1 / 24

Functions and Graphs

Functions and Graphs. 1.2. Symmetric about the y axis. F U N C T I O N S. Symmetric about the origin. -7. -2. -1. 1. 3. 5. 7. -6. -5. -4. -3. 0. 4. 6. 8. 2. Even functions have y -axis Symmetry. 8. 7. 6. 5. 4. 3. 2. 1. -2. -3. -4. -5. -6. -7.

garey
Download Presentation

Functions and Graphs

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. Functions and Graphs 1.2

  2. Symmetric about the y axis FUNCTIONS Symmetric about the origin

  3. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Even functions have y-axis Symmetry 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

  4. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Odd functions have origin Symmetry 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

  5. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 x-axis Symmetry We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7

  6. A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even. Is this function even? YES Is this function even? NO

  7. A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. Is this function odd? NO Is this function odd? YES

  8. If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER. Got f(x) back so EVEN.

  9. Library of Functions You should be familiar with the shapes of these basic functions.

  10. Linear Functions Equations that can be written f(x) = mx + b slope y-intercept The domain of these functions is all real numbers.

  11. f(x) = 3 f(x) = -1 f(x) = 1 Constant Functions f(x) = b, where b is a real number Would constant functions be even or odd or neither? The domain of these functions is all real numbers. The range will only be b

  12. If you put any real number in this function, you get the same real number “back”. f(x) = x Identity Function f(x) = x, slope 1, y-intercept = 0 Would the identity function be even or odd or neither? The domain of this function is all real numbers. The range is also all real numbers

  13. Square Function f(x) = x2 Would the square function be even or odd or neither? The domain of this function is all real numbers. The range is all NON-NEGATIVE real numbers

  14. Cube Function f(x) = x3 Would the cube function be even or odd or neither? The domain of this function is all real numbers. The range is all real numbers

  15. Square Root Function Would the square root function be even or odd or neither? The domain of this function is NON-NEGATIVE real numbers. The range is NON-NEGATIVE real numbers

  16. Reciprocal Function The domain of this function is all NON-ZERO real numbers. Would the reciprocal function be even or odd or neither? The range is all NON-ZERO real numbers.

  17. Absolute Value Function The domain of this function is all real numbers. Would the absolute value function be even or odd or neither? The range is all NON-NEGATIVE real numbers

  18. Recall: These are functions that are defined differently on different parts of the domain. WISE FUNCTIONS

  19. What do the graphs of these things look like?

  20. This means for x’s less than 0, put them in f(x) = -x but for x’s greater than or equal to 0, put them in f(x) = x2 What does the graph of f(x) = -x look like? What does the graph of f(x) = x2 look like? Remember y = f(x) so let’s graph y = - x which is a line of slope –1 and y-intercept 0. Remember y = f(x) so lets graph y = x2 which is a square function (parabola) Since we are only supposed to graph this for x  0, we’ll only keep the right half of the graph. Since we are only supposed to graph this for x< 0, we’ll stop the graph at x = 0. This then is the graph for the piecewise function given above.

  21. For x > 0 the function is supposed to be along the line y = - 5x. For x = 0 the function value is supposed to be –3 so plot the point (0, -3) For x values between –3 and 0 graph the line y = 2x + 5. Since you know the graph is a piece of a line, you can just plug in each end value to get the endpoints. f(-3) = -1 and f(0) = 5 Since you know this graph is a piece of a line, you can just plug in 0 to see where to start the line and then count a – 5 slope. open dot since not "or equal to" So this the graph of the piecewise function solid dot for "or equal to"

  22. You try one:Graph the function described by:

  23. Composite Functions “f of g of x”

  24. Essential Question How do you determine whether a function is even, odd or neither

More Related