This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
Identify the transformations to the graph of a quadratic function.
Change a function from general form to vertex form.
Identify the vertex, axis of symmetry, the domain, and the range of the function.
* Recognize characteristics of parabolas.
* Understand how the graph of a parabola is related to its quadratic function.
* Determine a quadratic function’s minimum or maximum value.
* Solve problems involving a quadratic function’s minimum or maximum value.
This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
Identify the transformations to the graph of a quadratic function.
Change a function from general form to vertex form.
Identify the vertex, axis of symmetry, the domain, and the range of the function.
* Recognize characteristics of parabolas.
* Understand how the graph of a parabola is related to its quadratic function.
* Determine a quadratic function’s minimum or maximum value.
* Solve problems involving a quadratic function’s minimum or maximum value.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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2. Concepts and Objectives
Graph a quadratic function in vertex form
Graph a quadratic function by completing the square
Identifying the axis of symmetry and vertex of a
parabola using the vertex formula
3. Quadratic Functions
A function f is a quadratic function if
where a, b, and c are real numbers, and a 0.
The graph of a quadratic function is a parabola whose
shape and position are determined by a, b, and c.
2
f x ax bx c
4. Vertex Form
The graph of gx = ax2 is a parabola with vertex at the
origin that opens up if a is positive and down if a is
negative. The magnitude (or absolute value) of a
determines the width of the parabola.
The vertex form of a quadratic function is written
The graph of this function is the same as that of gx
translated h units horizontally and k units vertically.
This means that the vertex of F is at h, k and the axis of
symmetry is x = h.
2
F x a x h k
5. Vertex Form (cont.)
Example: Graph the function and give its domain and
range.
21
4 3
2
F x x
6. Vertex Form (cont.)
Example: Graph the function and give its domain and
range.
Compare to : h = 4 and k = 3 (Notice
the signs!)
Vertex: 4, 3, axis of symmetry x = 4
We can graph this function by graphing the base
function and then shifting it.
21
4 3
2
F x x
2
F x a x h k
7. Vertex Form (cont.)
Example, cont.:
Let’s consider the graph of
Vertex is at 0, 0
Passes through 2, ‒2 and
4, ‒8.
(I picked 2 and 4 because of
the half.)
21
2
g x x
8. Vertex Form (cont.)
Example, cont.:
To graph F, we just shift everything over 4 units to the
right and 3 units up.
Domain: ‒,
Range: ‒, 3] or y 3
9. Completing the Square
If we are given a function that is not in vertex form, we
can “complete the square” to transform it into vertex
form. We do this by taking advantage of the additive
identify property (a + 0 = a).
For example, the function is not a
binomial square. We can add 0 in the form of 52 – 52
(5 is half of 10), and group the parts that factor to a
binomial square:
22 2
10 305 5f x x x
2
10 30f x x x
2 2 2
10 5 5 30x x
2
5 5x
10. Completing the Square (cont.)
Example: What is the vertex of the function?
2
6 7f x x x
11. Completing the Square (cont.)
Example: What is the vertex of the function?
The vertex is at 3, ‒2.
2
6 7f x x x
22 2
3 736x x
2
3 9 7x
2
3 2x
6
3
2
12. Practice
Find the vertex by completing the square.
1. fx = x2 +8x + 5
2. gx = x2 – 5x + 8
3. hx = 3x2 + 12x – 5
13. Practice (cont.)
1. fx = x2 +8x + 5
The vertex is at ‒4, ‒11.
22 2
8 4 4 5f x x x
2
4 16 5x
2
4 11x
15. Practice (cont.)
3. hx = 3x2 + 12x – 5
The vertex is at ‒2, ‒17.
2
4 53h x x x
2 2 2
5243 3 2x x
2
3 2 12 5x
3 2 17x
16. Vertex Formula
You may have noticed that we are doing the same
process each time we complete the square to produce
the vertex form of the function. We can generalize this
to create a formula for the vertex of a parabola.
Starting with the general quadratic form, we can
manipulate it by completing the square and end up with
a version that gives us a formula for the coordinates of
the vertex.
17. Vertex Formula (cont.)
Deriving the vertex formula:
2
f x ax bx c
2 2
2 1 1
2 2
b b b
a x x c a
a a a
2 2
2 4
b b
a x c
a a
18. Vertex Formula (cont.)
Comparing this to the vertex form of
shows that
and
We can simplify this further by substituting h for x:
So for any quadratic function,
2
f x a x h k
2
b
h
a
2
4
b
k c
a
2
f h a h h k k
2
,f x ax bx c
, , and is the axis of symmetry.
2
b
h k f h x h
a