Katie W's Math Blog!

Start a blog in which to record your thoughts about precalculus mathematics. Your first entry should discuss include the things listed here.

• Sketches of graphs from real-world information
• Familiar kinds of functions from previous courses and from this unit
• How to dilate and translate the graph of a function
• Any difficulties or misconceptions you have but overcame
• Any topics about which you are still unsure

10.03.06

Sketches of Graphs from Real-World Information

In class, we sketched graphs that could relate to real-world information. The goal was to take our knowledge of sketching graphs and combine it with a story problem. For example, one of our problems stated that a weightlifter is lifting weights over a period of time. In order to graph this problem, determining what the x and y axes are is necessary. The different weights the weightlifter is lifting is placed on the y axis in lbs, and the time the weightlifter is holding the weight up is placed on the x axis in sec. Using common knowledge, the sketched graph will show that the more weight that is held above the weightlifter's head, the less time he/she will be able to hold it up.

Familiar Kinds of Functions from Previous Courses and from this Unit

In class we reviewed familiar functions including linear functions, polynomial functions, and quadratic functions. Linear functions are usually straight lines that change at a constant rate. Polynomial functions are functions that cross the x-axis a certain amount of times and can have more than one vertex. Quadratic functions change directions one time and have only one vertex. All of these functions have all real numbers for domains.

In this unit we learned five new types of functions including rational functions, direct variation functions, inverse variation functions, exponential functions, and power functions.

A rational function is the quotient to two polynomial equations. In a rational function, vertical and horizontal asymptotes can be found. This is where the line almost touches the x or y axis. Also, holes can be found when seeing if the numerator and denominator both equal zero. Sometimes, when looking at rational functions, the x and y-intercepts can be found. To find the x-intercept, the numerator must equal zero while the denominator does not. To find the y-intercept, substitute x = 0 and solve.

A direct variation function is usually a straight line graph that goes through zero. The domain for this function is x is greater than or equal to zero.

An inverse variation function shows that both of the axes are asymptotes. This creates two palabra-looking lines. The domain is usually x greater than zero.

An exponential function features a line that crosses the y-axis and the x-axis is an asymptote.

Finally, power functions include a power in the equation. These functions usually include nonnegative real numbers.

How to dilate and translate the graph of a function

In this unit we learned how to dilate and translate graphs.

Vertical Dilation: When vertical dilating a graph, you must recognize the equation. Example: 2f(x). The number, 2, is on the outside of f(x), making it vertical. The graph stretches out, or sometimes shrinks in, but the x-intercepts stay the same.

Vertical Translation: Example of vertical translation equation: f(x) + 3. The number, 3, is on the outside of f(x) again. So it is concluded as a vertical equation. This graph moves up or down vertically the number indicated in the equation. In this case, the whole graph moves up 3 units up.

Horizontal Dilation: Example of horizontal dilation: f(1/2x). The number, 1/2, is inside f(x), making the figure move horizontally. Horizontal dilation stretches or shrinks the graph horizontally. In this case, because the number is f(1/2x) the graph will stretch two times wider than the original graph.

Horizontal Translation: Example of horizontal translation: g(x) = f (x-3). The number, -3, is inside f(x). Horizontal translation moves the graph horzontally either positive n units or negative n units. The graph moves the opposite amount of spaces as given. In this case, because the number is -3, we take the opposite, positive 3, and move all points positive 3 units.

Difficulties I have overcome

At the beginning of this unit, when learning the different types of functions, I had trouble remembering what the equations and graphs should look like. Although I still am not completely confident, I have a better idea of what each type of graph looks like by practicing.

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Posted by: period1kw    in: My entries

Modified on October 5, 2006 at 12:37 AM