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Exploring Linear and Piecewise Functions: Applications and Examples

Dive into the world of linear, quadratic, polynomial, rational, and power functions, with practical applications such as enzyme kinetics. Learn to graph piecewise functions effectively. Discover the significance of absolute value functions and solve for intercepts with ease.

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Exploring Linear and Piecewise Functions: Applications and Examples

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  1. §0.2 Some Important Functions

  2. Section Outline • Linear Equations • Applications of Linear Functions • Piece-Wise Functions • Quadratic Functions • Polynomial Functions • Rational Functions • Power Functions • Absolute Value Function

  3. Linear Equations

  4. Linear Equations CONTINUED

  5. Applications of Linear Functions EXAMPLE • (Enzyme Kinetics) In biochemistry, such as in the study of enzyme kinetics, one encounters a linear function of the form , where K and V are constants. • If f (x) = 0.2x + 50, find K and V so that f (x) may be written in the form, . • Find the x-intercept and y-intercept of the line in terms of K and V. SOLUTION (a) Since the number 50 in the equation f (x) = 0.2x + 50 is in place of the term 1/V (from the original function), we know the following. 50 = 1/V Explained above. 50V = 1 Multiply both sides by V. Divide both sides by 50. V = 0.02 Now that we know what V is, we can determine K. Since the number 0.2 in the equation f (x) = 0.2x + 50 is in place of K/V (from the original function), we know the following.

  6. Applications of Linear Functions CONTINUED 0.2 = K/V Explained above. 0.2V = K Multiply both sides by V. Replace V with 0.02. 0.2(0.02) = K 0.004 = K Multiply. Therefore, in the equation f (x) = 0.2x + 50, K = 0.004 and V = 0.02. (b) To find the x-intercept of the original function, replace f (x) with 0. This is the original function. Replace f (x) with 0. Solve for x by first subtracting 1/V from both sides.

  7. Applications of Linear Functions CONTINUED Multiply both sides by V/K. Simplify. Therefore, the x-intercept is -1/K. To find the y-intercept of the original function, we recognize that this equation is in the form y = mx + b. Therefore we know that 1/V is the y-intercept.

  8. Piece-Wise Functions EXAMPLE Sketch the graph of the following function . SOLUTION We graph the function f (x) = 1 + x only for those values of x that are less than or equal to 3. Notice that for all values of x greater than 3, there is no line.

  9. Piece-Wise Functions CONTINUED Now we graph the function f (x) = 4 only for those values of x that are greater than 3. Notice that for all values of x less than or equal to 3, there is no line.

  10. Piece-Wise Functions CONTINUED Now we graph both functions on the same set of axes.

  11. Quadratic Functions

  12. Polynomial Functions

  13. Rational Functions

  14. Power Functions

  15. Absolute Value Function

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