Writing Quadratic Equations In Vertex Form & Standard Form Given 3 Points

The Organic Chemistry Tutor

17 Oct 201642:23

EducationalLearning

32 Likes 10 Comments

TLDRThis educational video script offers a detailed guide on how to write quadratic equations in both standard and vertex form. It begins by demonstrating the process of finding the coefficients a, b, and c for a quadratic equation given three points, using a system of equations and elimination technique. The script then explains how to convert the standard form equation to vertex form by completing the square. Additionally, it covers alternative methods for deriving the quadratic equation when given specific points, including the vertex and intercepts. The video ensures understanding by verifying the derived equations with the given points and converting between forms, providing a comprehensive learning experience for students.

Takeaways

📚 The video focuses on writing quadratic equations in standard and vertex form.
📝 To find the equation in standard form, plug in given points to create a system of equations.
🔍 Use elimination or substitution to solve for the coefficients a, b, and c in the standard form equation.
📈 For converting to vertex form, complete the square to rearrange the standard form equation.
🔢 The process involves factoring and simplifying the equation to reveal the vertex form.
📐 The vertex form of a quadratic equation is given by \( y = a(x - h)^2 + k \), where (h, k) is the vertex.
📉 If given the vertex and another point, use these to derive the vertex form equation by solving for a.
🔄 Convert the vertex form back to standard form by expanding and simplifying the equation.
📊 Check the derived equations by substituting the given points to ensure they satisfy the equation.
✅ The script provides step-by-step methods to solve for quadratic equations using various techniques.
📈 The video also explains how to use x-intercepts and additional points to derive the quadratic equation.

Q & A

What is the standard form of a quadratic equation?
-The standard form of a quadratic equation is y = ax^2 + bx + c, where a, b, and c are constants, and x and y are variables.
How do you set up a system of equations to find the coefficients a, b, and c of a quadratic equation given three points?
-You substitute the x and y values of each point into the standard form equation to create a system of three equations. For example, with points (1, 0), (2, 1), and (3, 6), you get the equations 0 = a + b + c, 1 = 4a + 2b + c, and 6 = 9a + 3b + c.
What method can be used to solve the system of equations for a, b, and c?
-One can use either substitution or elimination methods to solve the system of equations. The script demonstrates the elimination method.
How do you use the elimination method to solve for a, b, and c in the context of the given quadratic equation?
-You manipulate the equations to cancel out one variable at a time. For instance, you can multiply the first equation by 1 to get -C and add it to the second equation to eliminate C, then use either the first or second equation with the third to eliminate C again and solve for a and b.
What is vertex form of a quadratic equation and how is it different from standard form?
-The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. It differs from the standard form as it explicitly shows the vertex coordinates and the direction in which the parabola opens.
How do you convert a quadratic equation from standard form to vertex form?
-To convert from standard form to vertex form, you complete the square for the x terms in the equation, then adjust the constant term to maintain equality.
What is the vertex of a parabola given by the vertex form y = a(x - h)^2 + k?
-The vertex of the parabola is the point (h, k), where h is the x-coordinate and k is the y-coordinate of the vertex as indicated in the vertex form of the equation.
How can you find the equation of a quadratic in vertex form if you know the vertex and one other point on the parabola?
-You start with the vertex form y = a(x - h)^2 + k, substitute the vertex coordinates (h, k) into it, and then use the other point to solve for the value of a.
What if you're given two x-intercepts and another point on a parabola, how can you write the equation in standard form?
-You can use the x-intercepts to write the equation in factored form as y = a(x - x1)(x - x2), and then use the third point to solve for a. Alternatively, you can set up a system of equations using the standard form and solve for a, b, and c.
How do you check if the quadratic equation you found is correct?
-You can check the correctness of the quadratic equation by substituting the x and y values of the given points into the equation and verifying that both sides of the equation are equal.

Outlines

00:00

📚 Writing Quadratic Equations in Standard and Vertex Form

This paragraph explains the process of writing quadratic equations given three points. It starts by substituting the points into the general quadratic equation (y = ax^2 + bx + c) to create a system of equations. The method of elimination is then used to solve for the coefficients a, b, and c. The paragraph demonstrates solving for the coefficients step by step, including the correction of an error made during the process. The final equation in standard form is confirmed by plugging the points back into the equation.

