Section 2.1 Properties of Linear Functions
A linear function is a function that when graphed produces a straight line. The slopeintercept form for the equation of a linear function is y  mx  b or f x  mx  b ,
where m is the slope of the line and (0, b) is the y-intercept of the function.
The slope of a line is a very important characteristic of the linear function. Because of its
importance, we need to find an algebraic way of calculating the slope of a line.
Given any two known ordered pairs on a line, say x1 , y1  and x 2 , y 2  , the slope of the
rise y y 2  y1
line, denoted by m, is found algebraically as follows: m 
.


run x x 2  x1
When we use a linear equation to represent a real-world situation, the slope can be a very
interesting, as well as an important, quantity. One real-world application of slope is the
grade of a road, which is the measure of how steep the road is. Grade is represented as a
percent. For example, a 5% grade means that for every horizontal distance of 100ft, the
road drops or rises vertically by 5ft.
Another real world application occurs when the independent variable, x, in a linear
equation represents time. In this case, the slope tells us how the dependent variable, y,
changes over a period of time. We call this the average rate of change of the dependent
variable. From the average rate of change, we can sometimes predict what value a
quantity will have in the future, or what value it had in the past.
Types of Lines
There are three types of lines created when we graph a linear function.
 Horizontal or constant lines – slope is zero (i.e. m  0 )
 Increasing lines – slope is positive (i.e. m  0 )
 Decreasing lines – slope is negative (i.e. m  0 )
There are two special points of a linear function that increases or decreases. They are the
x-intercept (also called the zero) and the y-intercept. The y-intercept can be read from
the function as described above. The x-intercept can be found algebraically as follows.
To find the x-intercept algebraically
1. Substitute y  0 into the equation.
2. Solve for x.
3. Record the intercept as an ordered pair in the form x, 0 .
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Graphing Linear Functions
There are several methods for graphing linear functions.
To graph a line using the y-intercept and the slope of the line:
1. Plot the y-intercept.
2. Count the rise and run from the located point.
a. For a positive rise, count upward; for a negative rise, count downward.
b. For a positive run, count to the right; for a negative run, count to the left.
3. Place a second point where the next ordered pair is located.
4. Draw a straight line connecting the two points.
To graph a horizontal line:
1. Locate the real number b on the y axis.
2. Draw a horizontal line through b.
To graph a linear equation using the intercepts:
1. Find the x and y intercepts
2. Plot these ordered pairs in the Cartesian coordinate plane.
3. Draw a single straight line through these points.
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Section 2.2 Building Linear Functions from Data: Direct Variation
Consider two sets of data: Set A = {A1, A2, … , An} and Set B = {B1, B2, …, Bn}. When
A1 corresponds to B1, A2 corresponds to B2, etc. we have a paired data set. In other
words, we can create a set of ordered pairs from the data: (A1, B1), (A2, B2), … , (An, Bn).
When we graph this set of ordered pairs we obtain a scatter plot of the data. Scatter plots
are a useful tool in determining if or how the data are related: linear, quadratic,
exponential, etc.
Once we know how the data are related we can create functions that “fit” the data. For
linear data we obtain equations of the form y  mx  b . This equation is called a
regression line. How well it “fits” the data is determined by a measure called the
correlation coefficient, which is denoted with the variable r. The value of r ranges from
negative one to positive one. The closer its absolute value is to one the better the fit.
Once we have an equation that fits the data we can predict a y-value for any given xvalue, or vice-versa.
To find a linear function that fits the data and the correlation coefficient using your TI-84
calculator, follow the steps outlined below.
1. Press 2nd, CATALOG, D
2. Scroll down using the ▼ key until you get to DiagnosticOn. Press ENTER
twice.
Note: Steps 1 & 2 only have to be done once.
3. Press STAT, Edit
4. Enter data wanted along the horizontal or x-axis under L1.
5. Enter data wanted along the vertical or y- axis under L2.
6. Press STAT, Calc
7. Choose LinReg(ax+b), then press ENTER
Variation
k
, or
x
y  kxz for some constant k. Such equations are called equations of variation where the
constant k is called the constant of proportionality or the variation constant.
There are many real world situations that lead to equations of the form y  kx , y 
Direct Variation
When a situation translates to an equation of the form y  kx , or f x  kx , with k a
nonzero constant, we say that y varies directly as x OR that y is proportional to x.
