Problem 3 Constructing a Quartic Function with Two Real Zeros

What is a quartic function with only two real zeros, x = 5 x equals 5  and x = 9 ? x equals 9 question mark

• Method 1 Use transformations.

  First, find a quartic with zeros at ± 2 . plus minus 2 .

  Translate the basic quartic 16 units down:

  y = x 4 → y = x 4 − 16 y equals , x to the fourth , rightwards arrow y equals , x to the fourth , minus 16

  9 is 7 units to the right of 2.

  Translate 7 units to the right.

  y = x 4 − 16 → y = ( x − 7 ) 4 − 16 y equals , x to the fourth , minus 16 rightwards arrow y equals . open x minus 7 close to the fourth . minus 16

  A quartic function with its only real zeros at 5 and 9 is y = ( x − 7 ) 4 − 16 . y equals . open x minus 7 close to the fourth . minus 16 .

  
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• Method 2 Use algebraic methods.

  Think

  What should you use for Q ( x ) ? q open x close question mark

  Choose Q ( x ) q open x close  to be a quadratic with no real zeros. Keep it simple, such as x 2 + 1 . x squared , plus 1 .

  
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  Another quartic function with its only real zeros at 5 and 9 is y = x 4 − 14 x 3 + 46 x 2 − 14 x + 45 . y equals , x to the fourth , minus , 14 x cubed , plus , 46 x squared , minus 14 x plus 45 .

✓ Got It?

3. 1. What is a quartic function f ( x ) f open x close  with only two real zeros, x = 0 x equals 0  and x = 6 ? x equals 6 question mark
   2. Reasoning Does the quartic function − f ( x ) negative f open x close  have the same zeros? Explain.

Take Note Key Concept: Power Functions

Definition

A power function is a function of the form y = a · x b , y equals eh middle dot , x to the b , comma  where a and b are nonzero real numbers.

Examples

y = 0.5 x 6 y = 1 2 x 2 y = − 4 x 2 3 y = x 0.25 table with 2 rows and 2 columns , row1 column 1 , y equals 0.5 , x to the sixth , column 2 y equals , 1 half , x squared , row2 column 1 , y equals negative 4 , x super 2 thirds end super , column 2 y equals , x to the 0.25 , end table