4-2 Standard Form of a Quadratic Function

Quick Review

The standard form of a quadratic function is 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 .  When a > 0 , eh greater than 0 comma  the parabola opens up. When a < 0 , eh less than 0 comma  the parabola opens down.

The axis of symmetry is the line x = − b 2 a . x equals negative , fraction b , over 2 eh end fraction , .  The vertex is ( − b 2 a , f ( − b 2 a ) ) , open . negative , fraction b , over 2 eh end fraction , comma f . open . negative , fraction b , over 2 eh end fraction . close . close . comma  and the y-intercept is (0, c).

Example

What are the vertex, the axis of symmetry and y-intercept of the graph of the function f ( x ) = x 2 − 6 x + 8 ? f open x close equals , x squared , minus 6 x plus 8 question mark

axis of symmetry: x = − ( − 6 2 ( 1 ) ) = 3 x equals negative . open . fraction negative 6 , over 2 , open 1 close end fraction . close . equals 3

vertex: ( 3 , − 1 ) open 3 comma negative 1 close

y-intercept: (0, 8)

Exercises

Graph each function.

13. f ( x ) = x 2 + 6 x + 5 f open x close equals , x squared , plus 6 x plus 5
14. f ( x ) = x 2 − 7 x − 18 f open x close equals , x squared , minus 7 x minus 18
15. f ( x ) = x 2 − 7 x + 12 f open x close equals , x squared , minus 7 x plus 12
16. f ( x ) = x 2 − 9 f open x close equals , x squared , minus 9

Write each function in vertex form.

17. f ( x ) = 4 x 2 − 8 x + 2 f open x close equals , 4 x squared , minus 8 x plus 2
18. f ( x ) = x 2 − 8 x + 12 f open x close equals , x squared , minus 8 x plus 12
19. f ( x ) = 8 x 2 + 8 x − 12 f open x close equals , 8 x squared , plus 8 x minus 12
20. f ( x ) = − 2 x 2 − 6 x + 10 f open x close equals negative 2 , x squared , minus 6 x plus 10
21. Physics The equation h = − 16 t 2 + 32 t + 9 h equals negative 16 , t squared , plus 32 t plus 9  gives the height of a ball, h, in feet above the ground, at t seconds after the ball is thrown upward. How many seconds after the ball is thrown will it reach its maximum height? What is its maximum height?

4-3 Modeling With Quadratic Functions

Quick Review

You can use quadratic functions to model real world data. You can find a quadratic function to model data that passes through any three non-collinear points given that no two of the points lie on a vertical line.

Example

Find the equation of the parabola that passes through the points ( − 2 , 8 ) , ( 0 , − 2 ) , open negative 2 comma 8 close comma open 0 comma negative 2 close comma  and (1, 2).

Exercises

Find the equation of the parabola that passes through each set of points.

22. (0, 5), ( 2 , − 3 ) , ( − 2 , 12 ) open 2 comma negative 3 close comma open negative 2 comma 12 close
23. (2, 0), ( 3 , − 2 ) , ( 1 , − 2 ) open 3 comma negative 2 close comma open 1 comma negative 2 close
24. (4, 10), ( 0 , − 18 ) , ( − 2 , − 20 ) open 0 comma negative 18 close comma open negative 2 comma negative 20 close
25. ( 0 , − 7 ) , ( 7 , − 14 ) , ( − 2 , − 19 ) open 0 comma negative 7 close comma open 7 comma negative 14 close comma open negative 2 comma negative 19 close

26. Track and Field The table shows the height of a javelin as it is thrown and travels across a horizontal distance. Use your calculator to find a quadratic model to represent the path of the javelin.

    Distance (m)	Height (m)
    5	2
    18	5
    33	8
    55	6
    68	4
    74	3

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