We can use these facts to create linear equations to predict future values.
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Section 2.3 Quadratic Functions and Their Zeros
A quadratic function is a polynomial function whose degree is two.
A quadratic function can be written in the following forms where a, b, and c are real
numbers and a  0 ..
y  ax 2  bx  c,
Relation form
f ( x)  ax  bx  c,
Function form
2
A quadratic equation is an equation which can be written in the form ax 2  bx  c  0 ,
where a, b, and c are real numbers and a  0 .
The zeros, or x-intercepts, of a quadratic function can be found by solving the equation
f x   0 .
Quadratic equations can be solved by at least one of the following methods.
1. Factoring
2. Square Root Method
3. Completing the square
4. Quadratic formula, which is x 
 b  b 2  4ac
2a
Steps to Factor a Quadratic Equation Completely
1. Factor out the common monomial factor from all terms, if any. In addition, if the
leading coefficient is negative one, factor out a –1.
2. Determine the type of quadratic equation remaining and factor as indicated for
each.
a. Binomial – A 2  B 2  ( A  B)( A  B)
b. Trinomial – There are several cases to consider. If none of them work, it
will not factor.
i. Check to see if it is a perfect-square trinomial. If it is, then
A 2  2 AB  B 2  ( A  B) 2
A 2  2 AB  B 2  ( A  B) 2
4
ii. Is it a quadratic trinomial of the form x 2  bx  c ? Then,
x 2  bx  c  ( x  m)( x  n), if and only if mn = c and n + m = b.
iii. Is it a quadratic trinomial of the form ax 2  bx  c ? Then, use the
ac method to factor it.
 Determine the product ac.
 Determine factors m and n of the product ac whose sum is b.
 Rewrite the trinomial as a four-term polynomial,
ax 2  mx  nx  c .
 Factor the resulting polynomial using grouping.
3. Check your factorization by
a. Making sure you factored completely, and then
b. Re-multiply the factors. The result should be the quadratic equation that
was factored.
The Square Root Method
Suppose you have an equation of the form x 2  p where p is a nonnegative number (i.e.
p  0 ). We can solve this type of equation by step 5 above or we can take a shortcut
using the square root method, which states that if x 2  p and p  0 , then x   p .
An equation is said to be reducible to quadratic (or of quadratic form) if the variable
factor of the leading term is the square of the variable factor in the second variable term.
We can solve these types of equations if we make an appropriate substitution to make
them appear quadratic. This process is often referred to as u substitution.
To solve equations of quadratic form:
1. Make an appropriate substitution so that the equation has been reduced to a
quadratic equation. (Make sure you note what substitution you have made.)
2. Solve the quadratic equation obtained in step 1.
3. Use the values obtained in step 2 to obtain the values of the original variable you
were asked to solve for.
4. Check your answers in the original equation. Discard any solutions which do not
make true equations.
When we solve f x  g x for two functions we are finding the x-values of the points of
intersection. Once we find these values, we need to find the y-values of the points of
intersection by substituting the x-values back into one of the original functions.
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Section 2.4 Properties of Quadratic Functions
All quadratic functions have a U-shaped graph called a parabola. This U may open
upwards or downwards.
The term ax2 in the function is called the quadratic term because it determines whether
or not the graph opens upwards or downwards. The highest point or lowest point of the
parabola is called the vertex of the parabola. Because the vertex is a point, it has an xcoordinate and a y-coordinate, and is recorded as a point (x, y).
Finding the Vertex of a Parabola
1. First, find the x-coordinate. The x-coordinate of the vertex is found by using the
coefficients of the quadratic function in its standard form. The formula for the xb
coordinate is x 
.
2a
2. Second, find the y-coordinate. Once the x-coordinate is found, the y-coordinate is
found by evaluating the quadratic function with the value obtained for the xcoordinate of the vertex.
3. Finally, record your answer as an ordered pair (x, y).
Notes:
 If the parabola opens upward, the vertex’s y-coordinate is the absolute minimum
of the function. Hence the range is [y, ∞).
 If the parabola opens downward, the vertex’s y-coordinate is the absolute
maximum of the function. Hence the range is (-∞, y].
 A vertical line can be drawn through a parabola’s vertex such that the parabola is
divided exactly in two. This line is called the axis of symmetry. The two halves
of the parabola are perfectly symmetrical and are considered to be mirror images
of each other.
Understanding the Effects of the Coefficients
The values of the coefficients a, b, and c in the standard form of a quadratic function
y  ax 2  bx  c , affect its graph.
Coefficient a:
 If a  0 (positive), then the graph opens upward and is said to be concave
upward.
6


