QUAD1 QUADRATIC FUNCTIONS AND EQUATIONS Student Packet 1: Introduction to Quadratic Functions

Transcription

Introduction to Quadratic Functions
QUAD1
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STUDENT PACKET
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QUADRATIC FUNCTIONS AND EQUATIONS
Student Packet 1: Introduction to Quadratic Functions
Functions Revisited
• Define the domain and range of a function.
• Learn some function notation
QUAD1.2
Modeling with Quadratic Functions 1
• Create mathematical models of an event.
• Determine the degree of accuracy appropriate for a given
situation.
• Graph different data sets that result in lines or parabolas.
• Use technology to display and analyze data.
• Understand some basic properties of a parabola.
QUAD1.3
Creating Parabolas from Lines
• Graph linear and quadratic functions
• Explore some basic properties of a quadratic function.
12
QUAD1.4
Vocabulary, Skill Builders, and Review
17
1
6
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QUAD1.1
Quadratic Functions and Equations Unit (Student Pages)
QUAD1 – SP0
Word or
Phrase
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WORD BANK (QUAD1)
Definition or Explanation
Example or Picture
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function
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domain of a
function
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parabola
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linear
function
vertex of a
parabola
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quadratic
function
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range of a
function
QUAD1 – SP0
1.1 Functions Revisited
FUNCTIONS REVISITED
Set (Goals)
• Define the domain and range of a
function.
• Learn some function notation
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We will revisit the concept of function and
graph of a function. We will define the
domain and range of a function, and learn
some function notation.
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Ready (Summary)
Fill in the t-tables and draw the graphs.
2.
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x
x = y2 – 1
y
3
2
1
0
-1
-2
-3
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x
3
2
1
0
-1
-2
-3
y = x2 – 1
y
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1.
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Go (Warmup)
NOTE: In this unit, the horizontal axis will always be the x-axis, the vertical axis will
always be the y-axis, and the input variable will always be x, unless otherwise noted.
QUAD1 – SP1
A REVISIT TO FUNCTIONS AND THEIR GRAPHS
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A function is a rule in which each input value is assigned to exactly one output value.
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The graph of a function is the set of ordered pairs, each consisting of an input and its
corresponding output. If the inputs and outputs are real numbers, then we can represent
the graph of a function as points on the coordinate plane.
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The vertical line test refers to a visual way to determine whether a set of ordered pairs
in the usual xy-plane is the graph of a function. The vertical line test simply states that if
a vertical line intersects a graph in more than one point, then it is a NOT a function.
1. Consider the equation and its graph in problem 1 of the warmup.
Why does it represent a function of x based upon:
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a. the definition of function?
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b. the vertical line test?
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c. the equation?
2. Consider the equation and it’s graph in problem 2 of the warmup.
Why does it not represent a function of x based upon:
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a. the definition of function?
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b. the vertical line test?
c. the equation?
QUAD1 – SP2
The domain of a function is the set of inputs of the function.
The range of a function is the set of outputs of the function.
Consider this set of ordered pairs: {(1, 10), (2, 20), (3, 30), (4, 40)}
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DOMAIN AND RANGE
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1. Do these ordered pairs represent a function? ______ Explain. ________________
2. The domain is { ___, ___, ___, ___ }.
3. The range is { ___, ___, ___, ___ }
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_____________________________________________________________________.
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4. What would be an ordered pair that, if added to the set, would make the relationship
NOT a function?
From smallest to largest:
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Consider this graph, which consists of ordered pairs in the form (x, y).
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3. The x-coordinates include all real values from ___ to ___.
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We can write this as the closed interval: [ ____, ____ ]
We can write this as an inequality: ____ ≤ x ≤ ____
4. The y-coordinates include all real values from ____ to ____.
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We can write this as a closed interval: [ ____, ____ ]
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We can write this as an inequality: ____ ≤ y ≤ ____
5. Does this graph represent a function of x? ________
If it is a function, what is the domain?_____________ the range?_____________
QUAD1 – SP3
DOMAIN AND RANGE (continued)
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Consider this graph, which consists of ordered pairs in the form (x, y). Assume that the
lines continue outside the graphing area to the left and right with no surprises, as
indicated by the arrows. From smallest to largest:
We can write this as an open interval: ( ____, ____ )
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We can write this as an inequality: ____ < x < ____
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6. The x-coordinates include all real values from ___ to ___.
