Chapter 9 Modeling With Functions Students will be able to transform quadratic and absolute value functions. Students will be able to graph quadratic and absolute value inequalities. Students will be able to solve system of equations involving quadratic equations.

Section 9.1 Students will learn how to shift, stretch, compress and reflect quadratic and absolute value functions.

Transforming the Quadratic Parent Function F(x)=x^2 As a group determine how to do the following Think of the vertex form of a quadratic function, or play around on your calculator Move it left or right Move it up or down Make it narrow or wide (stretch vertically or horizontally) Reflect it over the x axis Reflect it over the y axis

Move it left or right Add inside ( ) to move left Subtract inside the ( ) to move right Move it up or down Add outside ( ) to move up Subtract outside the ( ) to move down Make it narrow or wide (stretch vertically or horizontally) Number in front of the ( ) if >1 gets narrow or stretches vertically if <1 gets wide or stretches horizontally Reflect it over the x axis Number in front of ( ) is negative Reflect it over the y axis x inside the ( ) is negative Move first then reflect

Translations These shifts can be written in the parent function Parent Function Up and Down Translation Left and Right Translation – notice the opposite of what is done inside the ( ) is what is done to the graph Vertex form of a parabola (h,k) is the vertex

From Equation Can you state the translation if given the equation and sketch a graph

From Graph Can you state translation from a graph and give the equation

Other Parent Functions Do any of these ideas change for the other parent functions Absolute Value Cubic Square Root

Absolute value How would the transformations be different/ the same for the absolute value function F(x)=|x| All the transformations are identical

Examples

Homework Worksheet

Students will solve systems of equations and quadratic inequalities. Section 9.2 Students will solve systems of equations and quadratic inequalities.

Gold Digger’s Gulch Pg 499 work through the problem What equation did you get? How many inequalities did you get and how? When looking at the inequalities think of the warm-ups we did How did we solve the inequality How did we graph it

To Graph a Quadratic Inequality Compare quadratic to zero Factor the quadratic Plot each zero on a number line Test a point in each section of the number line (zeros break it into 3 parts) in the original equation On the number line shade the regions that are true then write the inequality Write the inequality

Number line and Terms The zeros break the number line into 3 parts Lower boundary is to the left of the lowest number Upper boundary is to the right of the highest number Between the boundary is between the 2 zeros

Examples

Factoring If a =1 Factors of c that add to b (x factor 1)(x factor 2) Divide everything by a negative then repeat above If a is not 1 A times c Factors that add to b Rewrite equation using factors Group 1st 2 terms, group 2nd 2 terms Factor the groups Group what is outside ( ) and multiply to what is inside the ( )

Systems of Quadratics Use a calculator to do the following How else could you solve systems of equations Substitution, Elimination

Systems of equations Solving systems of equations with quadratics Graphically – easiest using a calculator Substitution If linear and a quadratic solve linear for a variable and substitute into the quadratic If both quadratic solve for y and set both equal to each other Elimination – not used much If linear and quadratic does it work If both quadratic does it work

Example

Example

Homework Worksheet on Inequalities and Systems You can check systems on calculator but must show either substitution or elimination