Algebra Tutorials!

 Wednesday 18th of January

 Home Solving Linear Equations Systems of Linear Equations Solving Linear Equations Graphically Algebra Expressions Evaluating Expressions and Solving Equations Fraction rules Factoring Quadratic Trinomials Multiplying and Dividing Fractions Dividing Decimals by Whole Numbers Adding and Subtracting Radicals Subtracting Fractions Factoring Polynomials by Grouping Slopes of Perpendicular Lines Linear Equations Roots - Radicals 1 Graph of a Line Sum of the Roots of a Quadratic Writing Linear Equations Using Slope and Point Factoring Trinomials with Leading Coefficient 1 Writing Linear Equations Using Slope and Point Simplifying Expressions with Negative Exponents Solving Equations 3 Solving Quadratic Equations Parent and Family Graphs Collecting Like Terms nth Roots Power of a Quotient Property of Exponents Adding and Subtracting Fractions Percents Solving Linear Systems of Equations by Elimination The Quadratic Formula Fractions and Mixed Numbers Solving Rational Equations Multiplying Special Binomials Rounding Numbers Factoring by Grouping Polar Form of a Complex Number Solving Quadratic Equations Simplifying Complex Fractions Algebra Common Logs Operations on Signed Numbers Multiplying Fractions in General Dividing Polynomials Polynomials Higher Degrees and Variable Exponents Solving Quadratic Inequalities with a Sign Graph Writing a Rational Expression in Lowest Terms Solving Quadratic Inequalities with a Sign Graph Solving Linear Equations The Square of a Binomial Properties of Negative Exponents Inverse Functions fractions Rotating an Ellipse Multiplying Numbers Linear Equations Solving Equations with One Log Term Combining Operations The Ellipse Straight Lines Graphing Inequalities in Two Variables Solving Trigonometric Equations Adding and Subtracting Fractions Simple Trinomials as Products of Binomials Ratios and Proportions Solving Equations Multiplying and Dividing Fractions 2 Rational Numbers Difference of Two Squares Factoring Polynomials by Grouping Solving Equations That Contain Rational Expressions Solving Quadratic Equations Dividing and Subtracting Rational Expressions Square Roots and Real Numbers Order of Operations Solving Nonlinear Equations by Substitution The Distance and Midpoint Formulas Linear Equations Graphing Using x- and y- Intercepts Properties of Exponents Solving Quadratic Equations Solving One-Step Equations Using Algebra Relatively Prime Numbers Solving a Quadratic Inequality with Two Solutions Quadratics Operations on Radicals Factoring a Difference of Two Squares Straight Lines Solving Quadratic Equations by Factoring Graphing Logarithmic Functions Simplifying Expressions Involving Variables Adding Integers Decimals Factoring Completely General Quadratic Trinomials Using Patterns to Multiply Two Binomials Adding and Subtracting Rational Expressions With Unlike Denominators Rational Exponents Horizontal and Vertical Lines
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:

# Parent and Family Graphs

## Adding a Constant

Now consider a family of quadratic functions that are found by adding a constant to y = x 2 . Graph the following functions on graphing calculators.

y = x 2, y = x 2 - 2, y = x 2 + 2, y = x 2 + 4

Try to find out what happens to the graphs of quadratic functions of the form y = x 2 + c as c changes. Make sure your students notice the following.

1. The axis of symmetry for each parabola is the same, namely, the vertical line x = 0, which is the y -axis. This is because in y = x 2 + c = 1x 2 + 0x + c , a = 1, and b = 0. So, the axis of symmetry is the vertical line or 0.

2. The vertex of the parabola moves up or down, depending on the value of c in y = x 2 + c . Since the axis of symmetry is the line x = 0 (or the y -axis), the x -coordinate of the vertex is 0. To find the y -coordinate, substitute x = 0 into the equation.

y = x 2 + c

y = 0 2 + c Replace x with 0.

y = c

So the y -coordinate of the vertex is c . In other words, the vertex of the parabola given by the function y = x 2 + c is the point at (0, c ). So if we add c to y = x 2 , the vertex moves up by c . If we subtract c from y = x 2 , the vertex moves down by c .

3. The parabolas all have the same size. They have just been shifted up or down by c units because the y values of the function y = x 2 + c are exactly c units more than the y values of the function y = x 2 . This is called a vertical translation. You can see this pattern with a table like the one shown below.

## Adding a Constant Before Squaring

Now consider the family of quadratic functions of the form y = ( x + d ) 2 , where d is some constant. Graph the following examples on graphing calculators.

y = ( x - 2) 2 , y = ( x - 1) 2 , y = x 2 , y = ( x + 2) 2

Try to figure out what happens to the graphs of quadratic functions of the form y = (x + d) 2 as d changes. Make sure you understand the following.

1. The axis of symmetry shifts to the left or right, depending on whether d is positive or negative. To understand this algebraically, expand the expression y = ( x + d ) 2 = x 2 + 2dx + d 2 . In this expression, a = 1, b = 2d , and c = d 2. So the axis of symmetry has the equation . For example, if d is positive, the line x = - d is the vertical line shifted to the left d units from the y -axis.

2. The vertex of the parabola moves left or right, depending on the value of d in y = ( x + d ) 2 . Since the axis of symmetry is the line x = - d , the x-coordinate of the vertex is - d . To find the y -coordinate, substitute x = -d into the equation.

y = ( x + d ) 2

= ( - d + d ) 2 Replace x with - d .

= 0 2 or 0

So the y-coordinate of the vertex is 0. In other words, the vertex of the parabola given by the function y = ( x + d ) 2 is the point at ( - d , 0).

3. The parabolas all have the same size. They have just been shifted to the right or left by d units. This is called a horizontal translation .

 Copyrights © 2005-2017