4-1 Quadratic Functions and Transformations

Quick Review

You can write every quadratic function in the form f ( x ) = a x 2 + b x + c , f open x close equals eh , x squared , plus b x plus c comma  where a ≠ 0 . eh not equal to 0 .  A parabola is the graph of a quadratic function. Every parabola has a vertex and an axis of symmetry. Shown below is the graph of the quadratic parent function f ( x ) = x 2 . f open x close equals , x squared , .

The vertex form of a quadratic function is f ( x ) = a ( x − h ) 2 + k , f open x close equals eh . open x minus h close squared . plus k comma  where a ≠ 0 . eh not equal to 0 .  The vertex of the parabola formed by a quadratic function is (h, k).

If a > 0 , eh greater than 0 comma  k is the minimum value of the function.

If a < 0 , eh less than 0 comma  k is the maximum value of the function. The axis of symmetry is given by x = h.

Example

What is the vertex, axis of symmetry, maximum or minimum, and domain and range of the function f ( x ) = 5 ( x − 7 ) 2 + 2 ? f open x close equals 5 open x minus 7 , close squared , plus 2 question mark

Exercises

Identify the vertex, axis of symmetry, maximum or minimum, and domain and range of each function.

4. f ( x ) = 4 ( x + 2 ) 2 − 6 f open x close equals 4 open x plus 2 , close squared , minus 6
5. f ( x ) = − ( x − 3 ) 2 + 2 f open x close equals negative open x minus 3 , close squared , plus 2
6. f ( x ) = 10 ( x − 1 ) 2 + 5 f open x close equals 10 open x minus 1 , close squared , plus 5
7. f ( x ) = 2 ( x + 9 ) 2 − 4 f open x close equals 2 open x plus 9 , close squared , minus 4

Graph each function. Describe each transformation of the parent function f ( x ) = x 2 . f open x close equals , x squared , .

8. f ( x ) = x 2 + 4 f open x close equals , x squared , plus 4
9. f ( x ) = ( x − 9 ) 2 + 2 f open x close equals open x minus 9 , close squared , plus 2
10. f ( x ) = 1 2 ( x + 1 ) 2 − 5 f open x close equals , 1 half , open x plus 1 , close squared , minus 5

Write the equation of each parabola in vertex form.

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