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Quadratics, The Root Function & Inverses

Quadratics, The Root Function & Inverses. GROUP 1 – Unit A Shehzaad, Henil , Nirojan , Umesh. Important Terminology. Expression : It is a mixture of numbers and variables that can be calculated. Ex: 4x²- 2x +6

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Quadratics, The Root Function & Inverses

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  1. Quadratics, The Root Function & Inverses GROUP 1– Unit A Shehzaad, Henil, Nirojan, Umesh

  2. Important Terminology Expression: It is a mixture of numbers and variables that can be calculated. Ex: 4x²- 2x +6 Equation: A statement where two expressions are supposed to equal each other. It relies on the value of the variables.Ex: 4x²- 2x +6= 0 Function: It is a relationship or expression involving one or more variables. For every (x) value there is only one unique, possible (y) value. Ex: y= 4x² -2x +6

  3. Quadratic Equation A quadratic equation is an equation that contains x² as its highest degree or power. Base function is y= x²

  4. Factoring & Solving Quadratic Equations Solving Quadratic Equations Factoring Quadratic Formula The equations must be in “standard form” in order to use any of the methods listed above Standard form: ax²+bx+c =0

  5. How To Factor

  6. Factoring Problems Fully factor each equation: 1) 5a2– 15a + 10 2) 3)

  7. Discriminant Discriminant = b²-4acIt is a way to determine the number of possible solutions (or roots) of an equation. If b2– 4ac > 0  2 Distinct Real Roots If b2– 4ac = 0  2 Equal Real Roots If b2– 4ac< 0  NO Real Roots

  8. Determine Number of Solutions A) B)

  9. Quadratic Formula After using the discriminant to find number of solutions, one must proceed to using the quadratic formula. Solve each equation using the quadratic formula: 1) 2)

  10. Functions, Domain & Range Function: For every x value, there is only one y value. Vertical Line Test: a test which determines whether or not the equation is a function. Domain: It is the set of all first elements in a relation. The (x) values. For example: Range: The set of all second elements in a relation. The (y) values. For example: Function Notation y=f(x) For example: y=-2x+3 -> f(x) = -2x+3

  11. Domain and Range What is the domain and range for the following functions: 1) 2)

  12. Graphing Quadratic Functions Quadratic functions’ graph will always be a parabola. Vertex Form: y= a(x-h)² +k a> 0= minimum value of k when x is equal to h. a<0= maximum value of k when x is equal to h. *COMPLETE THE SQUARE*

  13. Complete the Square Completing the square is simply finding the missing term in order to have a perfect square. Given a quadratic function, to put it in vertex form, follow the following steps: 1. Factor the coefficient of x2 from the terms in x2and x. Don’t do anything with the constant. 2. Complete the square on the terms in x2 and x as shown above. Since adding the missing term would change the given expression, we add it and subtract it at the same time. 3. The first three terms will be a perfect square. 4. Distribute the coefficient factored in step 1.

  14. Complete the Square Example: y = x2 − 2x + 3 = x2 − 2x + 1 − 1 + 3 = x2 − 2x+ 1 − 1+ 3 = ( x − 1)2 + 2

  15. Graphing Quadratic Functions Mapping Notation: Equation must be in vertex form. Create a new set of points to get the key points of the function. Graphing by Factoring: Equation must be in standard form. You must factor, in order to find the x and y intercept as well as the roots. Step Pattern:Equation must be in vertex form. Use the vertex and transformations to find points.

  16. Inverses What is an inverse? An inverse is a relationship reflected through y=x An inverse has the same points in a graph but the x and y points are switched. Example (-2,4) would become (4,-2) Inverse Functions An inverse is not always a function. Inverse of a linear function will always be a function. Inverse of a quadratic function will never be a function. It is written as f¯¹(x).

  17. Inverse Problems Find the inverse of each of the following: 1) 2)

  18. Transformations Vertical Translations:y = f (x) + k • k > 0, graph shifts upk units. • k < 0, graph shifts downk units. Horizontal Translations:y = f (x – h) • h > 0, graph shifts righth units. • h < 0, graph shifts lefth units.

  19. Transformations Reflection on the x-axis y=-f(x)Multiply the y-values by -1 Reflection on the y-axis y=f(-x) Multiply the x-values by -1

  20. Transformations Vertical Stretches y=af(x) a > 1 = compression | 0>a>1 = stretch Horizontal Stretches y= f [b(x)] Use reciprocal to find value of x (1/b) b > 1 = compression | 0>b>1 = stretch

  21. Horizontal Transformations y= a f (b(x-h))+k • (a)= vertical stretch • (b)= horizontal stretch • (h)= horizontal translation • (k)= vertical translation

  22. Root Function It is written as f(x) = x We can never square root a negative number, so therefore, x >0

  23. Transformation Problems Graph each transformation: 1) 2)

  24. Transformation Problems Create an equation from each graph: 1)

  25. Transformation Problems 2)

  26. Hope Y’all Do Well On The Exam!

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