Precalculus
Functions and Graphs
Chapter 2 Section 1
Linear and Quadratic Functions and Modeling
Essential Question: How are linear and quadratic functions used in “real-world”
problems?
Student Objectives: The student will learn how to identify the parts of a polynomial.
The student will learn how to identify the parts of the graph of linear and quadratic functions.
The student will identify the slope as a rate change.
The student will learn how to calculate a linear regression equation.
The student will use the correlation coefficient to identify the strength of a regression equation.
The student will learn how to calculate a quadratic regression equation.
The student will learn how to use the method of completing the square to determine the vertex of a quadratic equation.
The student will apply to concepts of quadratic equations to solve “real-world” problems.
Terms:
Acceleration
Average rate of change
Coefficient
Complete the square
Constant function
Degree
Free fall equation
Gravity
Height
Horizontal line
Initial value
Leading coefficient
Linear
Negative correlation
No correlation
Oblique line
Positive correlation
Quadratic function
Slant line
Slope
Standard quadratic form
Velocity
Vertex
Vertex form
Vertical line
Graphing Calculator Skills:
The student is expect to understand how to do the following tasks on their calculator
a. Linear Regression
STAT
EDIT
Type data points into L1 and L2
2nd QUIT
STAT
CALC
LinReg (ax +b) —> Option 4
Xlist: L1
Ylist: L2
FreqList: (leave blank)
Store RegEQ: VAR, Y-VARS, FUNCTION, Y1
Highlight Calculate and press ENTER
b. Quadratic regression
STAT
EDIT
Type data points into L1 and L2
2nd QUIT
STAT
CALC
QuadReg (ax +b) —> Option 5
Xlist: L1
Ylist: L2
FreqList: (leave blank)
Store RegEQ: VAR, Y-VARS, FUNCTION, Y1
Highlight Calculate and press ENTER
c. Turn Diagonistics ON
2nd Catalog (0 key)
Scroll down until you get to DiagnosticOn
Press ENTER
Press ENTER
d. Graph a linear functions with an appropriate window
Make sure you can see the x-intercept and the y-intercept in your window.
e. Graph a quadratic function with an appropriate window
Make sure you can see the vertex
Make sure you can see the y-intercept
Make sure you can see the x-intercept (if applicable)
Theorems and Definitions
Polynomial Function
Let n be a nonnegative integer and let a0, a1, a2, a3, …, an-2, an-1, an be real
numbers where an ≠ 0, the function given by
is a polynomial function of degree n. The leading coefficient is an. The zero function is a polynomial function. It has no degree and no
leading coefficient.
Polynomial Functions of No and Low Degree
Name Form Degree
Zero function Undefined
Constant function 0
Linear function 1
Quadratic function 2
Theorem: Constant Rate of Change
A function defined on all real numbers is a linear function if and only if (iff) it
has a constant nonzero rate of change between any two points on its graph.
Hence, it must have a slope!
The Nature of a Linear Function
Point of View Characterization
Verbal Polynomial of degree 1
Algebraic
Graphical A slant (oblique) line with a slope of m and
a y-intercept of 0, b)
Analytical A function with a constant nonzero rate of change:
the function is increasing if m > 0;
the function is decreasing if m < 0; and
the initial value of the function is f(0) or b.
Properties of the Correlation Coefficient
1.
2. When , there is a positive correlation.
3. When , there si a negative correlation.
4. When or there is a perfect linear correlation.
5. When , there is no linear correlation
6. When , there is a very strong linear correlation.
7. When , there is a strong linear correlation.
8. When , there is a moderate linear correlation.
9. When , there is a weak linear correlation.
9. When , there is little to no linear or very weak linear correlation.
Process of Completing the Square
1. Factor the leading coefficient out of the first and second terms
2. Take have of the x value and square it. Then multiply the leading coefficient,
the squared value, and negative one. Place this value at the very end of the
function.
3. Simplify the expression.
4. Factor the expression.
The Nature of a Quadratic Function
Point of View Characterization
Verbal Polynomial of degree 2
Algebraic
Graphical A parabola with vertex (h, k) and axis of symmetry at
x = h; opens upward if a > 0 and downward if a < 0; and
initial value = y-intercept = f(0) = c.
Point of View Characterization
Verbal Polynomial of degree 2
Algebraic
Graphical A parabola with vertex (h, k) and axis of symmetry at
x = h; opens upward if a > 0 and downward if a < 0; and
initial value = y-intercept = f(0) = c.
Vertical Free-Fall Motion
The height, s, and the vertical velocity, v, of an object in free fall are given by
where t in time (in seconds), is the acceleration due to
gravity, is the initial vertical velocity of the object, and is its initial
height.
Sample Questions:
1. Determine the linear function that satisfies the given conditions. Graph the linear
function.
2. Use your calculator to determine the linear regression equation. Use the correlation
coefficient to describe the linear relationship. Plot the regression function and the data
points on the provided grid.
3. Complete the square to determine the vertex of the quadratic function. Sketch the
quadratic function on the provided grid.
4. Use your calculator to determine the linear regression equation. Use the correlation
coefficient to describe the linear relationship. Plot the regression function and the data
points on the provided grid. The calculator will provide r2 and not r. You will need to
take the square root of r2 to get the value of r.
5. Determine the equation of a quadratic function that has a vertex of (-3, 8) and goes
through the point (4, 158).
6. Susan is at the top of a 200-ft cliff. She throws her English book off the cliff with an
upward velocity of 44 feet per second.
(a) What is the maximum height that the book will reach?
(b) What is the height of the book at 4 seconds?
(c) What is the velocity of the book after 4 seconds?
(d) How long will it take the book to hit the ground?
(e) What is the velocity of the book when it impacts the ground in miles per
hour? You will need to convert the units of your answer.
Homework: Pages 169 - 173 Exercises: #9, 21, 31, 37, 43, 49, 61, 71, 73, and 75.
Exercises: #10, 22, 30, 36, 44, 50, 62, 72, 74, and 76.