Presentation on theme: "Quadratic & Polynomial Functions"— Presentation transcript:
1Quadratic & Polynomial Functions UNIT 1Quadratic & Polynomial Functions
2Properties 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.
3Properties 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.
4Properties of Quadratics We can find the x-intercepts of a quadratic function 3 ways:Using the quadratic formula:2. By factoring3. By using the “2nd trace” button and “zero” button on our calculator
5The 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 −4𝑎𝑐When , there are 2 real solutionsWhen there is 1 real solutionWhen there are 2 NONREAL solutions
6Quadratic Transformations Transformations of Functions: how functions move right/left, up/down, stretch/shrink, and reflectParent Function: a function before it transformsToday, the parent function was always y= 𝑥 2 . This was the original graph before it moved.
7Quadratic Transformations The Vertex Form of the quadratic function is: 𝐟 𝐱 = 𝒂(𝒙−𝒉) 𝟐 +𝒌When “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.
8Quadratic Transformations The Vertex Form of the quadratic function is: 𝐟 𝐱 = 𝒂(𝒙−𝒉) 𝟐 +𝒌When 𝐚>𝟏, the graph vertically stretches.When 𝐚<𝟏, the graph shrinks vertically.When “a” is negative, the graph reflects over the x-axis.
9Quadratic Transformations The Vertex Form of the quadratic function is: 𝐟 𝐱 = 𝒂(𝒙−𝒉) 𝟐 +𝒌NOTICE: (-h, k) is the vertex of the Quadratic Function!!!
10Systems of Equations & Quadratic Inequalities Unit 2Systems of Equations & Quadratic Inequalities
11SYSTEMS OF EQUATIONSSystem of Equations: a collection of two or more equations whose solution is the point where the functions intersectThe solution will make each equation a true statement
12Solving 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.
13Solving 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.
14Quadratic 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.
15Exponential Functions Unit 3Exponential Functions
16Exponential Functions An Exponential Function is a function with a base that is a number and an exponent that is a variable.y= 2 𝑥While a linear function increases by a constant rate (slope), an exponential function increases by a constant percentage.
17Exponentials Functions The Standard Form for an Exponential Function:𝒚= 𝒂𝒃 𝒙WHERE:a = the initial valueb = the growth factor(x, y) = points on the graphGrowth Factorb = 1 ± 𝒓When b is greater than 1 = growthWhen b is less than 1 = decay
18Solving 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.
19Graphing 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.
20Rational & Radical Functions Unit 4Rational & Radical Functions
21INVERSE 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.
22COMPARE 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.
23Solving 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.
24Rational 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
27TrianglesA 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.
28Triangle Congruence Congruent: means equal or the same Congruent Triangles: two triangles are congruent when ALL of their corresponding sides and angles are congruent.
30SSS: Side Side SideIf three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent
31SAS: Side Angle SideIf 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.
32ASA: Angle Side AngleIf 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.
33AAS: Angle Angle SideIf 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.
34HL: Hypotenuse LegIf the hypotenuse and leg of one right triangle are congruent to hypotenuse and leg of another right triangle, they are congruent.
35Geometric Transformations A transformation changes the position of a shape on a coordinate plane3 main types of transformations:1. translations2. rotations3. reflectionsTranslations, rotations, and reflections preserve congruence!!!
36TranslationA type of transformation that moves a point, line, or shapeTranslations preserve shape. Translations move objects, butdo not change them.
37Transformations 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’)
38ReflectionA reflection flips a figure over a line.yx
39Special 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)
40RotationA transformation in which a figure is turned about a fixed point, called the center of rotation.
41COUNTERCLOCKWISE 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 THESHAPE BEING ROTATED!!!
42ROTATION 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!!!
43Composition of Transformations Composition of Transformations: when two or more transformations are combined to form a new transformationIn a composition, the first transformation produces an image upon which the second transformation is then performed.
44Dig 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.
45Dig 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!!!
48Pythagorean 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
49B 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.ACHypotenuse = side opposite right angleOpposite side = the side that’s farthest away and not touching the angleAdjacent side = the side that is closest to the angle
50Trigonometry Ratios KEY CONCEPT: TRIG RATIOS RATIO WORDS SYMBOLS MODELSSINECOSINETANGENTsin = oppositehypotenusesinA = 4554A3 cos = adjacenthypotenuse5cosA = 354A3tan= oppositeadjacenttanA= 4354A3
51Shortcut! 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 TangentOpposite Adjacent OppositeHypotenuse Hypotenuse Adjacent
52A 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.
53SimilarityTwo polygons are similar when their corresponding angles are congruent and the lengths of the corresponding sides are proportional by a scale factor.
54Scale Factor: if two polygons are similar, the scale factor is the ratio of the lengths of two corresponding sides
55DilationA transformation that changes the size of the image.
56In 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.
60Intro to 3D ShapesPerimeter: 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
61Cross SectionIn geometry, a cross section is the 2 dimensional shape left on the surface when you cut a 3 dimensional shape.