Graphing Quadratic Functions Analyze nonlinear graphs including quadratic and exponential functions that model a contextual situations.

What do you need to know…. How to graph simple quadratic function. Today you are going to graph general quadratic functions.

Quadratic Functions A QUADRATIC FUNCTION will show up as a U-shaped graph called a PARABOLA. Parabolic curves can be seen in the steel cables of a suspension bridge or in the supporting arches of some highway or railroad bridges, for it is this shape that naturally holds up a uniformly weighted road surface.

Check out these real parabolas all over the world….

A baseball in flight or the arc of water at the lip of a waterfall both follow a parabolic path. Fireworks even follow the path of a parabola.

The STANDARD FORM of a quadratic function is: where . Why can’t a be zero?

Quadratic Equation -a -c Quadratic Term Linear Term Constant Term a c opens up y-intercept -a opens down +c shifts up -c skinny parabola shifts down wide parabola

3. If you graph a quadratic function on a piece of paper and fold it down the middle, the two sides will match exactly. The line down the middle of the parabola is called the AXIS OF SYMMETRY. The two symmetric parts are mirror images of each other. 4. The VERTEX is the lowest point (minimum) of a parabola that opens up or the highest point (maximum) of a parabola that opens down. Fold construction paper and cut a parabola. Open up to show axis of symmetry.

AXIS OF SYMMETRY is the vertical line VERTEX has an x-coordinate of Fold construction paper and cut a parabola. Open up to show axis of symmetry. To find the y-coordinate, substitute the x value in the equation

Finding Critical Features of Quadratics Vertex Axis of Symmetry Opens Up/Down Opens Up Y-intercept (0, 2)

Finding Critical Features of Quadratics Graph x y x y Axis of Symmetry -3 -2 -1 1 2 3 29 x = 3 18 9 2 -3 -6 Vertex (3,-7) -7 4 5 -6 -3

Find the critical features of the quadratic below. Vertex Axis of Symmetry Opens Up/Down Opens Up Y-intercept (0, 4)

Graph: Vertex=-2,-8 Axis=-2 Y-int= (0,4) Opens:A=3 up!

Find the critical features of the quadratic below. Vertex Axis of Symmetry Opens Up/Down Opens Down Y-intercept (0, 0)

Graphing Step 1: Find the axis of symmetry and the coordinates of the vertex. Determine if the parabola opens up or down. Step 2: Find two other points on the parabola. An easy point to determine is the y-intercept. Choose a value for x on the same side of the vertex as the y-intercept and solve for y. Step 3: Reflect your points across the axis of symmetry and draw the parabola.

Graph Step 1: Step 2: Step 3:

Graph Step 1: Step 2: Step 3:

Ariel fireworks follow a parabolic path. Suppose a particular star is projected from an aerial firework at a starting height of 520 ft with an initial upward velocity of 72 ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be? Define variables: Let h = height and t = time in seconds Use the function: Find the t-coordinate: Find the h-coordinate: Find the vertex:

Suppose you have 80 ft. of fence to enclose a rectangular garden Suppose you have 80 ft. of fence to enclose a rectangular garden. Use the function where x is the width in feet and A is the area of the garden. What width gives you the maximum gardening area? What is the maximum area? Write function in standard form: Find the vertex:

Reflection… Explain how to determine the axis of symmetry and its purpose. Explain how to determine the vertex and its purpose.

Extended Writing… Describe the steps you would take to graph a quadratic equation in standard form. Be specific!

Practice p. 623 - 624 #9-16 *find the axis of symmetry, vertex, open up/down, y-int.