7. The y-coordinates include all real values from ____ to ____.
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We can write this interval: ( ____, ____ ]
We can write this as an inequality: ____ < y ≤ ____
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Consider this graph, which consists of ordered pairs in the form (x, y). Assume that the
entire graph is shown here.
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9. The x-coordinates include all real values
from ___ to ___ (but not including ____).
Write this using interval notation: _______________
Write this as an inequality: ____________________
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10. The y-coordinates include all real values from ___ to ___.
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Write this using interval notation: _______________
Write this as an inequality: ____________________
QUAD1 – SP4
FUNCTION NOTATION
Error alert: f(x) does NOT mean f multiplied by x.
Example of a linear function:
Example of a linear function using the new notation:
To evaluate this function at x = 5, we can simply write:
Given function: g(x) = x2 – 7
Evaluate the function g
for the given inputs
2.
g(1) =
3.
g(0) =
4.
g(-2) =
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g(3) =
Write the ordered pair
this represents
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1.
y = 3x + 2
f(x) = 3x + 2.
f(5) = 3(5) + 2 = 17
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The notation f(x) (read “f of x”) is used to denote a function in which x is an element of
the domain (the input value). Therefore, f(x) is the output of the function f that
corresponds to the input x.
Given function: h(x) = 2x2 – 3x + 1
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Evaluate the function h
for given inputs
h(3) =
6.
h(0) =
7.
h(-2) =
8.
challenge: h(a) =
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5.
Write the ordered pair
this represents
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9. Graph the function f(x) = x2 – 2x + 1.
First make a table.
QUAD1 – SP5
1.2 Modeling with Quadratic Functions 1
MODELING WITH QUADRATIC FUNCTIONS 1
Set (Goals)
We will conduct an experiment with
a flashlight and collect data. We will
use multiple representations to
organize, display, and analyze
data. We will create mathematical
models to fit our data.
• Create mathematical models of an event.
• Determine the degree of accuracy appropriate
for a given situation.
• Graph different data sets that result in lines or
parabolas.
• Use technology to display and analyze data.
• Understand some basic properties of a
parabola.
Go (Warmup)
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Ready (Summary)
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A function is a rule that assigns to each input x a unique output y = f(x).
Fill in the t-table and draw the graph for each function.
Use a different color for each graph.
Not all values will fit on the graph.
3
y=x +3
y = 2x2 + 3
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y=x
3.
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x
2.
2
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1.
1
0
-1
-2
-3
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color
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4. How are the graphs for #1 and #2 the same? How are they different?
5. How are the graphs for #2 and #3 the same? How are they different?
QUAD1 – SP6
Your teacher will perform a flashlight demonstration.
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FLASHLIGHT INVESTIGATION: RECORDING PAGE
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1. Write your observations about what quantities are changing and how you might
measure those changes.
2. You will design and conduct a flashlight investigation. Record the data you collect
on this page.
Write a short description of this investigation. Underline variables in your
description (what you are measuring).
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• What is a reasonable degree of accuracy for your measurements and
calculations?
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Label columns as needed and record your data here.
4. Perform other flashlight investigations (or repeat the one you just did), and collect
additional data. Record results on blank paper.
QUAD1 – SP7
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FLASHLIGHT INVESTIGATION:
A LINEAR FUNCTION
Compare two variables from your experiment that you think will result in a linear function.
Record your data in the table, or use data provided by your teacher.
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2. What variables are being compared?
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1. Briefly describe the experiment that
was performed to gather the data.
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3. What will go on the horizontal axis?
How will you scale it?
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4. What will go on the vertical axis? How
will you scale it?
5. What is a good title for your graph?
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6. What does the y-intercept on your
graph represent?
7. Estimate a “line of best fit” for your
graph
QUAD1 – SP8
FLASHLIGHT INVESTIGATION:
A QUADRATIC FUNCTION
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Compare a variable that represents a length measurement to a variable that represents
an area measurement. Record your data in the table, or use data provided by your
teacher.
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2. What variables are being compared?
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1. Briefly describe the experiment that
was performed to gather the data.
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3. What will go on the horizontal axis?