05:01

🔍 Converting Standard Form to Vertex Form

The paragraph introduces how to convert a quadratic equation from standard form to vertex form using the method of completing the square. It details the steps of factoring out the coefficient of x^2, adding the square of half the coefficient of x, and adjusting the constant term to keep the equation balanced. The process is illustrated with an example, and the resulting vertex form is verified by checking it against given points.

10:01

📈 Using Points to Write Equations in Standard and Vertex Form

This section discusses using three points, including the vertex, to write the equation of a parabola in both standard and vertex forms. It begins by writing the equations based on the points and solving for the coefficients a, b, and c. After finding the standard form equation, the paragraph shows the process of converting it to vertex form by completing the square and adjusting terms. The correctness of the vertex form equation is confirmed by substituting the given points.

15:02

📝 Writing Equations from X-Intercepts and a Point

The paragraph explains how to write the equation of a parabola when given two x-intercepts and an additional point. It presents two methods: the initial method using the general quadratic equation and a new method using the x-intercepts directly. The process involves creating equations from the given points, solving for the coefficients, and then using the third point to find the value of 'a'. The resulting equations in both factored form and standard form are demonstrated with examples.

20:02

🔧 Converting Factored Form to Standard Form and Vertex Form

This section describes the process of converting a parabola's equation from factored form to standard and vertex forms. It starts by using the x-intercepts to write the equation in factored form and then uses a given point to solve for the coefficient 'a'. The paragraph then demonstrates converting the factored form to standard form by expanding and combining like terms. Finally, it shows the conversion to vertex form by completing the square and adjusting the equation accordingly.

25:06

📉 Writing Equations from Vertex and a Point

The paragraph focuses on writing the equation of a parabola in vertex form when given the vertex and another point on the graph. It begins with the vertex form equation and uses the given point to solve for the coefficient 'a'. After finding 'a', the vertex form equation is fully determined. The paragraph then explains how to convert this vertex form to standard form by expanding the squared term and combining like terms. The resulting standard form equation is checked against the given points for accuracy.

30:15

📌 Writing Equations from X-Intercepts and Additional Points

This section explains how to write the equation of a parabola in standard form when given two x-intercepts and another point. It presents a method involving creating a system of equations from the given points and solving for the coefficients a, b, and c. The paragraph demonstrates solving the system using elimination and substitution techniques, finding the values of the coefficients, and then writing the final equation in standard form. The correctness of the equation is verified by checking it against the given points.

35:16

📐 Converting Standard Form to Vertex Form Using Completing the Square

The paragraph details the process of converting a quadratic equation from standard form to vertex form using the completing the square method. It involves factoring out the coefficient of x^2, determining the missing term, and adjusting the equation to maintain equality. The process is illustrated with an example, and the final vertex form equation is simplified and checked for correctness by substituting the given points.

40:20

📉 Writing Equations from X-Intercepts and a Given Point

This section explains how to write the equation of a parabola in both factored and standard forms when given the x-intercepts and another point. It starts by writing the equation in factored form using the x-intercepts and then uses the additional point to solve for the coefficient 'a'. The paragraph then demonstrates converting the factored form to standard form by expanding and combining like terms. The resulting standard form equation is verified by checking it against the given points.

📈 Converting Standard Form to Vertex Form by Completing the Square

The final paragraph discusses converting a quadratic equation from standard form to vertex form using the completing the square method. It explains factoring out the coefficient of x^2, adding the square of half the coefficient of x, and adjusting the constant term. The paragraph illustrates the process with an example, showing the steps to simplify the equation and convert it to vertex form. The final vertex form equation is presented, completing the explanation of the conversion process.

Mindmap

Keywords

💡Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form of y = ax^2 + bx + c, where a, b, and c are constants, and x and y are variables. In the video, the main theme revolves around writing and converting quadratic equations, making it a central concept. The script discusses finding the coefficients a, b, and c when given three points on the graph of the equation.

