Warm Up Identify the following and then sketch the graph: Direction of opening Y-intercept Vertex AOS Zeros (factor) Max or Min f(x) = 2x2 – 10x + 12

Quadratic Properties: Increasing, Decreasing, & Interval Notation Sept. 15th

Number of Solutions (zeros, roots) If the parabola is completely above or below the x-axis, we say there are NO Real Solutions. If the parabola sits on the x-axis, we say there is ONE Real Solution. If the parabola is on both sides of the x-axis (crosses the axis twice), we say there is TWO Real Solutions.

Review (quadratics): What can we get … From the Graph: From the Equation:

Domain and Range Domain: The set of x-values that exist on the function What values of the x-axis are covered by the graph? For Quadratics, this is ALWAYS: -∞ to ∞ Range: The set of y-values that exist on the function - What values of the y-axis are covered by the graph?

What are the Domain & Range?

What are the Domain & Range? Give the domain and range for these graphs …

Increasing vs. Decreasing Properties to consider: The direction of the opening of a parabola Don’t forget about your end behavior!!

Intervals We can show the region of the graph that is increasing and decreasing by an interval. Intervals describe the range of x-values that meet the given requirement (where the graph is increasing or decreasing).

Increasing vs. Decreasing In this graph, the interval where the parabola is increasing is from - (remember the end behavior) to -1. The graph is decreasing from _____ to _____.

Increasing vs. Decreasing Discuss with a partner where the graph is increasing and decreasing. Describe in the way we just did where the graphs are increasing and where they are decreasing.

Interval Notation We use interval notation to abbreviate the description. List the starting and ending points of your interval, separated by a comma. - to -1 will look like: -, -1 Then we decide if there should be parentheses ( ) or brackets [ ]

Interval Notation **Parentheses indicate that the graph does not include the endpoint **Brackets indicate that the graph does include the endpoint On a graph, we can see this with open and closed circles Open Circles indicate we are NOT including the point: ( ) Closed Circles indicate that we ARE including the point: [ ]

Let’s Try using Inequalities: 1. What does x ≤ -2 look like on a number line? Using the number line, how could we write this in interval notation? 2. Now lets try x > 5

Partner Activity Graph the inequality and Write in Interval Notation: x ≤ -5 x > 10 x < 2 Draw each interval on a number line: 1. (-∞, -8] 2. [5, ∞) 3. (1, 5)

Interval notation Now, use interval notation to describe where the graphs are increasing and decreasing (the graph you discussed with your partner).

Try These Graphs

Now Let’s Use Interval Notation for Domain and Range

Now find the Domain and Range of these graphs

Translation Sometimes we have graphs that increase/decrease in more than one place. Rather than write out the word “and” we use the symbol “” We call this a Union.

Let’s look again… Using our new math vocabulary and our knowledge of interval notation, describe the increasing and decreasing parts of the graph. And the domain and range.

Homework Quadratics Properties Worksheet

Poster Project You will be making 2 Posters Tomorrow! You will work TOGETHER with 1 – 2 other people This is NOT an individual task that multiple people put their name on I will have up what is needed for the poster I will come around and give you the materials you need (2 posters for each group) This will be a QUIZ GRADE so use the time I am giving you and work you hardest!

Posters Project Equation Table x and y values Sketch of the Graph Find and Label the y-intercepts Find and Label the vertex Give the Axis of Symmetry Give the min/max Value Give the Direction of Opening Zeroes if they can be Identified Interval Notation for Increasing and Decreasing Curves Interval Notation for Domain and Range

Flex: Factoring Review Factor the following expressions: x2 – 15x + 50 12xy – 4y + 9x – 3 x2y - 3x2 – 8y + 24 x2 – 36 20x2 – 16x 4x2 - 24x + 36 2x3 + 12x2 + 16x x2 - 5x- 14 15ab – 5b + 9a – 3