Unit 4 Quadratics

Quadratic Functions Any function that can be written in the form

Put in Standard Form and Find a, b, and c  

Is it quadratic?  

Quadratic Functions Graph forms a parabola concave up concave down or

Determine whether a parabola opens up or down  

Up or Down? Max or Min?  

Using a graphing calculator, find vertex, line of symmetry, max/min, and zeros and where the function increases and decreases..  

Another  

And another  

Average Rate of Change of a Quadratic  

Example  

Finding Average Rate of Change  

Average Rate of Change  

To find the axis of symmetry When

Find the vertex and los

Vertex (h,k) form of a Quadratic Standard Form:

Parent Function

Transformations You can tell what the graph of the quadratic will look like if the eq. is in (h,k) form

Sketch the graph

Sketch the graph

Sketch the graph

Sketch the graph

Sketch the graph

Identifying Important Parts on Calculator 2nd calc—then select max or min

Completing the Square Used to go from standard form to (h,k) form or to get the equation in the form of a perfect square to solve Steps: Move the constant Factor out the # in front of x2 Take ½ of middle term and square it Write in factored form for the perfect sq. trinomial Add to both sides (multiply by # in front) Move constant back to get in (h,k) form

Solve  

Solve  

Complete the Square

Complete the Square

Example

Example

Example

Solving Quadratics You can solve by graphing, factoring, square root method, and quadratic formula Solutions, roots, or zeros

Solving by Graphing Graph the parabola Look for where is crosses the x-axis (where y=0) May have 2 real, 1 real, or no real solutions (Show on calculator) Review finding the vertex

Solve the following by graphing

Solving Quadratics by Factoring Factor the quadratic Set each factor that contains a variable equal to zero and solve (zero product property)

More solving by factoring

You Try

Writing the Quadratic Eq. Write the quadratic with the given roots of ½ and -5

Write the quadratic with Roots of 2/3 and -2

More about solving Graphing—not always best unless you have exact answers Factoring—not every polynomial can be factored Quadratic Formula—always works Square Root method—may have to complete the square first

Solving using Quadratic formula Must be in standard form Identify a, b, and c

Examples

Examples

Examples

Examples

Discriminant Used to identify the “type” of solutions you will have (without having to solve)

If the discriminant is… A perfect square---2 rational solutions A non-perfect square—2 irrational sol. Zero—1 rational sol. Negative—2 complex sol.

Identify the nature of the solution

Identify the nature of the solution(s)  

Solving Quadratics using the Sq. Rt. method Useful when you have x2 = constant or a perfect sq. trinomial ex. (x-3)2=constant Get the x2 by itself Take the square rt. of both sides Don’t forget + or – in your answer!!!

Examples

Examples

Examples

Quadratic Inequalities Graphing quadratic inequalities in 2 variables: Steps: Graph the related equation Test a point not on the graph of the parabola Shade region that contains the point if it makes the inequality true or shade the other region if it does not make the inequality true Ex. Ex.

Graphing Quadratic Inequalities

Solving Quadratic Inequalities Solving Quadratic Inequalities in one variable: May be solved by graphing or algebraically. To solve by graphing: Steps: Put the inequality in standard form Find the zeros and sketch the graph of the related equation identify the x values for which the graph lies below the x-axis if the inequality sign is < or identify the x values for which the graph lies above the x-axis if the inequality sign is > or

Solve by graphing Solutions:_______________________

To solve algebraically: Steps: Solve the related equation Plot the zeros on a number line—decide whether or not the zeros are actually included in the solution set Test all regions of the number to determine other values to include in the solution set

Solve Algebraically

Solving Quadratic Inequalities

Word Problems