Linear Functions Quadratic Functions and Models

linear function s

A linear function is a function of the form
f ( x) = mx + b

Note: The graph of a linear function is a line with the
slope m and the y-intercept b.
The domain of a linear function is the set of all real
numbers.

Example: Graph the function f ( x) = −2x + 4.

Identifying Linear Functions by Using Average Rate of
Change

The average rate of change Δy/Δx of a linear function
f ( x) = mx + b is constant and equal to the slope m.

Proof:

Important Note: A function f ( x) is linear if and only if
the average rate of change is constant .

Example: Find the average rate of change Δy/Δx of
f( x ) = (2/9)x − 3 from x₁ to x₂ (x₁ ≠ x₂ ).

Example: A manufacturer has been selling 1000
television sets a week at $450 each. A market survey
indicates that for each $10 rebate offered to the buyer, the
number of sets sold will increase by 100 per week.
Determine whether the relation between the price p and
demand x is linear and, if so, ex press p as a linear
function of the demand x.

Increasing, Decreasing, and Constant Linear Functions

Theorem: The linear function f (x) = mx + b over its
entire domain is
increasing if m > 0
decreasing if m < 0
constant if m = 0

Example: Determine which of the fol lowing linear
functions is increasing, decreasing, and constant.

Mathematical Models

When using a graphing utility to construct a model that
approximates the data most accurately, we refer to linear,
quadratic, exp onential , and etc. regression feature. The
correlation coefficient r , −1≤ r ≤1, (or R) shows how
well the model fits the data. The closer the value of |r| (or
|R| ) to 1, the better the fit.

Steps of Constructing a Model by Using Data

Example: The table below represents average weight (in
pounds) of American women of age 20-29 years since
year of 1960.

Step 1. Plot ordered pairs using rectangular coordinates –
draw a scatter diagram. Identify, if possible, what type
of relation exists between the variables .

Step 2. Use a graphing utility to draw a scatter diagram.
Use the regression feature to find the function that fits the
data. Use the correlation coefficient to justify how
accurately the model fits the data.

Step 3. Graph the line of best fit in the same window
with the scatter diagram.

According to your model, what would be the average
weight in year 2009 ( x = 49):

Quadratic Function

A quadratic function is a function of the form
f (x) = ax² + bx + c
where a, b, and c are real numbers and a ≠ 0.

Example. Use the graph of f (x) = x² as a reference to
draw the graphs of the following functions:

By completing the square, the equation of a
quadratic function
f (x) = ax² + bx + c , a ≠ 0,

can be put in the form
f (x) = a(x − h)² + k ,

where or k = f (h).

The Graph of f (x) = ax² + bx + c (a ≠ 0) is _________
Vertex:
Axis:

Parabola opens up if _______; f (h) = k is the
minimum value of f.
Parabola opens down if _______; f (h) = k is the
maximum value of f.

Compared with the graph of f (x) = x², the graph of
f (x) = ax² + bx + c is:
stretched vertically if a >1;
compressed vertically if 0 < a <1.

Example: Find the vertex, axis, intercepts, domain,
and range. Graph the parabola.
f (x) = 2x² +12x +10

Example: Find the equation of the quadratic function
that has the indicated vertex and whose graph passes
through the given point.
Vertex: (−2,−3)
Point: (−1,0)

Example: Suppose that a baseball is tossed straight
up, and its height s (in feet) as a function of time t (in
seconds) is given by s(t) = −16t² + 64t + 6, with t = 0
corresponding to the instant when the ball is released.

When does the ball reach the maximum height?

What is the maximum height of the ball?

Enclosing the Most Area:
A builder has 800 feet of fencing left over from a job.
He wants to fence in a rectangular plot of land except
for a 20-foot strip to be used as a driveway.

Express the area A of the plot as a function of x.

What is the domain of A?

For what x is the area A a maximum?

Using a graphing utility to find a quadratic function of
best fit
Example: The following data shows the growth in
myspace.com participation over a period of 31 months.

(a) Use a graphing utility to draw a scatter diagram and
comment on the type of relation that might exist between
the variables.

(b) Use the quadratic regression feature to fit a quadratic
equation to the data.