythagorean triples: 3 4 5 5 12 13 7 24 25 8 15 17 9 40 41 look at page 6

02.03

Translation: Slide Reflection: Flip Rotation: Spin Rigid transformation: Movement that changes the position of a figure but not its size; translations, reflections, and rotations. Congruent: Having the same size and shape; expressed with the symbol ≅. Remember! If all of the transformations in a series are rigid transformations, the pre-image and image will be congruent.

02.07

Triangle Angle Sum Property: The sum of the measures of the interior angles in a triangle is 180 degrees. The Triangle Angle Sum Property says that if you add up the angles inside a triangle they will equal 180 degrees. It doesn't matter how large or small the triangle is, the angles inside always add up to 180. Angles that are located outside the triangle are called exterior angles. Go to page 4. Rule The sum of the exterior angles of a triangle is 360 degrees. Remote interior angles: The two angles inside the triangle that do not share a vertex with the exterior angle. Go to page 5. Rule Exterior Angle Theorem: The measure of an exterior angle is equal to the sum of the two remote interior angles Congruent angles: Angles with the same measure. Rule Angle-Angle Criterion (AA): If two angles of a triangle are congruent to two angles of another triangle, then the two triangles are similar.

03.02

Using the Pythagorean Theorem in 2D The Pythagorean Theorem is also a tool that you can use to find out information in the real world. The following steps will help you tackle real-world problems: Identify important information in the problem. Draw a diagram of what you know, and label parts. Find the right triangle in your drawing, and apply Two-dimensional (2D) An object that has width and length but no thickness (depth); examples include squares, circles, and triangles. Three-dimensional (3D) An object that has width, length, and depth; examples include cubes, prisms, spheres, and cones.