How will you scale it?
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4. What will go on the vertical axis? How
will you scale it?
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5. What is a good title for your graph?
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6. What does the y-intercept on your
graph represent?
7. What is the shape of your graph?
QUAD1 – SP9
FINDING AN EQUATION FOR THE DATA
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The graph of a quadratic function is a curve called a parabola. Functions whose graphs
are parabolas may open upward or downward and vary in "width" or "steepness", but
they all have the same basic "U" shape. Parabolas that have symmetry about the y-axis
have the form:
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y = ax2 + k
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This parabola has its vertex at (0, k). The value of a determines its “steepness”.
We will estimate a quadratic function to fit the data on the previous page.
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1. Estimate a value for k. The minimum point on the graph is (0, ____). This
coordinate is called the vertex of the parabola. Use this as a value for k.
y = ax2 + _____
y = ax2 + k
For ( ____, ____ )
For ( ____, ____ )
y = ax2 + k
y = ax2 + k
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For ( ____, ____ )
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2. Substitute (x, y) coordinates from the graph on the previous page to calculate some
possible values for a.
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_____ = (a)(____)2 + _____ _____ = (a)(____)2 + _____
a = _______
For ( ____, ____ )
For ( ____, ____ )
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For ( ____, ____ )
QUAD1 – SP10
3. Describe a strategy for estimating a value for a.
4. Use your strategy to estimate a value for a.
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FINDING AN EQUATION FOR THE DATA (continued)
+
k
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y = ax2
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5. Using your calculations and estimations, write a quadratic function to fit your data
y = ____x2 + _____
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7. Locate a program on the internet, or use a graphing calculator that finds a quadratic
function to best fits a set of data. Find the equation of a parabola that best fits your
data.
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8. Compare your estimated quadratic function to the one generated by technology.
Discuss discrepancies.
9. Locate a program on the internet or use a graphing calculator that finds a linear
function to best fit a set of data. Find the equation of a line that best fits the linear
function you created from flashlight investigation data.
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10. Compare your estimated linear function to the one generated by technology.
Discuss discrepancies.
QUAD1 – SP11
1.3 Creating Parabolas from Lines
CREATING PARABOLAS FROM LINES
Set (Goals)
By multiplying two linear expressions, we
will generate a quadratic expression that
defines a quadratic function. Graphically,
we will observe that multiplying the ycoordinates of the two linear functions
yields the y-coordinate of the quadratic
function. We will learn some basic
properties of parabolas.
• Graph linear and quadratic functions
• Explore some basic properties of a
quadratic function.
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Go (Warmup)
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Ready (Summary)
x
f(x)
g(x) = x – 3
g(x)
x
-3
-2
-2
-1
-1
0
0
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1
2
2
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-3
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f(x) = 2x + 2
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1. Use two different colored pencils. Graph the following lines:
3
4
4
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2. Graph h(x) = f(x) • g(x) by following
your teacher’s directions.
QUAD1 – SP12
PRODUCTS OF LINEAR EXPRESSIONS
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Consider the two lines you graphed in the warmup: f(x) = 2x + 2 and g(x) = x – 3.
1. Find the product h(x) = f(x) • g(x). That is, h(x) = (2x + 2)(x – 3) = ___________
2. Fill in the in the table.
f(x) =
g(x) =
h(x) =
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x
-2
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-1
0
1
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2
3
4
f(x) =
g(x) =
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( _____ , _____ )
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x-intercept
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3. Write ordered pairs for the intercepts for f(x) and g(x).
y-intercept
( _____ , _____ )
h(x) =
( _____ , _____ )
( _____ , _____ )
( _____ , _____ )
( _____ , _____ )
( _____ , _____ )
4. Compare the ordered pairs for (x, h(x)) to the points graphed for the parabola on the
warmup page. What do you notice?
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5. How do the x-intercepts of the lines compare to the x-intercept(s) of the parabola?
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6. How do the y-intercepts of the lines compare to the y-intercept(s) of the parabola?
7. Does the parabola have a minimum point or maximum point? _______________
What are its coordinates? _____
QUAD1 – SP13
CREATING PARABOLAS FROM LINES 1
g(x) = -x + 6 using two different colors.
