Quadratic & Polynomial Functions UNIT 1 Quadratic & Polynomial Functions

Properties of Quadratics An Extreme is a maximum or minimum of a quadratic function. We also know this as the vertex. We can find the vertex of a quadratic function using the formula: ( −𝑏 2𝑎 , 𝑓( −𝑏 2𝑎 )) On the y-axis, the value of x is “0”. SO, we can find the y-intercept of a quadratic function by plugging in “0” for x.

Properties of Quadratics The Axis of Symmetry is the line that cuts a quadratic function directly in half. The Axis of Symmetry is the vertical line that runs through the “x” coordinate of the vertex.

Properties of Quadratics We can find the x-intercepts of a quadratic function 3 ways: Using the quadratic formula: 2. By factoring 3. By using the “2nd trace” button and “zero” button on our calculator

The Quadratic Formula NOTES The DISCRIMINANT of the quadratic formula tells us all about the types of solutions the function has. The DISCRIMINANT of the quadratic formula is: 𝑏 2 −4𝑎𝑐 When , there are 2 real solutions When there is 1 real solution When there are 2 NONREAL solutions

Quadratic Transformations Transformations of Functions: how functions move right/left, up/down, stretch/shrink, and reflect Parent Function: a function before it transforms Today, the parent function was always y= 𝑥 2 . This was the original graph before it moved.

Quadratic Transformations The Vertex Form of the quadratic function is: 𝐟 𝐱 = 𝒂(𝒙−𝒉) 𝟐 +𝒌 When “k” is positive, the graph translates up. When “k” is negative, the graph translates down. When “h” is positive, the graph translates left. When “h” is negative, the graph translates right.

Quadratic Transformations The Vertex Form of the quadratic function is: 𝐟 𝐱 = 𝒂(𝒙−𝒉) 𝟐 +𝒌 When 𝐚>𝟏, the graph vertically stretches. When 𝐚<𝟏, the graph shrinks vertically. When “a” is negative, the graph reflects over the x-axis.

Quadratic Transformations The Vertex Form of the quadratic function is: 𝐟 𝐱 = 𝒂(𝒙−𝒉) 𝟐 +𝒌 NOTICE: (-h, k) is the vertex of the Quadratic Function!!!

Systems of Equations & Quadratic Inequalities Unit 2 Systems of Equations & Quadratic Inequalities

SYSTEMS OF EQUATIONS System of Equations: a collection of two or more equations whose solution is the point where the functions intersect The solution will make each equation a true statement

Solving Systems in a Calculator 1. Put both equations into the calculator under the “y=“ button. 2. Hit graph to see if the functions intersect. Remember, an intersection means that the system has a solution. 3. Hit “2nd”, then “Trace”. Hit the “Intersect” button. Get close to an intersection and hit “Enter” three times. This will give you a solution, as long as the graphs cross.

Solving Systems Algebraically Isolate one equation for “y”. Plug your isolated equation into the second equation for “y”. Solve for “x”. Plug your “x” solutions back into one of the equations to find the “y” values. Remember, the solution to a system of equations is one or more coordinate points, so you need an “x” and a “y” for each solution.

Quadratic Inequalities Move all terms to one side of the inequality. Determine if your solution requires the part of the graph over or under the x-axis. This will aid you in writing your solution. Find the x-intercepts of the quadratic in your calculator. Remember, hit”2nd”, then “Trace”. Use the “Zero” button. Be careful to figure out where the left and right bounds are. Write your solution in inequality notation.

Exponential Functions Unit 3 Exponential Functions

Exponential Functions An Exponential Function is a function with a base that is a number and an exponent that is a variable. y= 2 𝑥 While a linear function increases by a constant rate (slope), an exponential function increases by a constant percentage.

Exponentials Functions The Standard Form for an Exponential Function: 𝒚= 𝒂𝒃 𝒙 WHERE: a = the initial value b = the growth factor (x, y) = points on the graph Growth Factor b = 1 ± 𝒓 When b is greater than 1 = growth When b is less than 1 = decay

Solving Exponential Equations To solve an exponential equation, remember to use reverse PEMDAS. The inverse of an exponential is a logarithm. SO, to solve exponential equation, we need to LOG both sides of the equation.

