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Chapter 2

Modeling with Linear and Quadratic Functions

Section 2.1 deals with linear functions.

- A linear function has a constant rate of
change.

Explain what that means in your own words.

- Slope

What does the slope of a line tell you about the graph?

How can you use two points on a line to calculate the slope of a line?

How can the slope tell you if the function is increasing or decreasing?

How can slope be interpreted as a rate of change in a story problem?

- Equations

What is slope-intercept form?

How can you use slope-intercept form to determine the slope and y-intercept of the graph?

What is point-slope form?

How are parallel lines related?

How are perpendicular lines related?

How can you use a linear equation in a story problem?

How can you write the equation for a horizontal line?

How can you write the equation for a vertical line?

Is a horizontal line an example of a function?

Is a vertical line an example of a function?

How can you write the equation of a line from its graph?

Section 2.2 deals with linear equations and models.

- You will need to know how to recognize each of the following special cases: identity, conditional, and contradiction.
- You will have to be able to use algebra as well as graphing to solve equations.
- You will have to be able to recognize extraneous solutions.
- You will have to be able to solve story problems involving distance/rate/time, traveling in wind or current, mixture problems, and other applications.

Section 2.3 deals with **quadratic
functions**. Fill
in the blanks below. Try to do it without using your book or notes.

- Quadratic
functions are transformations on f(x) = x
^{2 }.

- Standard form
for a quadratic function is f(x) = ________________.

- Vertex form
for a quadratic function is f(x) = __________________.

What does the value of "a" tell you? _________________________.

Where is the vertex located? _______________________________.

- You can convert a quadratic function from
standard form to vertex form by completing the square.
There are examples in your text starting on page 153. You need to be
able to do this without looking at an example or using your notes, book, or
calculator.

1. Factor out the leading coefficient.

2. Add and subtract (b/2)^{2 }.

3. Simplify.

Try
an example now by converting the following equation to vertex form:

(The answer is
here; don't look until you've worked this
problem for yourself!)

f(x) = -2x^{2 }+ 9x - 5

Section 2.4 introduces complex numbers.

- What is standard form for a complex number?
- What is the imaginary unit?
- You will need to know how to perform operations on complex numbers.
- What is the
conjugate for a given complex number?

Section 2.5 deals with
Quadratic Equations and Models

- The graph of any quadratic function is symmetric about the vertical line through its vertex. Know what this vertical line is called and be able to find its equation.
- You will need to know how to determine
the x-intercepts of the graph of a quadratic
function. These values are also known as zeros
or roots of the function.

1. Begin by moving all terms to one side of the equation. There will be a

zero on the other side of the equation.

2. You may be able to factor. Then set each factor equal to zero and solve.

3. If you can't use factoring, you can always complete the square.

Then solve by isolating the square; taking the square root (plus/minus);

then solving for the variable.

4. Using the quadratic formula is also always an option.

Memorize it; you will find it written at the top of page 185. - You will need to know how to determine the formula for a quadratic function when you are given points on the graph.
- What is the discriminant and what does it tell you?
- You will need to know how to use a
quadratic function to solve story problems. These typically involve
finding a maximum or minimum
of some quantity. Note that the max or min of a quadratic function
(without domain restrictions) will be located at the vertex.

Section 2.6 introduces some additional equation-solving techniques.

- How can you use algebra (and how can you use a graph) to solve equations involving one or more than one radical?
- How can you recognize an equation that is of quadratic type?
- How can you use algebra (and how can you use a graph) to solve equations of quadratic type?
- What is an extraneous solution?
- Why is it important to check your answer?

Section 2.7 deals with solving inequalities.

- How can you use a graph to solve an inequality?
- How can you use algebra to solve an inequality?
- How do you solve an inequality involving absolute value?
- How do you solve an inequality involving a quadratic?
- How can you use an inequality to solve a story problem?