2. Find the intercepts for each function.
g(x) =
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f(x) =
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1. Graph the lines f(x) = x and
( _____ , _____ )
( _____ , _____ )
y-intercept
( _____ , _____ )
( _____ , _____ )
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x-intercept
3. Graph a parabola by multiplying y-coordinates from the lines. Use a third color.
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4. Find an equation for the parabola.
h(x) = f(x) • g(x) = _______________
5. Identify the following coordinates for
the parabola:
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c. vertex
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b. y-intercept(s)
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a. x-intercept(s)
6. Is the vertex a minimum or a
maximum point?
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7. Where is the vertex located in
relation to the x-intercepts?
QUAD1 – SP14
f(x) =
g(x) =
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1. Graph the lines f(x) = -2x – 4 and g(x)= -x – 2 using two different colors.
( _____ , _____ )
( _____ , _____ )
y-intercept
( _____ , _____ )
( _____ , _____ )
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x-intercept
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h(x) = f(x) • g(x) = _______________
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the parabola:
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c. vertex
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b. y-intercept(s)
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a. x-intercept(s)
maximum point?
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QUAD1 – SP15
and
g(x) = -x + 1 using two different colors.
g(x) =
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f(x) =
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1. Graph the lines f(x) = 2x
( _____ , _____ )
( _____ , _____ )
y-intercept
( _____ , _____ )
( _____ , _____ )
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x-intercept
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h(x) = f(x) • g(x) = _______________
the parabola:
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c. vertex
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b. y-intercept(s)
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a. x-intercept(s)
maximum point?
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QUAD1 – SP16
1.4 Vocabulary, Skill builders, and Review
1.3
Complete the flow chart to show how this week’s vocabulary is related.
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FOCUS ON VOCABULARY (QUAD1)
2a.
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2b.
3.
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4.
6.
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5.
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1. A rule where each input matches exactly one output
2a. The set of input values for a function
2b. The set of output values for a function
3. The type of function described by y = 2x + 3
4. The type of function described by y = 2x2 + 4
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5. The shape of y = 2x + 3
(a word not on the vocabulary list)
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6. The shape of y = 2x2 + 4
QUAD1 – SP17
1.3
SKILL BUILDER 1
2. (x + 6) (x – 6)
3. (-2x - 1) (3x + 7)
4. (5x + 3) (5x + 3)
5. (2x - 5) (3x)
6. -10(x + 1)
Factor:
7.
3x -6
8.
11.
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6x2 + 7x + 2
9.
x4 - 625
12.
x2 – 8x + 16
3x2 – 15x + 18
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10.
x2 - 9
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Multiply:
1. (x + 3) (x – 2)
ax2 + c
16.
2ax2 – bx + c
14.
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13.
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Evaluate each expression when a = 1, b = 4, c = -2, and x =
ac + b
15.
18.
28
2a
b 2 − 4ac
−b −
2a
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17.
(5c - 2) (5c + 2)
1
:
2
QUAD1 – SP18
1.3
SKILL BUILDER 2
a.
18 ? _____ and _____.
b.
-
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1. Between which two consecutive integers is:
18 ? _____ and _____.
2+
18 ? ______ and ______.
d. 2 + 180 ? ______ and ______.
e.
2-
18 ? ______ and ______.
f.
4.
6 9
25
10
2 - 180 ? ______ and ______.
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Simplify each radical expression.
2.
3.
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c.
6.
−
8.
9.
3 12
4
17
81
10.
31− 32
25
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36
2 18
50
32
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8
49
7.
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5.
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6− 9
(−2 + 36 )(−2 − 36 )
QUAD1 – SP19
1.3
SKILL BUILDER 3
3
7
2
10
1
14
0
19
1
25
2
32
3
40
2.
x
y
(input)
(output)
3
9
2
4
1
1
0
0
-1
1
-2
4
-3
9
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(input)
(output)
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1.
x
y
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Which of the following input-output tables illustrate functions? If it is not a function, state
why not.
3. (1,5), (2,6), (3,5), (4,6)
4. (10,8), (20,16), (30,24), (8,10)
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Which of the following sets of ordered pairs illustrate functions? If it is not a function,
state why not.
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5. (1.5,5), (2.8,12), (1.5,7), (2.8,15)
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Which of the following mapping diagrams illustrate functions? If it is not a function, state
why not.