Graphing Exponentials! We can find the y-intercept by plugging in “0” for X and solving for Y. We can find x-intercepts by plugging in a “0” for Y and solving for X. End behavior describes the Y value that a graph approaches as X gets really big and positive and as X gets really big and negative.

Rational & Radical Functions Unit 4 Rational & Radical Functions

INVERSE VARIATION!!! “k” is a number called the constant of variation. In these equations, as “x” gets bigger, “y” gets smaller. LIKEWISE, when “y” gets bigger, “x” gets smaller.

COMPARE TO DIRECT VARIATION In these equations, as “x” gets bigger, “y” gets bigger. LIKEWISE, when “x” gets smaller, “y” gets smaller too. “k” is STILL called the constant of variation. The equation for direct variation is y = kx.

Solving Radical Functions Use reverse PEMDAS. When we solve a radical function, remember that the inverse or opposite of a square root is a square! Always check your answers. Answers that do not work are called extraneous solutions.

Rational Exponents Rational Exponent: a fractional exponent We can re-write any radical expression as a rational exponent. We can also re-write any rational exponent as a radical expression

Properties of Rational Exponents

Unit 5 Congruence

Triangles A triangle is a closed three-sided shape whose interior angles sum to 180 degrees. A straight line has 180 degrees. When two lines cross, the opposite angles are congruent and are called vertical angles.

Triangle Congruence Congruent: means equal or the same Congruent Triangles: two triangles are congruent when ALL of their corresponding sides and angles are congruent.

Triangle Congruence

SSS: Side Side Side If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent

SAS: Side Angle Side If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

ASA: Angle Side Angle If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.

AAS: Angle Angle Side If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of another triangle, then the two triangles are congruent.

HL: Hypotenuse Leg If the hypotenuse and leg of one right triangle are congruent to hypotenuse and leg of another right triangle, they are congruent.

Geometric Transformations A transformation changes the position of a shape on a coordinate plane 3 main types of transformations: 1. translations 2. rotations 3. reflections Translations, rotations, and reflections preserve congruence!!!

Translation A type of transformation that moves a point, line, or shape Translations preserve shape. Translations move objects, but do not change them.

Transformations Pre-image: the original image (ex: ∆ABC) Image: The image after the transformation. It is usually represented with the same letters as the pre-image but we add an apostrophe. (ex: ∆A’B’C’)

Reflection A reflection flips a figure over a line. y x

Special Reflections Rules Reflection across x – axis: Reflection across y – axis: Reflection over origin: Reflection over line y = x: APPLY THESE RULES TO ALL POINTS ON THE SHAPE BEING REFLECTED!!! (x, y )  (x, -y) (x, y )  (-x, y) (x, y )  (-x, -y) (x, y )  (y, x)

Rotation A transformation in which a figure is turned about a fixed point, called the center of rotation.

COUNTERCLOCKWISE ROTATION RULES 90 degrees counterclockwise around origin:(x, y)  (-y, x) 180 degrees around the origin: (x, y)  (-x, -y) 270 degrees counterclockwise around origin: (x, y) (y, -x) APPLY THESE RULES TO ALL POINTS ON THE SHAPE BEING ROTATED!!!

ROTATION RULES CLOCKWISE 90 degrees clockwise around origin:(x, y) (y, - x) 180 degrees around the origin: (x, y)  (-x, -y) 270 degrees clockwise around origin: (-y, x) APPLY THESE RULES TO ALL POINTS ON THE SHAPE BEING ROTATED!!!

Composition of Transformations Composition of Transformations: when two or more transformations are combined to form a new transformation In a composition, the first transformation produces an image upon which the second transformation is then performed. 

Dig Deep Into Transformations! WHAT WE ALREADY KNOW: Transformations are essentially functions that take points in the coordinate plane as inputs and give other points as outputs. Two figures are congruent if they have the same size (sides & angles) and shape.

