COLLEGE ALGEBRA2.3 Linear Functions

2.4 Quadratic Functions

3.1 Polynomial and Rational Functions

2.3 Slopes of LinesA 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 and with is given by

2.3 Slopes of LinesFind the slop of the line passing through

Find the slop of the line passing through

2.3 Slopes of LinesFind the slop of the line passing through

Vertical Lines

Find the slop of the line passing through

Horizontal Lines

2.3 Slopes of LinesGraph:

2.3 Slopes of LinesGraph:

2.3 Finding the Equation of a LineWe can find the equation of a line provided we know its slope and at least one point on the line….then we can use Point-Slope Form

Find the equation of the line with slope -3 that passes through the (-1, 4).

2.3 Finding the Equation of a LineFind the equation of the line that passes through A (-2,4) and B(2,-1)

2.3 Parallel and Perpendicular LinesTwo 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 90 angle. ⁰Their slopes are the opposite and

reciprocal of one another

2.3 Parallel and Perpendicular LinesFind 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 LinesFind the equation of the line whose graph is perpendicular to the graph of and passes through the point P(-4,1).

2.3 Applications of Linear FunctionsThe 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 FunctionsA 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 FunctionsA quadratic function of x is a function that can be represented by an equation of the from

Where a, b, and c are real numbers and

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 FunctionsA 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 FunctionsEvery quadratic function f is given by can be written in the standard form of a quadratic function,

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:

4

Parabola opens: Line of Symmetry: x =

Vertex is at:

2.4 Quadratic FunctionsUse the technique of completing the square to find the standard form of . Sketch the graph.

2.4 Quadratic FunctionsSometimes 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 are

2.4 Quadratic FunctionsUse the vertex formula to find the vertex and standard form of

2.4 Max and Min of Quad FunctionNote 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

What is the range of the parabola to the

left?

2.4 Max and Min of Quad FunctionMaximum or Minimum of a Quadratic Function

If a > 0 then the vertex (h, k) is the lowest point on the graph of

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

, and the y – coordinate k of the vertex is the maximum value of the function f.

2.4 Max and Min of Quad FunctionFind the maximum or minimum value of each quadratic function. State whether the value is a maximum or a minimum.

a.

2.4 Max and Min of Quad FunctionFind the maximum or minimum value of each quadratic function. State whether the value is a maximum or a minimum.

b. G

2.4 Applications of Quadratic FunctionsA 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 PolynomialsWe 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.

3.1 Division of PolynomialsWhat 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.**

3.1 Division of Polynomials** Make sure that each polynomial is written in descending order.**

3.1 Synthetic DivisionA 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…

3.1 Remainder Theorem

divided by which is the same as

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