6.
1
2
7.
1
m
7
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8.
1
8
4
5
4
10
8
9
6
6
-1
-1
-3
-3
-5
9.
0
-1
3
-7
9
13
5
QUAD1 – SP20
1.3
SKILL BUILDER 4
x
2.
y=x
3.
y=x+3
y= 2x + 3
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2
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1
0
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-1
-2
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Fill in the t-table and draw the graph for each equation.
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-3
color
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Solve each inequality. Then draw a number line and graph the solution.
4. 2x + 1 < -7
5. -2x + 1 < -7
Graph:
6. 7 − 3x ≤ 28
7. 7 − 3x ≥ 28
Graph:
Graph:
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Graph:
QUAD1 – SP21
1.3
1.
y = -x
3.
y = -x + 1
y = -2x + 1
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x
2.
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SKILL BUILDER 5
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3
2
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1
0
-1
N
-2
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-3
color
4. Evaluate 4x2 + 2x – 3xy + 5y – 6y2 for x = -5, y = 10.
6.
(x + 3)(x – 7)
7.
8.
(x – 3)(x – 7)
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Multiply.
5.
(x + 3)(x + 7)
(x – 3)(x + 7)
QUAD1 – SP22
1.3
SKILL BUILDER 6
2.
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1.
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On each graph below, the x-axis is the horizontal axis and the y-axis is the vertical axis.
Which of the following graphs illustrate functions of x? If it is not a function, state why
not. If it is a function, state the domain and range.
4.
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3.
6. Sketch a linear graph that does NOT
represent a function
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5. Sketch a nonlinear graph that
represents a function
QUAD1 – SP23
1.3
2. x2 + 5x – 14
3. x2 + 4x – 32
4. x2 – 4x – 32
5. x2 – 4x + 45
6. x2 – 10x – 60
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Factor.
1. 3x2 + 2x + 24
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SKILL BUILDER 7
8. 10x2 – 15
N
7. 3x2 + 4x
11. −b −
b 2 − 4ac
2a
14. −b −
b 2 − 4ac
2a
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Evaluate if a = 15, b = -1, and c = -6
9. b 2 − 4ac
b 2 − 4ac
10. −b +
2a
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Evaluate if a = 2, b = 0, and c = -6
12. b 2 − 4ac
b 2 − 4ac
13. −b +
2a
QUAD1 – SP24
1.3
1.
f(x) = x2
3.
g(x) = x2 – 3
h(x) = 2x2 – 3
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x
2.
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SKILL BUILDER 8
3
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2
1
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0
-1
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color
o
-3
N
-2
4. In order to ride a rollercoaster, all riders must be at least 60 inches tall. Using
“h” to represent the height, write an inequality that shows the height of all the
riders on the rollercoaster. Then draw a number line and graph the solution.
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Inequality: ______________
Graph:
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5. In order to ride on the ponies at a fair, all riders must weigh less than 50 pounds.
Using “w” to represent the weight, write an inequality that shows the weight of all
the riders on the ponies. Then draw a number line and graph the solution.
Inequality: ______________
Graph:
QUAD1 – SP25
1.3
1.
f(x) = -x2
3.
g(x) = -x2 + 3
h(x) = -2x2 + 3
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x
2.
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SKILL BUILDER 9
3
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2
1
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0
-1
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color
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-3
N
-2
4. Draw a coordinate plane on the graph below and then graph the following system
of linear inequalities. Don’t forget to shade the region that represents all the
coordinates that are solutions to the system.
y < −2x + 1
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y ≥ x−3
QUAD1 – SP26
1.3
SKILL BUILDER 10
1
1
x + 3 and g(x) =
2
x – 3 using two different colors.
2
g(x) =
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f(x) =
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1. Graph the lines f(x) =
( _____ , _____ )
( _____ , _____ )
y-intercept
( _____ , _____ )
( _____ , _____ )
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x-intercept
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h(x) = f(x) • g(x) = _______________
the parabola:
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c. vertex
o
b. y-intercept(s)
N
a. x-intercept(s)
maximum point?
Sa
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QUAD1 – SP27
1.3
TEST PREPARATION (QUAD 1)
1. What is the domain of the set {(-4, 1), (1, -2), (-1, -3)}
{-4, 1, -1}
B.
