Warm Up 1. Evaluate x 2 + 5x for x = 4 and x = –3. 2. Generate ordered pairs for the function y = x with the given x VALUES. 36; – 6 { – 2, – 1, 0, 1, 2} x –2–2 –1–1 012 y PLUG IT IN, PLUG IT IN

QUADRATIC FUNCTIONS AND THEIR GRAPHS NC GOAL: 4.02 Graph, factor, and evaluate quadratic functions to solve problems. 7-1

ESSENTIAL QUESTIONS WHAT IS THE STANDARD FORM OF A QUADRATIC FUNCTION? WHAT IS A “VERTEX”? WHAT ROLE WOULD AN “AXIS OF SYMMETRY” PLAY? HOW IMPORTANT ARE THE SOLUTIONS? WHAT EXACTLY ARE SOLUTIONS TO A QUADRATIC FUNCTION? WHAT IS A “PARABOLA”? WHAT CAN THE LEADING COEFFICIENT IN A QUADRATIC FUNCTION TELL ME? 7- 1

VOCABULARY PARABOLA VERTEX SOLUTIONS AXIS OF SYMMETRY STANDARD FORM MAXIMUM MINIMUM 7-1

WHAT IS A “PARABOLA”?

The graph of a quadratic function is a curve called a parabola. To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. Then connect the points with a smooth curve. 7-1

WHAT IS THE STANDARD FORM OF A QUADRATIC FUNCTION? ax 2 + bx + c = 0 BUT a= 0

WHAT IS A “VERTEX”? THE VERTEX IS THE HIGHEST OR LOWEST POINT ON THE GRAPH. NOTE: A POINT IS AN (x,y) COORDINATE.

LETS TALK MAX AND MIN A PARABOLA HAS EITHER A MAXIMUM POINT OR A MINIMUM POINT, ALSO KNOWN AS THE VERTEX MINIMUM POINT/LOWEST POINT MAXIMUM POINT/HIGHEST POINT

WHAT CAN THE LEADING COEFFICIENT IN A QUADRATIC FUNCTION TELL ME? IF THE LEADING COEFFICIENT IS POSITIVE, THE GRAPH SMILES. IF THE LEADING COEFFICIENT IS NEGATIVE, THE GRAPH FROWNS. 2x 2 + 3x – 4 MINIMUM POINT -2x 2 + 3x – 4 MAXIMUM POINT

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Example 1: Identifying the Vertex and the Minimum or Maximum Identify the vertex of each parabola. Then give the minimum or maximum value of the function. The vertex is (–3, 2), and the minimum is 2. The vertex is (2, 5), and the maximum is 5. A.B. 7-1

Check It Out! Example 2 Identify the vertex of each parabola. Then give the minimum or maximum value of the function. The vertex is (3, –1), and the minimum is –1. The vertex is (–2, 5) and the maximum is 5. a. b. 7-1

WHAT IS THE “AXIS OF SYMMETRY”? THE IMAGINARY LINE DOWN THE MIDDLE OF THE PARABOLA MAKING BOTH SIDES MIRROR IMAGES. AXIS OF SYMMETRY THE AXIS OF SYMMETRY IS ALWAYS THE x VALUE IN THE VERTEX.

WHAT EXACTLY ARE SOLUTIONS TO A QUADRATIC FUNCTION? THE SOLUTIONS ARE THE X VALUES WHERE THE PARABOLA CROSSES THE x AXIS. I AM A SOLUTION SO AM I

A quick review on plotting points The first number in an ordered pair is your x value and the second one is the y value. Example: (2,3) x,y You move left and right to graph an x value and up and down to graph a y value. Lets try a few. ALWAYS BEGIN FROM THE ORIGIN

TODAY WE WILL LEARN HOW TO FIND THE VERTEX IN THE CALCULATOR Lets use the function 3x 2 + 5x + 5. Put this in your calculator under y=. Push 2 nd, trace, : notice #3 and 4? What does our graph have? Push 3 NOW IT GETS TRICKY USE YOUR ARROWS TO MOVE TO THE LEFT OF THE VERTEX, STAY VERY CLOSE THOUGH, AND HIT ENTER. NOW MOVE TO THE RIGHT OF THE VERTEX, BUT STAY CLOSE, AND HIT ENTER 2 TIMES. THE ORDERED PAIR AT THE BOTTOM OF THE SCREEN IS YOUR VERTEX.

LETS TRY ANOTHER ONE -x 2 + 4x + 3 does this graph have a minimum point or a maximum point? 2 nd, trace, 4, left bound, enter, right bound, enter enter. And the vertex is: (2,7) B E SURE TO PUT THE VERTEX IN PARENTHESIS OR IT IS WRONG.

NOW FIND THE SOLUTIONS Use 2 nd, trace, 5, enter, enter, enter. Find both of them and write them down. We can now graph this parabola with this information. But what if we needed a few more points to make a good graph. If you push 2 nd graph you will have a gozillion points to choose from. Pick a few. DRAW THIS GRAPH ON THE PAPER PROVIDED. BE SURE TO LABEL EVERYTHING.

Use a table of values to graph each quadratic function. Check It Out! Example 3 y = x x –2 –2 –1 – y Make a table of values. Choose values of x and use them to find values of y. Graph the points. Then connect the points with a smooth curve. 7-1

Example 4: Graphing Quadratic Functions by Using a Table of Values Use a table of values to graph the quadratic function. y = – 4x 2 x –2 –2 –1 – y 0 –4–4 – 16 –4–4 Make a table of values. Choose values of x and use them to find values of y. Graph the points. Then connect the points with a smooth curve. 7-1

WHAT IS THE STANDARD FORM OF A QUADRATIC FUNCTION? 7-1

WHAT IS A “VERTEX”? 7-1

WHAT CAN THE LEADING COEFFICIENT IN A QUADRATIC FUNCTION TELL ME? 7-1

WHAT EXACTLY ARE SOLUTIONS TO A QUADRATIC FUNCTION? 7-1

WHAT IS AN AXIS OF SYMMETRY? 7-1

CAN YOU NOW FIND THE VERTEX OF A QUADRATIC FUNCTION, AND IDENTIFY THE AXIS OF SYMMETRY AND SOLUTIONS. 7-1

CAN YOU GRAPH A PARABOLA ON THE COORDINATE PLANE USING THE VERTEX, SOLUTIONS, AND OTHER ORDERED PAIRS. 7-1