💡Standard Form

The standard form of a quadratic equation is y = ax^2 + bx + c, where the highest power of x is squared and the terms are ordered from the highest power to the constant. The video script provides a method to find the values of a, b, and c by using three given points, illustrating the process of setting up and solving a system of equations to arrive at the standard form.

💡Vertex Form

Vertex form is another representation of a quadratic equation, written as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. The script explains how to convert a quadratic equation from standard form to vertex form by completing the square, a technique used to make the equation easier to analyze and graph.

💡System of Equations

A system of equations refers to a collection of two or more equations that need to be solved simultaneously. In the context of the video, the script describes using a system of equations to find the coefficients a, b, and c of a quadratic equation by substituting the x and y values of three given points into the standard form equation.

💡Elimination Method

The elimination method is a technique used to solve a system of equations by adding or subtracting the equations to eliminate one variable and solve for the other. The script demonstrates using the elimination method to find the values of a and b by strategically combining the equations derived from the given points.

💡Completing the Square

Completing the square is a process used to rewrite a quadratic equation in vertex form. It involves manipulating the equation to form a perfect square trinomial, which makes the equation easier to graph and analyze. The video script shows step-by-step how to complete the square to convert the standard form to vertex form.

💡Vertex

The vertex of a parabola is the point where the parabola turns; it is the maximum or minimum point on the graph. In the video, the vertex form of a quadratic equation is discussed, and the vertex (h, k) is used to describe the location of this turning point in the parabola's graph.

💡X-intercepts

X-intercepts, also known as roots or zeros, are the points where the graph of a quadratic equation intersects the x-axis. The script mentions using x-intercepts to write the quadratic equation in factored form, which is a useful method when the x-intercepts are known.

💡Factored Form

Factored form is a way to express a quadratic equation when it is factored into the product of two binomials, such as y = a(x - r)(x - s), where r and s are the x-intercepts. The video script explains how to use the x-intercepts to write the equation in factored form and then convert it to standard form.

💡Coefficients

Coefficients are the numerical factors in a term of an algebraic expression. In the context of a quadratic equation, the coefficients a, b, and c multiply the terms x^2, x, and the constant, respectively. The video script focuses on determining these coefficients based on given points to form the quadratic equation.

Highlights

Introduction to writing quadratic equations in standard and vertex form.

Demonstration of setting up a system of equations using given points (1,0), (2,1), and (3,6).

Explanation of plugging in points to form equations for a, b, and c in the standard form equation y = ax^2 + bx + c.

Use of elimination method to solve for variables a, b, and c.

Multiplication of equations to facilitate the cancellation of variables.

Derivation of equations 1 = 3a + b and 6 = 8a + 2b to solve for a and b.

Correction of an error in writing the equation and the importance of accuracy in mathematical processes.

Solving for the variable 'a' using the method of elimination and finding a = 2.

Finding the value of 'b' by substituting the value of 'a' into the derived equation.

Determining the value of 'c' using the initial equation and the found values of 'a' and 'b'.

Writing the final quadratic equation in standard form y = 2x^2 - 5x + 3.

Verification of the derived equation using the original three points.

Introduction to converting the standard form equation to vertex form by completing the square.

Explanation of the process to convert y = 2x^2 - 5x + 3 to vertex form.

Finding the vertex form equation and its corresponding vertex coordinates.

Verification of the vertex form equation using the provided points.

Demonstration of writing equations in vertex form given the vertex and another point on the graph.

Conversion of vertex form to standard form for a parabola opening upward with vertex (3,4) and point (4,6).

Method of using x-intercepts to write the equation in factored form and then converting to standard form.

Solving for the coefficient 'a' using a third point when given x-intercepts.

Final conversion of the factored form to standard form and verification with given points.

Illustration of converting standard form to vertex form by completing the square for a different set of points.

Summary of the process to convert a standard form equation to vertex form and verify with given points.

Transcripts

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Related Tags

Quadratic Equations Standard Form Vertex Form Mathematics Algebra Educational Solving Systems Equation Techniques Complete the Square Graphing Parabolas