If a  0 (negative), then the graph opens downward and is said to be concave
downward.
If a  1 , then the graph is narrow compared to when a = 1.

If a  1 , then the graph is wide compared to when a = 1.
Coefficient c:
The coefficient c is the y-coordinate of the y-intercept of the graph.
Coefficient b:
b
b
, and the y-coordinate is f 
.
2a
 2a 
b
The axis of symmetry is the line graphed by x 
.
2a
The x-coordinate of the vertex is x 
To graph a quadratic function:
 Find the vertex and plot it. Sketch a dotted vertical line through the vertex.
 Find the x-intercepts, if any, and the y-intercept and plot them.
 Find other points and plot them if necessary to find the trend of the graph.
 Connect the points with a smooth curve, as they occur, left to right.
Alternate Form of a Quadratic Function
Let h, k  represent the coordinates of the vertex of a quadratic function. Then an
alternate form of the quadratic function is f x   ax  h   k .
2
If we know the vertex and one other point, say x1 , y1  , on the graph of a quadratic
function, we can find its corresponding function using this alternate form. How?
1. Substitute h, k  and x1 , y1  into the function f x   ax  h   k as follows:
2
y1  a  x1  h   k .
2. Solve for a.
2
3. Write the equation f x   ax  h   k with a, h, and k.
4. Simplify.
2
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Section 2.5 Inequalities Involving Quadratic Equations
A quadratic inequality in one variable is written by replacing the equal symbol in a
quadratic equation in one variable with one of the following order symbols: <, >, ≤, or ≥.
A quadratic inequality in standard form is written in one of the following ways:
ax 2  bx  c  0
ax 2  bx  c  0
ax 2  bx  c  0
ax 2  bx  c  0
where a, b, and c, are real numbers and a ≠ 0.
To solve a quadratic inequality algebraically:
1. Rewrite the inequality in its standard form, if necessary.
2. Replace the order symbol with the equal sign, and then determine the solutions of
the resulting quadratic equation.
3. From the values obtained above, divide the x-axis into appropriate intervals.
4. Take one value from inside of each and every interval formed above and
determine if the inequality is true or false for each of those values.
a. If the inequality is true, then the inequality is true for all values in that
interval. Hence, that interval is a solution set of the inequality.
b. If the inequality is false, then the inequality is false for all values in that
interval. Hence, that interval is not a solution set of the inequality.
5. Write the solution set as a union of all intervals that are valid solution sets of the
inequality. (Keep in mind what type of order symbol is in the inequality when
writing the intervals.)
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Section 2.6 Quadratic Models
Optimization Problems
Optimization is the process of finding the best value of something, which is usually the
largest or smallest quantity the object can be. For example: one may want to maximize
profit or minimize expenses.
There are many real world situations which can be modeled with a quadratic function. In
these situations, the vertex often has an important significance. Why? Because the
minimum or maximum of a quadratic function occurs at the vertex. Thus finding it and
interpreting it correctly is important.
For a vertex (x, y), the y-value is either the minimum or maximum and the x-value is the
quantity that produces the minimum or maximum.
Mathematical Modeling
For a paired data set that shows a quadratic trend, we can find an equation that “fits” the
data. To find this equation in your TI-84 calculator, follow the steps outlined below.
1.
2.
3.
4.
5.
Press STAT, Edit
Enter data wanted along the horizontal or x-axis under L1.
Enter data wanted along the vertical or y- axis under L2.
Press STAT, Calc
Choose QuadReg, then press ENTER
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Section 2.7 Complex Zeros of a Quadratic Model
Solving Equations with Imaginary-Number Solutions
Recall, that for any positive real-number b, if x 2  b , then x   b .
Well, for any negative real-number b, if x 2  b , then x   b  i .
We can use any of the three following methods to solve quadratic equations which have
imaginary-number solutions:
1. Principle of square roots.
2. Completing the square.
3. Quadratic Formula Once in its standard form of ax 2  bx  c  0
x
 b  b 2  4ac
2a
Note: Whenever a complex number is a solution to an equation, its conjugate is also a
solution. In other words, complex numbers always come in conjugate pairs as solutions.
Character of the Solutions of a Quadratic Equation
The radicand of the quadratic formula, b 2  4ac , is called the discriminant because its
value determines what type (or character) of solutions you will obtain.
 If b 2  4ac  0 , the equation has two unequal real-number solutions.
 If b 2  4ac  0 = 0, the equation has one repeated real-number solution, called a
double root.
 If b 2  4ac < 0, the equation has two complex solutions that are not real numbers.
The complex solutions are conjugates of each other.
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Section 2.8 Equations and Inequalities Involving the Absolute Value Function
Absolute Value Equations
Recall that the definition of the absolute value of an expression is the distance of the
expression from zero. Although the expression inside of the absolute value symbol may
be positive or negative, the result (taking the absolute value of the expression) is always
nonnegative (positive or zero).
This holds true when we have an expression containing a variable term inside of the
absolute value symbol.
The standard form of an absolute value equation in one variable is written as f x  c ,
where c  0 .
To solve a linear absolute value equation in one variable algebraically:
1. Set f x  c and f x   c
2. Solve both equations.
3. Record your solutions.
Inequalities Involving Absolute Value
If a is a real number such that a  0 and f x  is any algebraic function, then

f x   a is equivalent to  a  f x  a

f x   a is equivalent to  a  f x  a

f x   a is equivalent to f x  a or f x   a

f x   a is equivalent to f x   a or f x   a
To solve an inequality involving absolute value, write its equivalent form(s) and then
solve.
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