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ALGEBRA 2, VERTEX FORM OF A LINEAR EQUATION
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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) = x2
.
- 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) = -2x2 + 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?
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