Algebra-Trigonometry-CynthiaYoung-3ed42.pdf - c04a.qxd 3:23 PM Page 381 4.1 Quadratic Functions 381 Typically quadratic functions are expressed in

Algebra-Trigonometry-CynthiaYoung-3ed42.pdf - c04a.qxd 3:23...

This preview shows page 1 - 3 out of 10 pages.

4.1 Quadratic Functions 381 EXAMPLE 3 Graphing a Quadratic Function Given in General Form Graph the quadratic function f ( x ) x 2 6 x 4. Solution: Express the quadratic function in standard form by completing the square. Write the original function. f ( x ) x 2 6 x 4 Group the variable terms together. ( x 2 6 x ) 4 Complete the square. Half of 6 is 3; 3 squared is 9. Add and subtract 9 within the parentheses. ( x 2 6 x 9 9 ) 4 Write the 9 outside the parentheses. ( x 2 6 x 9) 9 4 Write the expression inside the parentheses as a perfect square and simplify. ( x 3) 2 5 Now that the quadratic function is written in standard form, f ( x ) ( x 3) 2 5, we follow our step-by-step procedure for graphing a quadratic function in standard form. S TEP 1 The parabola opens up. a 1, so a 0 S TEP 2 Determine the vertex. ( h , k ) (3, 5) S TEP 3 Find the y -intercept. f (0) (0) 2 6(0) 4 4 (0, 4) corresponds to the y -intercept S TEP 4 Find any x -intercepts. f ( x ) 0 f ( x ) ( x 3) 2 5 0 ( x 3) 2 5 ( , 0 ) and ( , 0 ) correspond to the x -intercepts. S TEP 5 Plot the vertex and intercepts (3, 5), (0, 4), ( , 0 ) , and ( , 0 ) . Connect the points with a smooth parabolic curve. Note: and YOUR TURN Graph the quadratic function f ( x ) x 2 8 x 14. 3 - 1 5 L 0.76. 3 + 1 5 L 5.24 3 - 1 5 3 + 1 5 3 - 1 5 3 + 1 5 x = 3 ; 1 5 x - 3 = ; 1 5 Study Tip Although either form (standard or general) can be used to find the inter- cepts, it is often more convenient to use the general form when finding the y -intercept and the standard form when finding the x -intercept. Technology Tip Use a graphing utility to graph the function f ( x ) x 2 6 x 4 as y 1 . x y (3, –5) (0, 4) (0.76, 0) (5.24, 0) Answer: x y (5.4, 0) (2.6, 0) (4, –2) Typically, quadratic functions are expressed in general form and a graph is the ultimate goal, so we must first express the quadratic function in standard form. One technique for transforming a quadratic function from general form to standard form was introduced in Section 1.3 and is called completing the square.
Image of page 1
382 CHAPTER 4 Polynomial and Rational Functions EXAMPLE 4 Graphing a Quadratic Function Given in General Form with a Negative Leading Coefficient Graph the quadratic function f ( x ) 3 x 2 6 x 2. Solution: Express the function in standard form by completing the square. Write the original function. f ( x ) 3 x 2 6 x 2 Group the variable terms together. ( 3 x 2 6 x ) 2 Factor out 3 in order to make the coefficient of x 2 equal to 1 inside the parentheses. 3 ( x 2 2 x ) 2 Add and subtract 1 inside the parentheses to create a perfect square. 3 ( x 2 2 x 1 1 ) 2 Regroup the terms. 3 ( x 2 2 x 1 ) 3( 1 ) 2 Write the expression inside the parentheses as a perfect square and simplify. 3( x 1) 2 5 Now that the quadratic function is written in standard form, f ( x ) 3( x 1) 2 5, we follow our step-by-step procedure for graphing a quadratic function in standard form. S TEP 1 The parabola opens down. a 3, therefore a 0 S TEP 2 Determine the vertex. ( h , k ) (1, 5) S TEP 3 Find the y -intercept using f (0) 3(0) 2 6(0) 2 2 the general form. (0, 2) corresponds to the y -intercept S TEP 4 Find any x -intercepts using the standard form. f ( x ) 3( x 1) 2 5 0 3( x 1) 2 5 x - 1 = ; A 5 3 ( x - 1) 2 = 5 3 x y (1, 5) (0, 2) (–0.3, 0) (2.3, 0) Answer: x y –5 5 5
Image of page 2
Image of page 3

You've reached the end of your free preview.

Want to read all 10 pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture