Presentation on theme: "EXAMPLE 1 Write a cubic function"— Presentation transcript:
1EXAMPLE 1Write a cubic functionWrite the cubic function whose graph is shown.SOLUTIONSTEP 1Use the three given x - intercepts to write the function in factored form.f (x) = a (x + 4)(x – 1)(x – 3)STEP 2Find the value of a by substituting the coordinates of the fourth point.
2EXAMPLE 1Write a cubic function–6 = a (0 + 4) (0 – 1) (0 – 3)–6 = 12a– = a21ANSWER21The function is f (x) = – (x + 4)(x – 1)(x – 3).CHECK Check the end behavior of f. The degree of f is odd and a < 0. So f (x) +∞ as x → –∞ and f (x) → –∞ asx → +∞ which matches the graph.
3EXAMPLE 2Find finite differencesThe first five triangular numbers are shown below. A formula for the n the triangular number isf (n) = (n2 + n). Show that this function has constant second-order differences.12
4EXAMPLE 2Find finite differencesSOLUTIONWrite the first several triangular numbers. Find the first-order differences by subtracting consecutive triangular numbers. Then find the second-orderdifferences by subtracting consecutive first-order differences.
5EXAMPLE 2Find finite differencesEach second-order difference is 1, so the second-order differences are constant.ANSWER
6GUIDED PRACTICEfor Examples 1 and 2Write a cubic function whose graph passes through the given points.(– 4, 0), (0, 10), (2, 0), (5, 0)41The function is f (x) = (x + 4) (x – 2) (x – 5).ANSWERy = 0.25x3 – 0.75x2 – 4.5x +10
83. GEOMETRY Show that f (n) = n(3n – 1), a 1 2 GUIDED PRACTICEfor Examples 1 and 2GEOMETRY Show that f (n) = n(3n – 1), a12formula for the nth pentagonal number, has constant second-order differences.ANSWERWrite function values forequally-spaced n - values.First-order differencesSecond-order differencesEach second-order difference is 3, so the second-order differences are constant.