Transcript 2.3_3.1 - MSBMoorheadMath

COLLEGE ALGEBRA
2.3 Linear Functions
2.4 Quadratic Functions
3.1 Polynomial and Rational Functions
2.3 Slopes of Lines
A function that can be written in the form f(x) = mx + b is called
a linear function because its graph is a straight line.
Linear functions have a constant rise or fall. This rise or fall is
called the slope.
The slope m of the line pass through the points 𝑃1 𝑥1 , 𝑦1 and
𝑃2 𝑥2 , 𝑦2 with 𝑥1 ≠ 𝑥2 is given by
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
2.3 Slopes of Lines
Find the slop of the line passing through 𝑃1 1,2 , 𝑃2 3,6
Find the slop of the line passing through 𝑃1 −3,4 , 𝑃2 1, −2
2.3 Slopes of Lines
Find the slop of the line passing through 𝑃1 3,7 , 𝑃2 3,2
Vertical Lines
Find the slop of the line passing through 𝑃1 1,5 , 𝑃2 4,5
Horizontal Lines
2.3 Slopes of Lines
Graph: 𝑓 𝑥 = 2𝑥 − 1
2.3 Slopes of Lines
Graph: 3𝑥 + 2𝑦 = 4
2.3 Finding the Equation of a Line
We can find the equation of a line provided we know its slope
and at least one point on the line….then we can use PointSlope Form
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Find the equation of the line with slope -3 that passes through
the (-1, 4).
2.3 Finding the Equation of a Line
Find the equation of the line that passes through A (-2,4) and
B(2,-1)
2.3 Parallel and Perpendicular Lines
Two nonintersecting lines in a plane
are parallel. Their slopes are
equivalent to one another.
Two lines are perpendicular if and
only if they intersect at a 90⁰ angle.
Their slopes are the opposite and
reciprocal of one another
2.3 Parallel and Perpendicular Lines
Find the equation of the line whose graph is parallel to the
graph of 2x – 3y = 7 and passes through the point P(-6, -2)
2.3 Parallel and Perpendicular Lines
Find the equation of the line whose graph is perpendicular to
4
the graph of 𝑦 = 𝑥 − 2 and passes through the point P(-4,1).
3
2.3 Applications of Linear Functions
The bar graph on page 193 is based on data from the Nevada
Department of Motor Vehicles. The graph illustrates the
distance d (in feet) a car travels between the time a driver
recognizes an emergency and the time the brakes are applied
for different speeds.
a. Find a linear function that models the reaction distance in
terms of speed of the car by using the ordered pairs (25, 27)
(55, 60).
b. Find the reaction distance for a car traveling at 50 miles per
hour.
2.3 Applications of Linear Functions
A rock attached to a string is whirled horizontally about the
origin in a circular counter-clockwise path with radius 5 feet.
When the string breaks, the rock travels on a linear path
perpendicular to the radius OP and hits a wall located at
y = x + 12
Where x and y are measured in feet. If the string breaks when
the rock is at P(4,3), determine the point at which the rock hits
that wall.
2.4 Quadratic Functions
A quadratic function of x is a function that can be represented
by an equation of the from
𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Where a, b, and c are real numbers and 𝑎 ≠ 0
When graphed if a > 0 the graph will open upwards.
When graphed if a < 0 the graph will open downwards.
The vertex of a parabola is the either the lowest/highest point
depending upon which way the graph opens.
2.4 Quadratic Functions
A graph is symmetric with respect to a line L if for each point
P on the graph there is a point H on the graph such that the line
L is the perpendicular bisector of the line segment PH.
2.4 Quadratic Functions
Every quadratic function f is given by 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 can
be written in the standard form of a quadratic function,
𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 + 𝑘, 𝑎 ≠ 0
The graph of f is a parabola with vertex (h, k). The parabola
opens up if a > 0, and it opens down if a < 0. The vertical line x
= his the axis of symmetry of the parabola.
Example:
𝑓 𝑥 = (𝑥 − 3)2 −4
Parabola opens:
Vertex is at:
Line of Symmetry: x =
2.4 Quadratic Functions
Use the technique of completing the square to find the standard
form of 𝑔 𝑥 = 2𝑥 2 − 12𝑥 + 19. Sketch the graph.
2.4 Quadratic Functions
Sometimes finding the vertex by completing the square can get
tricky so, on page 202 there is a proven formula on how to find
the vertex with proving the ending formula as follows:
Vertex Formula:
The coordinates of the vertex 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 are
𝑏
𝑏
− ,𝑓 −
2𝑎
2𝑎
2.4 Quadratic Functions
Use the vertex formula to find the vertex and standard form of
𝑓 𝑥 = 2𝑥 2 − 8𝑥 + 3
2.4 Max and Min of Quad Function
Note that the previous example the graph opens up and we
notice that the lowest point of the graph is the vertex of the
parabola. Which also means the y-coordinate is the minimum
value of that function. We can use this to determine the range.
The range of the previous function is 𝑦 𝑦 ≥ −5
What is the range of the parabola to the
left?
2.4 Max and Min of Quad Function
Maximum or Minimum of a Quadratic Function
If a > 0 then the vertex (h, k) is the lowest point on the graph of
𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 + 𝑘, and the y – coordinate k of the vertex is
the minimum value of the function f.
If a < 0 then the vertex (h, k) is the highest point on the graph of
𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 + 𝑘, and the y – coordinate k of the vertex is
the maximum value of the function f.
2.4 Max and Min of Quad Function
Find the maximum or minimum value of each quadratic
function. State whether the value is a maximum or a minimum.
a. 𝐹 𝑥 = −2𝑥 2 + 8𝑥 − 1
2.4 Max and Min of Quad Function
Find the maximum or minimum value of each quadratic
function. State whether the value is a maximum or a minimum.
b. G 𝑥 = 𝑥 2 − 3𝑥 + 1
2.4 Applications of Quadratic Functions
A long sheet of tin 20 inches wide is to be made into a trough by
bending up two sides until they are perpendicular to the bottom.
How many inches should be turned up so that the trough will
achieve its maximum carrying capacity?
3.1 Division of Polynomials
We will spend most of Chapter 3 with finding zeros of
polynomial functions. We can first use division of polynomials.
To divide a polynomial by a monomial…we divide each term of
the polynomial by the monomial.
16𝑥 3 − 8𝑥 2 + 12𝑥
4𝑥
3.1 Division of Polynomials
What if we have a polynomial divided by a binomial?
We will then need to divide using a long division method which
will result in a remainder and quotient. ** Make sure that each
polynomial is written in descending order.**
(6𝑥 3 − 16𝑥 2 + 23𝑥 − 5) ÷ (3𝑥 − 2)
3.1 Division of Polynomials
** Make sure that each polynomial is written in descending
order.**
(−5𝑥 2 − 8𝑥 + 𝑥 4 + 3) ÷ (𝑥 − 3)
3.1 Synthetic Division
A procedure called synthetic division can make the division
process more quick. In synthetic division we do not use
variables, but rather just the coefficients.
We can write the ending result from synthetic division in
fractional form.
Use synthetic division to divide…
𝑥 4 − 4𝑥 2 + 7𝑥 + 15 ÷ 𝑥 + 4
3.1 Remainder Theorem
𝑃 𝑥 = 𝑥 2 + 9𝑥 − 16 divided by 𝑥 − 3 which is the same as 𝑃(3)
3.1 Factor Theorem
3.1 Reduced Polynomials
The previous answer we just found of (x+5) is called a
reduced polynomial or a depressed polynomial.
3.1 Reduced Polynomials