Dig Deep Into Transformations! WHAT WE NEED TO UNDERSTAND: MORE SPECIFICALLY, two shapes are congruent if and only if one can be obtained from the other by rigid motions, meaning a sequence of reflections, translations, and/or rotations can carry one shape on top of another. Well what is a rigid motion??? A motion that preserves shape congruence!!!

Distance Formula & Midpoint Formula

Right Triangle Trigonometry & Similarity Unit 6 Right Triangle Trigonometry & Similarity

Pythagorean Theorem: is a mathematical relationship between the three sides of a RIGHT triangle Where: a = shortest side (leg) b = 2nd shortest side (leg) c = longest side (hypotenuse) 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐

B A C Hypotenuse = side opposite right angle We’re starting with identification. Can someone first remind me what a hypotenuse is, and what legs are? Hypotenuse is across from the right angle, and the other sides are legs. They only apply to right triangles. Draw massive triangle: if I am talking about angle A right now, where is the opposite side? We remember this from doing triangle inequalities; it’s the side that’s the farthest away and not touching. Now where is the adjacent side? It’s the other leg, the one that’s closer. Where’s the hypotenuse? Good. Let’s switch our focus to angle C. Where’s the opposite side? Where’s the adjacent side? Where’s the hypotenuse? Why didn’t it change when my perspective changed? Because the hypotenuse is always the one that’s across from the right angle. A C Hypotenuse = side opposite right angle Opposite side = the side that’s farthest away and not touching the angle Adjacent side = the side that is closest to the angle

Trigonometry Ratios KEY CONCEPT: TRIG RATIOS RATIO WORDS SYMBOLS MODELS SINE   COSINE TANGENT    sin = opposite hypotenuse sinA = 4 5 5 4 A 3  cos = adjacent hypotenuse 5 cosA = 3 5 4 A 3 tan= opposite adjacent tanA= 4 3 5 4 A 3

Shortcut! SOH CAH TOA Sin Cos Tangent Opposite Adjacent Opposite So the sine of an angle, as you wrote into your notecards, can be calculated by taking the ratio of the opposite side over the hypotenuse. In this diagram, we could write that the sine of angle A is opposite/hypotenuse. In notation, this looks like sinA= (fraction). The cosine is adjacent/hypotenuse, and the tangent is the opposite/adjacent. How are we going to remember this? That’s where SOHCAHTOA comes in. We like shortcuts; we spent all of yesterday learning shortcuts, and SOHCAHTOA is going to be our trigonometry wonder. This is what it stands for.   Sin Cos Tangent Opposite Adjacent Opposite Hypotenuse Hypotenuse Adjacent

A Quick Trg NOTE!!! When you are using SOH CAH TOA, law of sines, or law of cosines, your calculator needs to be in DEGREE mode. When you graph sine, cosine, and tangent, your calculator needs to be in RADIAN mode.

Similarity Two polygons are similar when their corresponding angles are congruent and the lengths of the corresponding sides are proportional by a scale factor.

Scale Factor: if two polygons are similar, the scale factor is the ratio of the lengths of two corresponding sides

Dilation A transformation that changes the size of the image.

In Unit 5, translations, reflections, and rotations did not change the shapes. The pre-image and image were CONGRUENT. Dilations change the size of the shapes proportionally. So, dilations create SIMILAR shapes.

3. The mid-segment of a triangle divides a triangle creates two similar triangles.

Surface Area, Volume, & Cross Sections Unit 7 Surface Area, Volume, & Cross Sections

Intro to 3D Shapes Perimeter: the total distance around a shape Area: the 2 dimensional or surface space that a shape takes up Surface Area: the sum of the area of all of the surfaces of a 3 dimensional shape Volume: the amount of 3 dimensional space that an object takes up

Cross Section In geometry, a cross section is the 2 dimensional shape left on the surface when you cut a 3 dimensional shape.

FORMULAS Part I

FORMULAS Part II