9-1 and 9-2 Graphing Quadratic Functions

Quick Review

A function of the form y = a x 2 + b x + c , y equals , eh x squared , plus b x plus c comma  where a ≠ 0 , eh not equal to 0 comma  is a quadratic function. Its graph is a parabola. The axis of symmetry of a parabola divides it into two matching halves. The vertex of a parabola is the point at which the parabola intersects the axis of symmetry.

Example

What is the vertex of the graph of y = x 2 + 6 x − 2 ? y equals , x squared , plus 6 x minus 2 question mark

The x-coordinate of the vertex is given by x = − b 2 a . x equals . fraction negative b , over 2 eh end fraction . .

x = − b 2 a = − 6 2 ( 1 ) = − 3 x equals . fraction negative b , over 2 eh end fraction . equals . fraction negative 6 , over 2 , open 1 close end fraction . equals negative 3

Find the y-coordinate of the vertex.

y = ( − 3 ) 2 + 6 ( − 3 ) − 2 Substitute  − 3  for  x . y = − 11 Simplify . table with 2 rows and 3 columns , row1 column 1 , y , column 2 equals . open , negative 3 , close squared . plus 6 . open , negative 3 , close . minus 2 , column 3 cap substitute . minus 3 , for , x . , row2 column 1 , y , column 2 equals negative 11 , column 3 cap simplify , . , end table

The vertex is ( − 3 , − 11 ) . open negative 3 comma negative 11 close .

Exercises

Graph each function. Label the axis of symmetry and the vertex.

5. y = 2 3 x 2 y equals , 2 thirds , x squared
6. y = − x 2 + 1 y equals negative . x squared , plus 1
7. y = x 2 − 4 y equals . x squared , minus 4
8. y = 5 x 2 + 8 y equals , 5 x squared , plus 8
9. y = − 1 2 x 2 + 4 x + 1 y equals negative , 1 half , x squared , plus 4 x plus 1
10. y = − 2 x 2 − 3 x + 10 y equals negative , 2 x squared , minus 3 x plus 10
11. y = 1 2 x 2 + 2 x − 3 y equals , 1 half , x squared , plus 2 x minus 3
12. y = 3 x 2 + x − 5 y equals , 3 x squared , plus x minus 5

Open-Ended Give an example of a quadratic function that matches each description.

13. Its graph opens downward.
14. The vertex of its graph is at the origin.
15. Its graph opens upward.
16. Its graph is wider than the graph of y = x 2 . y equals , x squared , .

9-3 and 9-4 Solving Quadratic Equations

Quick Review

The standard form of a quadratic equation is a x 2 + b x + c = 0 , eh x squared , plus b x plus c equals 0 comma  where a ≠ 0 . eh not equal to 0 .  Quadratic equations can have two, one, or no real-number solutions. You can solve a quadratic equation by graphing the related function and finding the x-intercepts. Some quadratic equations can also be solved using square roots. If the left side of a x 2 + b x + c = 0 eh x squared , plus b x plus c equals 0  can be factored, you can use the Zero-Product Property to solve the equation.

Example

What are the solutions of 2 x 2 − 72 = 0 ? 2 x squared , minus 72 equals 0 question mark

2 x 2 − 72 = 0   2 x 2 = 72 Add  72  to each side . x 2 = 36 Divide each side by  2. x = ± 36 Find the square roots of each side . x = ± 6 Simplify . table with 5 rows and 3 columns , row1 column 1 , 2 , x squared , minus 72 , column 2 equals 0 , column 3 , row2 column 1 , 2 , x squared , column 2 equals 72 , column 3 cap add , 72 . toeachside . . , row3 column 1 , x squared , column 2 equals 36 , column 3 cap divideeachsideby . 2. , row4 column 1 , x , column 2 equals plus minus square root of 36 , column 3 cap findthesquarerootsofeachside . . , row5 column 1 , x , column 2 equals plus minus 6 , column 3 cap simplify , . , end table

Exercises

Solve each equation. If the equation has no real-number solution, write no solution.

17. 6 ( x 2 − 2 ) = 12 6 open , x squared , minus , 2 close equals 12
18. − 5 m 2 = − 125 negative , 5 m squared , equals negative 125
19. 9 ( w 2 + 1 ) = 9 9 open , w squared , plus , 1 close equals 9
20. 3 r 2 + 27 = 0 3 r squared , plus 27 equals 0
21. 4 = 9 k 2 4 equals , 9 k squared
22. 4 n 2 = 64 4 n squared , equals 64

Solve by factoring.

23. x 2 + 7 x + 12 = 0 x squared , plus 7 . x plus 12 equals 0
24. 5 x 2 − 10 x = 0 5 x squared , minus 10 x equals 0
25. 2 x 2 − 9 x = x 2 − 20 2 x squared , minus 9 x equals . x squared , minus 20
26. 2 x 2 + 5 x = 3 2 x squared , plus 5 x equals 3
27. 3 x 2 − 5 x = − 3 x 2 + 6 3 x squared , minus 5 x equals negative , 3 x squared , plus 6
28. x 2 − 5 x + 4 = 0 x squared , minus 5 . x plus 4 equals 0
29. Geometry The area of a circle A is given by the formula A = π r 2 , eh equals pi , r squared , comma  where r is the radius of the circle. Find the radius of a circle with area 16  in. 2 . 16 , in. squared , .  Round to the nearest tenth of an inch.

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