Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Parent and Family Graphs


Use a graphing calculator to see how the graphs of quadratic functions change as the coefficients change.

In this lesson, we will start with the parent function y = x 2 and study how the graphs change in shape or in position when the coefficients are changed. You should use graphing calculators or computers to graph the families of functions, and try to find similarities and differences between them.


Multiplying x2 by a Positive Number

Graph y = x 2 and then the following family of functions.

Notice how the sizes of these parabolas change as the coefficient of x 2 increases or decreases. Make sure you understand the following characteristics.

1. The vertices and the axes of symmetry of all these parabolas are the same. Namely, the axis of symmetry is the line x = 0, and the coordinates of the vertex are (0, 0). This is because in a quadratic function of the form y = ax 2 + 0 x + 0, b = 0, so the axis of symmetry is given by or 0, which is the y -axis.

2. The graphs of the quadratic functions y = ax 2 , where a > 0, get more narrow as the value of a increases. The graphs get wider as the value of a decreases.

Why do parabolas change sizes in this way? The larger the value of a in y = ax 2 , the faster the y variable increases as x grows. To see this, make a table of values of the functions , y = x 2 , and y = 5 x 2 . It is clear from this table that the larger the coefficient of x 2 , the faster the y value grows. This results in a steeper incline of the graph, or a “narrower” parabola. Similarly, the smaller the coefficient of x 2 , the more slowly the y value grows. This results in a more gradual incline of the graph, or a “wider” parabola.

What happens as the coefficient a in y = ax 2 gets smaller and smaller (but remains positive), until it approaches 0? As a approaches 0, the parabolas get wider and wider until they approach a flat line, namely the graph of y = 0x or 0, which is the x-axis.


Multiplying x 2 by a Negative Number

Now what happens to the graphs of y = ax 2 when a is negative? Graph the following family of functions.

When the a is negative in y = ax 2 , the parabolas open downwards. This is because as x grows, the y variable becomes more and more negative. So the graphs incline downward. Notice that these graphs have the same characteristics that we saw before.

1. These parabolas all have the same axes of symmetry and vertices. The axis of symmetry is still the vertical line x = 0 (or the y -axis), and the vertices are still at (0, 0). This is because in these functions y = - ax 2 + 0x + 0, or 0. Notice however that the vertices are now maximum points of the graphs, where in the earlier examples they were minimum points.

2. Similar to previous examples, as the coefficient a in y = ax 2 becomes smaller (more negative), the parabola becomes more narrow. As a gets closer to zero, the parabola gets wider. This is because the smaller the coefficient a is, the faster the y value decreases (gets more negative) as the x value grows. Have your students make tables of values of the functions graphed above to verify this.

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