1 / 32

College algebra

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.

kassia
Download Presentation

College algebra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. College algebra 2.3 Linear Functions 2.4 Quadratic Functions 3.1 Polynomial and Rational Functions

  2. 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 slopem of the line pass through the points and with is given by

  3. 2.3 Slopes of Lines Find the slop of the line passing through Find the slop of the line passing through

  4. 2.3 Slopes of Lines Find the slop of the line passing through Vertical Lines Find the slop of the line passing through Horizontal Lines

  5. 2.3 Slopes of Lines Graph:

  6. 2.3 Slopes of Lines Graph:

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

  8. 2.3 Finding the Equation of a Line Find the equation of the line that passes through A (-2,4) and B(2,-1)

  9. 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

  10. 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)

  11. 2.3 Parallel and Perpendicular Lines Find the equation of the line whose graph is perpendicular to the graph of and passes through the point P(-4,1).

  12. 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. • 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). • Find the reaction distance for a car traveling at 50 miles per hour.

  13. 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.

  14. 2.4 Quadratic Functions A 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.

  15. 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.

  16. 2.4 Quadratic Functions Every 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:

  17. 2.4 Quadratic Functions Use the technique of completing the square to find the standard form of . Sketch the graph.

  18. 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 are

  19. 2.4 Quadratic Functions Use the vertex formula to find the vertex and standard form of

  20. 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 What is the range of the parabola to the left?

  21. 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 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.

  22. 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.

  23. 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

  24. 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?

  25. 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.

  26. 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.**

  27. 3.1 Division of Polynomials ** Make sure that each polynomial is written in descending order.**

  28. 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…

  29. 3.1 Remainder Theorem divided by which is the same as

  30. 3.1 Factor Theorem

  31. 3.1 Reduced Polynomials The previous answer we just found of (x+5) is called a reduced polynomial or a depressed polynomial.

  32. 3.1 Reduced Polynomials

More Related