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Fields Medal Prize Winners (1998)




TUTORIALS:


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Fractions
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Roots
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Powers
Rewriting Algebraic Fractions
Exponents
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Axis of Symmetry and Vertex of a Parabola

Axis of Symmetry

Parabolas are symmetric. Draw any parabola on a paper and fold it along a vertical line that goes right through the middle of the parabola. This line is called the axis of symmetry of the parabola. The two halves of the parabola match up after you folded along this line.

Axis of Symmetry

The equation of the axis of symmetry for the graph of the quadratic function y = ax 2 + bx + c , where a 0, is .

The reason why the axis of symmetry has this as its equation will be explained in a future lesson.

Example 1

Find the equation of the axis of symmetry for the graph of y = x 2.

Solution

Write this function as y = 1x 2 + 0x + 0. So, a = 1, b = 0, and c = 0.

Therefore, the equation of the axis of symmetry is x = 0, which is the y-axis.

 

Example 2

Find the equation of the axis of symmetry for the graph of y = x 2 - 2x .

Solution

Write this function as y = x 2 - 2x + 0, so in this case a = 1, b = -2, and c = 0.

The equation of the axis of symmetry is x = 1.

 

Example 3

Find the equation of the axis of symmetry for the graph of y = -2x 2 - 4x .

Solution

Write this function as y = -2x 2 - 4x + 0 so that in this case a = -2, b = -4, and c = 0.

So the equation of the axis of symmetry is x = -1.

 

Vertex of a Parabola

The axis of symmetry intersects the parabola at exactly one point. This is called the vertex of the parabola. This point is either the minimum point on the parabola (as in Examples 1 and 2) or the maximum point (as in Example 3). Graphically, the vertex is the tip of the parabola. If we know the equation of the axis of symmetry , then since the vertex lies on the axis of symmetry, we know the x-coordinate of the vertex . To find the y-coordinate, substitute the x value into the equation of the function.

 

Example 6

Find the vertex of the graph of y = x 2 - 2x .

Solution

In Example 4, we found that the axis of symmetry of this parabola is x = 1. So the vertex has an x-coordinate equal to 1. To find the y-coordinate, substitute x = 1 into the equation.

y = x 2 - 2x

y = (1) 2 - 2(1)

y = -1

So the y-coordinate of the vertex is -1. The vertex of this parabola is at (1, -1). This point is the minimum since, as the graph shows, the tip of the parabola is at the bottom.

 

Example 7

Find the vertex of the graph of y = - 2x 2 - 4 x .

Solution

In Example 5, we found that the axis of symmetry of this parabola is the line x = -1. So the vertex has an x -coordinate equal to -1. To find the y-coordinate, substitute x = -1 into the equation.

y = -2x 2 - 4x

y = -2(-1) 2 - 4(-1)

y = 2

So 2 is the y-coordinate of the vertex. The vertex of this parabola is at ( -1, 2). This point is the maximum since, as the graph shows, the tip of this parabola is at the top.

If you are given the coordinates of the vertex, how can you find the axis of symmetry of the parabola?

Since the vertex lies on the axis of symmetry, which is vertical, the x-coordinate of the vertex will tell you the equation of the axis of symmetry. For example, if the vertex of a parabola is at the point (2, 3), then the axis of symmetry has equation x = 2.