{1, -2, -3}
C.
{-4, 1, -2, -1, 3}
2. Is the set of ordered pairs {(-4, 1), (1, -2), (1, -3)} a function?
No
B.
C.
There aren’t enough points to tell
D.
f parentheses x
2 2
f before x
b 2 − 4ac if a = -1, b = 4, c = 2 and write in simplest form.
B.
2 6
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A.
f of x
N
4. Evaluate f(x) =
D.
o
C.
Can’t tell without a coordinate graph
ot
R
3. How do you read the function notation “f(x)”
A. f multiplied by x
B.
Yes
Cannot
determine
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r
A.
D.
od
A.
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Show your work on a separate sheet of paper and choose the best answer.
5. What are the x-intercepts of the function
D.
C.
24
Not enough
information to
evaluate
h(x) = (x + 3) (x + 2)
A.
3 and 2
B.
6
C.
-3 and -2
D.
Cannot tell from the function.
m
6. Geanie is collecting data about her commute to school. What variable is probably NOT
relevant?
Distance from school to home
B.
Geanie’s walking pace
C.
Time it takes to get to school
D.
Geanie’s height
Sa
A.
QUAD1 – SP28
1.3
KNOWLEDGE CHECK (QUAD1)
uc
e
Show your work on a separate sheet of paper and write your answers on this page.
QUAD1.1 Functions Revisited
Does the graph represent a function of x?
Explain how you know.
2.
If it is a function, state the domain and range.
QUAD 1.2 Modeling with Quadratic Functions 1
ep
r
1.
od
Refer to the graph at the right.
Matty put a pencil on a projector and then moved the projector away from the wall.
What do you think will happen as he moves the projector? What are two variables
that Matty might measure? Do you think the variables you selected are related? If
so, how?
4.
Graph y = x2 + 3 and y = x2 – 3. How are these graphs the same? How are these
graphs different?
o
N
ot
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3.
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QUAD 1.3 Creating Parabolas From Lines
For the function f(x) = (x – 1)(x):
Name the x-intercept(s).
6.
Name the y-intercept(s).
7.
Find the vertex.
m
5.
Graph the function.
9.
Describe the shape of the function.
Sa
8.
QUAD1 – SP29
1.3
Sa
m
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D
o
N
ot
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ep
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od
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This page is left blank intentionally.
QUAD1 – SP30
1.3
Sa
m
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D
o
N
ot
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ep
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od
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QUAD1 – SP31
1.3
Sa
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o
N
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QUAD1 – SP32
Here are some questions to review with your young mathematician.
For the function g(x) = (x – 1)(x + 7)
od
1. Graph the function.
ep
r
2. Name the x-intercept(s).
3. Name the y-intercept(s).
4. Find the vertex.
ot
R
5. Graph the function.
uc
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HOME-SCHOOL CONNECTION (QUAD1)
6. State the domain
N
7. State the range.
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D
o
Parent (or Guardian) signature _____________________________
QUAD1 – SP33
SELECTED COMMON CORE MATHEMATICS STANDARDS
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is
the set of ordered pairs consisting of an input and the corresponding output.
8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change
and initial value of the function from a description of a relationship or from two (x, y) values, including
reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function
in terms of the situation it models, and in terms of its graph or a table of values.
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a
rough graph of the function defined by the polynomial.
uc
e
8.F.1
od
A-APR-3
Create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales.
A-REI-10
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the
coordinate plane, often forming a curve (which could be a line).
F-IF-1
Understand that a function from one set (called the domain) to another set (called the range) assigns to
each element of the domain exactly one element of the range. If f is a function and x is an element of its
domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the
equation y = f(x).
F-IF-7a
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases: Graph linear and quadratic functions and show intercepts,
maxima, and minima.
N
ot
R
ep
r
A-CED-2
STANDARDS FOR MATHEMATICAL PRACTICE
o
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
pl
e:
D
MP1
MP2
MP3
MP4
MP5
MP6
MP7
MP8
copyright 2012
Sa
m
First printing
QUAD1 – SP34

QUAD1 QUADRATIC FUNCTIONS AND EQUATIONS Student Packet 1: Introduction to Quadratic Functions

Transcription

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