Warm Up Given the function y = x2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane. X Y -3 -2 -1 1 2 3

Properties of Quadratic Equations - Graphs September 24th

Parabola This is the math term for the u-shape of a quadratic function. Any quadratic function (one with an x2 term) will have this same basic shape.

What information can I find from the graph? Direction of Opening: which way the open side of the parabola is facing y – intercept: where the parabola crosses the y – axis

What information can I find from the graph? Vertex: the point where the parabola changes directions (the min/max value) – represented (h, k) Axis of Symmetry: vertical line through the vertex that cuts the parabola in half

What information can I find from the graph? Maximum or Minimum Value: highest or lowest point of the parabola The maximum/minimum is the y – value of the vertex

We can also use the vertex to find the axis of symmetry The axis of symmetry is the vertical line that cuts the parabola in half. The equation of the axis of symmetry is the x – value of the vertex AOS: Min/Max Value:

Let’s Practice Direction of Opening: y – intercept: Vertex: Axis of Symmetry: Max/Min Value:

Let’s Practice Direction of Opening: y – intercept: Vertex: Axis of Symmetry: Max/Min Value:

Let’s Practice Direction of Opening: y – intercept: Vertex: Axis of Symmetry: Max/Min Value:

Partner Activity – Post-its Direction of Opening: y – intercept: Vertex: Axis of Symmetry: Max/Min Value:

End Behavior What would the graph do if we expanded the view further left and further right? We use the parabola’s direction of opening to see the end behavior. Do the ends up to infinity? Or down to negative infinity?

End Behavior If a>0, the end behavior will be that the graph goes “up to the left and up to the right” If a<0, the end behavior will be that the graph goes “down to the left and down to the right”

Zeros, Roots, x – intercepts, Solutions The x – values that show where the parabola crosses the x – axis We will find these values by graphing, factoring, or using the quadratic formula

Zeros, Roots, x – intercepts, Solutions Graphing: visually identify the intersections of the parabola and x – axis. Factoring: factor the equation then set each set of parentheses equal to 0 and solve for x Quadratic Formula: plug a, b, and c into the formula and simplify

Number of Solutions 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 (in one spot), we say there is 1 REAL solution If the parabola is on both sides of the x – axis (crosses twice), we say there are 2 REAL solutions

Identify the Solutions

Post – It Direction of Opening: y – intercept: Vertex: Axis of Symmetry: Max/Min Value: End Behavior: Zeros:

Domain and Range Domain: the set of x – values that exist on the function Which x – values (the horizontal axis) are covered by the quadratic? For quadratics this is ALWAYS (- ∞, ∞) Range: the set of y – values that exist on the function - Which y – values (the vertical axis) are covered by the quadratic?

Domain and Range?

Increasing Vs. Decreasing Describe where the graphs are increasing and where they are decreasing:

Intervals We can show the region of the graph that is increasing or decreasing by an interval Intervals describe the range of x – values that meet the given requirement

Interval Notation We use interval notation to abbreviate the description List the starting and ending points of your interval separated by a comma Example: -∞ to -1 will look like: -∞, -1

Interval Notation Then we decide if there should be parentheses ( ) or brackets [ ] ( ) indicated that the graph does not include the endpoint [ ] indicate that the graph does include the endoint

Interval Notation On a graph, we can see this with open and closed circles Open Circles indicate that we are NOT including that point – so we are using ( ) Closed Circles indicate that we ARE including that point – so we are using [ ]

Interval Notation - Practice 1. Draw the inequality and 2. Draw each interval write in interval notation: x < 6 (-∞, -4] x ≥ -2 [-4, 5)

Interval Notation - Practice Pierre the Mountain Climbing Ant

Increasing vs. Decreasing In this graph the interval where the graph is decreasing is from -∞ to 1 The graph is increasing from _______ to _______

Increasing vs. Decreasing – Interval Notation

Domain and Range – Interval Notation

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

Let’s Look at this Graph Tell where the graph is increasing & decreasing and the domain & range:

Group Work! For your groups parabola: Sketch the graph on your poster and list: a. Direction of Opening: b. y – intercept: c. Vertex: d. Axis of Symmetry: e. Max/Min Value: f. End Behavior: g. Number of Solutions and the Zeros: h. Interval Notation: increasing/decreasing, domain/range

Homework Worksheet