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Cool Math: Finding Roots and Graphs of Quadratics

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Cool Math: Finding Roots and Graphs of Quadratics
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Morven

@morvenwebb_uaud

·

7 Followers

Follow

A comprehensive guide to solving quadratic equations and understanding their graphical representations, focusing on transformations, roots, and the discriminant method.

  • The guide explains how quadratic functions create parabolas with either U-shapes (minimum turning points) or n-shapes (maximum turning points)
  • Covers methods for finding roots of a quadratic equation including factoring, graphing, and the quadratic formula
  • Details transformations of quadratic graphs and how different terms affect the shape and position
  • Explains the discriminant and its role in determining the nature of roots
  • Provides step-by-step instructions for sketching quadratic graphs and finding key points

14/02/2023

170

VA min T PARABOLA Anything with a "t". has a "u-shape" with a min turning point. The Quadratic Function f(x) = ax²+bx+c min Tr Anything with

View

Solving Quadratic Equations

This section explores the three main methods for solving quadratic equations: graphical method, factoring, and the quadratic formula.

Definition: The roots of a quadratic equation are the x-values where the parabola intersects the x-axis.

Example: For x² - 2x - 8 = 0, factoring gives (x+2)(x-4) = 0, leading to solutions x = -2 or x = 4.

Highlight: The quadratic formula x = (-b ± √(b² - 4ac))/2a can be used when factoring is difficult.

VA min T PARABOLA Anything with a "t". has a "u-shape" with a min turning point. The Quadratic Function f(x) = ax²+bx+c min Tr Anything with

View

The Discriminant and Nature of Roots

This section details how the discriminant determines the nature and number of roots in a quadratic equation.

Definition: The discriminant is the expression b² - 4ac under the square root in the quadratic formula.

Highlight: The discriminant determines three possible scenarios:

  • b² - 4ac > 0: Two real and distinct roots
  • b² - 4ac = 0: One repeated real root
  • b² - 4ac < 0: No real roots

Example: When b² - 4ac = 0, the parabola touches the x-axis at exactly one point, indicating a repeated root.

VA min T PARABOLA Anything with a "t". has a "u-shape" with a min turning point. The Quadratic Function f(x) = ax²+bx+c min Tr Anything with

View

Understanding Basic Quadratic Functions

This section introduces the fundamental concepts of quadratic functions and their graphical representations. The quadratic function f(x) = ax² + bx + c forms the basis of all quadratic equations.

Definition: A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a ≠ 0.

Highlight: The coefficient 'a' determines whether the parabola opens upward (a > 0) creating a U-shape, or downward (a < 0) creating an n-shape.

Example: For the function y = kx², when given the point (2,12), we can find k by substituting: 12 = k(2²), therefore k = 3, making the equation y = 3x².

Vocabulary: The vertical line of symmetry is a line that divides the parabola into two identical halves.

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Lena, iOS user

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Subjects

/

Maths

/

Algebra

/

Solving quadratic equations

Cool Math: Finding Roots and Graphs of Quadratics

user profile picture

Morven

@morvenwebb_uaud

·

7 Followers

Follow

A comprehensive guide to solving quadratic equations and understanding their graphical representations, focusing on transformations, roots, and the discriminant method.

  • The guide explains how quadratic functions create parabolas with either U-shapes (minimum turning points) or n-shapes (maximum turning points)
  • Covers methods for finding roots of a quadratic equation including factoring, graphing, and the quadratic formula
  • Details transformations of quadratic graphs and how different terms affect the shape and position
  • Explains the discriminant and its role in determining the nature of roots
  • Provides step-by-step instructions for sketching quadratic graphs and finding key points

14/02/2023

170

 

S4

 

Maths

6

VA min T PARABOLA Anything with a "t". has a "u-shape" with a min turning point. The Quadratic Function f(x) = ax²+bx+c min Tr Anything with

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Solving Quadratic Equations

This section explores the three main methods for solving quadratic equations: graphical method, factoring, and the quadratic formula.

Definition: The roots of a quadratic equation are the x-values where the parabola intersects the x-axis.

Example: For x² - 2x - 8 = 0, factoring gives (x+2)(x-4) = 0, leading to solutions x = -2 or x = 4.

Highlight: The quadratic formula x = (-b ± √(b² - 4ac))/2a can be used when factoring is difficult.

VA min T PARABOLA Anything with a "t". has a "u-shape" with a min turning point. The Quadratic Function f(x) = ax²+bx+c min Tr Anything with

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

The Discriminant and Nature of Roots

This section details how the discriminant determines the nature and number of roots in a quadratic equation.

Definition: The discriminant is the expression b² - 4ac under the square root in the quadratic formula.

Highlight: The discriminant determines three possible scenarios:

  • b² - 4ac > 0: Two real and distinct roots
  • b² - 4ac = 0: One repeated real root
  • b² - 4ac < 0: No real roots

Example: When b² - 4ac = 0, the parabola touches the x-axis at exactly one point, indicating a repeated root.

VA min T PARABOLA Anything with a "t". has a "u-shape" with a min turning point. The Quadratic Function f(x) = ax²+bx+c min Tr Anything with

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Understanding Basic Quadratic Functions

This section introduces the fundamental concepts of quadratic functions and their graphical representations. The quadratic function f(x) = ax² + bx + c forms the basis of all quadratic equations.

Definition: A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a ≠ 0.

Highlight: The coefficient 'a' determines whether the parabola opens upward (a > 0) creating a U-shape, or downward (a < 0) creating an n-shape.

Example: For the function y = kx², when given the point (2,12), we can find k by substituting: 12 = k(2²), therefore k = 3, making the equation y = 3x².

Vocabulary: The vertical line of symmetry is a line that divides the parabola into two identical halves.

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Can't find what you're looking for? Explore other subjects.

Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the #1 education app in five European countries

4.9+

Average app rating

15 M

Pupils love Knowunity

#1

In education app charts in 12 countries

950 K+

Students have uploaded notes

Still not convinced? See what other students are saying...

iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

Philip, iOS User

The app is very simple and well designed. So far I have always found everything I was looking for :D

Lena, iOS user

I love this app ❤️ I actually use it every time I study.