Hitting a golf ball out of a sand trap: https://

www.youtube.com/watch?v=FLR4V_lXJYE The path of the ball can be approximated

by:◦ y = -x2 + 5x◦ x is the horizontal distance travelled by the ball in

meters◦ y is the height of the ball in meters

3.2 Properties of Graphs of Quadratic Relations

What is the greatest height achieved by the ball?

Table of Values

2 Ways to Solve – Table & Graph

x 0 1 2 3 4 5

y 0 4 6 6 4 0

NOTICE:x values are all

positive, since ball moves forward,

not backward and a y-value of 0 is where the ball stops (ground

level)• So the maximum height is y = 6m,

and it occurs for 2 seconds• Wait, how can the maximum height

occur for more than an instant?• Perhaps there’s a more accurate

way to see the maximum height

Graph

2 Ways to Solve – Table & Graph

• Using the graph, we can see the 2 points where maximum height of 6m occurs

• The maximum height actually occurs between these points, at x = 2.5, which is y = 6.25

• A table of values is also correct, but you need more data points to have more accuracy

Sketch the graph of the relation y = x2 – 6x. Determine the equation of the axis of symmetry, the coordinates of the vertex, the y-intercept and the x-intercepts. Relationship is quadratic since highest

degree is 2 a = 1(parabola opens upward), b = -6, c = 0 In order to sketch the graph, create a table

of values and then plot the points

Example #2

x y

-1 7

0 0

1 -5

2 -8

3 -9

4 -8

5 -5

6 0

7 7

Example #2 - y = x2 – 6xOnce you make your table of values, plot the points on a set of axes.

Axis of Symmetry

Equation of Axis of Symmetry: x = 3

VertexCoordinates of Vertex: (3, -9)

x y

-1 7

0 0

1 -5

2 -8

3 -9

4 -8

5 -5

6 0

7 7

Example #2 - y = x2 – 6xLocating the x and y intercepts.

x-intercepts: when y = 0(0,0) and (6,0)

y-intercept: when x = 0(0,0)

You can either read this off the table of values, or see it

from the graph.

Pied Kingfisher catching fish in a split second

https://www.youtube.com/watch?v=6YRM0sy3xIYThe underwater path of the bird is described by a parabola with the equation y = 0.5x2 – 3x, where x is the horizontal position of the bird relative to its entry point, and y is the depth of the bird underwater (both in meters).

Quadratic Relations in Nature

The vertex of a parabola has the:◦ Min y-value if parabola opens up◦ Max y-value if parabola opens down

A parabola is symmetric with respect to a vertical line through its vertex

x-intercept is when y=0 y-intercept is when x=0

In Summary…