FreeAlgebra                             Tutorials! Home Polynomials Finding the Greatest Common Factor Factoring Trinomials Absolute Value Function A Summary of Factoring Polynomials Solving Equations with One Radical Term Adding Fractions Subtracting Fractions The FOIL Method Graphing Compound Inequalities Solving Absolute Value Inequalities Adding and Subtracting Polynomials Using Slope Solving Quadratic Equations Factoring Multiplication Properties of Exponents Completing the Square Solving Systems of Equations by using the Substitution Method Combining Like Radical Terms Elimination Using Multiplication Solving Equations Pythagoras' Theorem 1 Finding the Least Common Multiples Multiplying and Dividing in Scientific Notation Adding and Subtracting Fractions Solving Quadratic Equations Adding and Subtracting Fractions Multiplication by 111 Adding Fractions Multiplying and Dividing Rational Numbers Multiplication by 50 Solving Linear Inequalities in One Variable Simplifying Cube Roots That Contain Integers Graphing Compound Inequalities Simple Trinomials as Products of Binomials Writing Linear Equations in Slope-Intercept Form Solving Linear Equations Lines and Equations The Intercepts of a Parabola Absolute Value Function Solving Equations Solving Compound Linear Inequalities Complex Numbers Factoring the Difference of Two Squares Multiplying and Dividing Rational Expressions Adding and Subtracting Radicals Multiplying and Dividing Signed Numbers Solving Systems of Equations Factoring Out the Opposite of the GCF Multiplying Special Polynomials Properties of Exponents Scientific Notation Multiplying Rational Expressions Adding and Subtracting Rational Expressions With Unlike Denominators Multiplication by 25 Decimals to Fractions Solving Quadratic Equations by Completing the Square Quotient Rule for Exponents Simplifying Square Roots Multiplying and Dividing Rational Expressions Independent, Inconsistent, and Dependent Systems of Equations Slopes Graphing Lines in the Coordinate Plane Graphing Functions Powers of Ten Zero Power Property of Exponents The Vertex of a Parabola Rationalizing the Denominator Test for Factorability for Quadratic Trinomials Trinomial Squares Solving Two-Step Equations Solving Linear Equations Containing Fractions Multiplying by 125 Exponent Properties Multiplying Fractions Adding and Subtracting Rational Expressions With the Same Denominator Quadratic Expressions - Completing Squares Adding and Subtracting Mixed Numbers with Different Denominators Solving a Formula for a Given Variable Factoring Trinomials Multiplying and Dividing Fractions Multiplying and Dividing Complex Numbers in Polar Form Power Equations and their Graphs Solving Linear Systems of Equations by Substitution Solving Polynomial Equations by Factoring Laws of Exponents index casa mÃ­o Systems of Linear Equations Properties of Rational Exponents Power of a Product and Power of a Quotient Factoring Differences of Perfect Squares Dividing Fractions Factoring a Polynomial by Finding the GCF Graphing Linear Equations Steps in Factoring Multiplication Property of Exponents Solving Systems of Linear Equations in Three Variables Solving Exponential Equations Finding the GCF of a Set of Monomials

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# The Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction.

 â€¢ If the parabola opens up, the vertex is the lowest point on the graph. This is called the minimum. â€¢ If the parabola opens down, the vertex is the highest point on the graph. This is called the maximum. A vertical line drawn through the vertex is called the axis of symmetry.

If you fold the graph along the axis of symmetry one side of the graph will lie on top of the other.

We can use the following procedure to find the vertex of a parabola.

Procedure â€” To Find the Vertex of a Parabola y = Ax2 + Bx + C

Step 1 Find the x-coordinate. It is given by the formula .

Step 2 Find the y-coordinate. It is found by substituting the x-coordinate into y = Ax2 + Bx + C and then simplifying.

Note:

It can be shown that the y-coordinate of the vertex of a parabola is given by Example 1

Find the vertex of the parabola: y = x2 - 8x + 12

Solution

Here, A = 1, B = -8, and C = 12.

 Step 1 Find the x-coordinate. x Substitute 1 for A and -8 for B. x Simplify. x = 4 Step 2 Find the y-coordinate. Substitute 4 for x. Simplify. yy y y = x2 - 8x + 12= (4)2 - 8(4) + 12 = 16 - 32 + 12 = -4
So, the vertex of y = x2 - 8x + 12 is (4, -4).

Note:

When a parabola has two x-intercepts, the x-coordinate of the vertex always lies halfway between the x-intercepts. Here the x-intercepts are x =  2 and x = 6. The x-coordinate of the vertex, 4, is halfway between 2 and 6.

Example 2

Find the vertex of the parabola: f(x) = -2x2 - 12x - 18

Solution

Here, A = -2, B = -12, and C = -18.

 Step 1 Find the x-coordinate. x Substitute -2 for A and -12 for B. x Simplify. x -3 Step 2 Find the y-coordinate. Substitute -3 for x. Simplify. f(x)y y y = -2x2 - 12x - 18 = -2(-3)2 - 12(-3) - 18 = -2(9) + 36 - 18 = 0
So, the vertex of = -2x2 - 12x - 18  is (-3, 0).

Note:

When a parabola has one x-intercept, the x-intercept is the x-coordinate of the vertex. Here the x-intercept is x = -3 and the x-coordinate of the vertex is x = -3.