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![Page 1: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/1.jpg)
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-3-2-10123
![Page 2: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/2.jpg)
Properties of Quadratic Equations - GraphsSeptember 24th
![Page 3: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/3.jpg)
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.
![Page 4: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/4.jpg)
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
![Page 5: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/5.jpg)
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
![Page 6: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/6.jpg)
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
![Page 7: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/7.jpg)
We can also use the vertex to find the axis of symmetry
Vertex (1 , 2)
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:
![Page 8: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/8.jpg)
Let’s Practice
Direction of Opening:
y – intercept:
Vertex:
Axis of Symmetry:
Max/Min Value:
![Page 9: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/9.jpg)
Let’s Practice
Direction of Opening:
y – intercept:
Vertex:
Axis of Symmetry:
Max/Min Value:
![Page 10: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/10.jpg)
Let’s Practice
Direction of Opening:
y – intercept:
Vertex:
Axis of Symmetry:
Max/Min Value:
![Page 11: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/11.jpg)
Partner Activity – Post-its
Direction of Opening:
y – intercept:
Vertex:
Axis of Symmetry:
Max/Min Value:
![Page 12: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/12.jpg)
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?
![Page 13: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/13.jpg)
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”
![Page 14: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/14.jpg)
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
![Page 15: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/15.jpg)
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
![Page 16: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/16.jpg)
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
![Page 17: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/17.jpg)
Identify the Solutions
![Page 18: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/18.jpg)
Post – It
Direction of Opening:
y – intercept:
Vertex:
Axis of Symmetry:
Max/Min Value:
End Behavior:
Zeros:
![Page 19: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/19.jpg)
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?
![Page 20: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/20.jpg)
Domain and Range?
![Page 21: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/21.jpg)
Increasing Vs. Decreasing
Describe where the graphs are increasing and where they are decreasing:
![Page 22: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/22.jpg)
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
![Page 23: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/23.jpg)
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
![Page 24: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/24.jpg)
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
![Page 25: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/25.jpg)
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 [ ]
![Page 26: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/26.jpg)
Interval Notation - Practice
1. Draw the inequality and 2. Draw each intervalwrite in interval notation:
x < 6 (-∞, -4]
x ≥ -2 [-4, 5)
![Page 27: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/27.jpg)
Interval Notation - Practice
Pierre the Mountain Climbing Ant
![Page 28: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/28.jpg)
Increasing vs. Decreasing
In this graph the interval where the graph is decreasing is from -∞ to 1
The graph is increasing from_______ to _______
![Page 29: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/29.jpg)
Increasing vs. Decreasing – Interval Notation
![Page 30: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/30.jpg)
Domain and Range – Interval Notation
![Page 31: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/31.jpg)
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
![Page 32: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/32.jpg)
Let’s Look at this Graph
Tell where the graph is increasing & decreasing and the domain & range:
![Page 33: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/33.jpg)
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
![Page 34: Warm Up Given the function y = x 2, copy and complete the table below for the values of this function. Then sketch these points on a coordinate plane.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649eff5503460f94c14bda/html5/thumbnails/34.jpg)
Homework
Worksheet